gradient computes a vector and then multiplies it by $ c _ { exp } $. hellinger distance $ $ c _ { hd } ( w, b, s ^ r, e ^ r ) = \ frac { 1 } { \ sqrt { 2 } } \ sum \ limits _ j ( \ sqrt { a ^ l _ j } - \ sqrt { e ^ r _ j } ) ^ 2 $ $ you can find more about this here. this needs to have positive values, and ideally values between $ 0 $ and $ 1 $. the same is true for the following divergences. the gradient of this cost function with respect to the output of a neural network and some sample $ r $ is : $ $ \ nabla _ a c = \ frac { \ sqrt { a ^ l } - \ sqrt { e ^ r } } { \ sqrt { 2 } \ sqrt { a ^ l } } $ $ kullback – leibler divergence also known as information divergence, information gain, relative entropy, klic, or kl divergence ( see here ). kullback – leibler divergence is typically denoted $ $ d _ { \ mathrm { kl } } ( p \ | q ) = \ sum _ i p ( i ) \, \ ln \ frac { p ( i ) } { q ( i ) } $ $, where $ d _ { \ mathrm { kl } } ( p \ | q ) $ is a measure of the information lost when $ q $ is used to approximate $ p $. thus we want to set $ p = e ^ i $ and $ q = a ^ l $, because we want to measure how much information is lost when we use $ a ^ i _ j $ to approximate $ e ^ i _ j $. this gives us $ $ c _ { kl } ( w, b, s ^ r, e ^ r ) = \ sum \ limits _ je ^ r _ j \ log \ frac { e ^ r _ j } { a ^ l _ j } $ $ the other divergences here use this same idea of setting $ p = e ^ i $ and $ q = a ^ l $. the gradient of this cost function with respect to the output of a neural network and some sample $ r $ is : $ $ \ nabla _ a c = - \ frac { e ^ r


} { a ^ l } $ $ generalized kullback – leibler divergence from here. $ $ c _ { gkl } ( w, b, s ^ r, e ^ r ) = \ sum \ limits _ j e ^ r _ j \ log \ frac { e ^ r _ j } { a ^ l _ j } - \ sum \ limits _ j ( e ^ r _ j ) + \ sum \ limits _ j ( a ^ l _ j ) $ $ the gradient of this cost function with respect to the output of a neural network and some sample $ r $ is : $ $ \ nabla _ a c = \ frac { a ^ l - e ^ r } { a ^ l } $ $ itakura – saito distance also from here. $ $ c _ { gkl } ( w, b, s ^ r, e ^ r ) = \ sum _ j \ left ( \ frac { e ^ r _ j } { a ^ l _ j } - \ log \ frac { e ^ r _ j } { a ^ l _ j } - 1 \ right ) $ $ the gradient of this cost function with respect to the output of a neural network and some sample $ r $ is : $ $ \ nabla _ a c = \ frac { a ^ l - e ^ r } { \ left ( a ^ l \ right ) ^ 2 } $ $ where $ \ left ( \ left ( a ^ l \ right ) ^ 2 \ right ) _ j = a ^ l _ j \ cdot a ^ l _ j $. in other words, $ \ left ( a ^ l \ right ) ^ 2 $ is simply equal to squaring each element of $ a ^ l $.


language designers face many choices. ken kennedy emphasized two : ( 1 ) better abstractions and ( 2 ) higher - or lower - level ( less or more machine - like ) code. while functional languages like haskell and scheme focus on the former, traditional scientific - computing languages like fortran and c / c + + focused on the latter. saying that one language is faster than another is usually quite misleading : each language has a problem domain for which it excels. fortran fares better in the domain of array - based numerical codes than other languages for two basic reasons : its array model and its explicitness. array model fortran programmers largely do array manipulations. for that, fortran facilitates several compiler optimizations that are not available in other languages. the best example is vectorization : knowing the data layout enables the compiler to invoke assembly - level intrinsics over the array. language explicitness while it seems that a simpler language should compile " better " than a more complex one, that really isn't the case. when one writes in an assembly language, there isn't much a compiler can do : all it sees are very - fine - grained instructions. fortran requires explicitness ( thus, more work by the programmer ) only in cases that yield real rewards for array - based computing. fortran uses simple data types, basic control flow, and limited namespaces ; by contrast, it does not tell the computer how to load registers ( which might be necessary for real - time ). where fortran is explicit, it enables things like complete type inference, which helps novices to get started. it also avoids one thing that often makes c slow : opaque pointers. fortran can be slow fortran is not fast for every task : that's why not many people use it for building guis or even for highly unstructured scientific computing. once you leave the world of arrays for graphs, decision trees, and other realms, this speed advantage quickly goes away. see the computer language benchmarks for some examples and numbers.


first, here are some quick comments : the $ p $ - values of a kolmogorov - smirnov - test ( ks - test ) with estimated parameters can be quite wrong because the p - value does not take the uncertainty of the estimation into account. so unfortunately, you can't just fit a distribution and then use the estimated parameters in a kolmogorov - smirnov - test to test your sample. there is a normality test called lilliefors test which is a modified version of the ks - test that allows for estimated parameters. your sample will never follow a specific distribution exactly. so even if your $ p $ - values from the ks - test would be valid and $ > 0. 05 $, it would just mean that you can't rule out that your data follow this specific distribution. another formulation would be that your sample is compatible with a certain distribution. but the answer to the question " does my data follow the distribution xy exactly? " is always no. the goal here cannot be to determine with certainty what distribution your sample follows. the goal is what @ whuber ( in the comments ) calls parsimonious approximate descriptions of the data. having a specific parametric distribution can be useful as a model of the data ( such as the model " earth is a sphere " can be useful ). but let's do some exploration. i will use the excellent fitdistrplus package which offers some nice functions for distribution fitting. we will use the functiondescdist to gain some ideas about possible candidate distributions. library ( fitdistrplus ) library ( logspline ) x < - c ( 37. 50, 46. 79, 48. 30, 46. 04, 43. 40, 39. 25, 38. 49, 49. 51, 40. 38, 36. 98, 40. 00, 38. 49, 37. 74, 47. 92, 44. 53, 44. 91, 44. 91, 40. 00, 41. 51, 47. 92, 36. 98, 43. 40, 42. 26, 41. 89, 38. 87, 43. 02, 39. 25, 40. 38, 42. 64, 36. 98, 44. 15, 44. 91, 43. 40, 49. 81, 38. 87, 40. 00, 52. 45, 53. 13, 47. 92, 52. 45, 44. 91, 29. 54, 27. 13,


35. 60, 45. 34, 43. 37, 54. 15, 42. 77, 42. 88, 44. 26, 27. 14, 39. 31, 24. 80, 16. 62, 30. 30, 36. 39, 28. 60, 28. 53, 35. 84, 31. 10, 34. 55, 52. 65, 48. 81, 43. 42, 52. 49, 38. 00, 38. 65, 34. 54, 37. 70, 38. 11, 43. 05, 29. 95, 32. 48, 24. 63, 35. 33, 41. 34 ) now let's use descdist : descdist ( x, discrete = false ) the kurtosis and squared skewness of your sample are plotted as a blue point named " observation ". it seems that possible distributions include the weibull, lognormal and possibly the gamma distribution. let's fit a weibull distribution and a normal distribution : fit. weibull < - fitdist ( x, " weibull " ) fit. norm < - fitdist ( x, " norm " ) now inspect the fit for the normal : plot ( fit. norm ) and for the weibull fit : plot ( fit. weibull ) both look good but judged by the qq - plot, the weibull maybe looks a bit better, especially in the tails. correspondingly, the aic of the weibull fit is lower compared with the normal fit : fit. weibull $ aic [ 1 ] 519. 8537 fit. norm $ aic [ 1 ] 523. 3079 kolmogorov - smirnov test simulation i will use @ aksakal's procedure explained here to simulate the ks - statistic under the null. n. sims < - 5e4 stats < - replicate ( n. sims, { r < - rweibull ( n = length ( x ), shape = fit. weibull $ estimate [ " shape " ], scale = fit. weibull $ estimate [ " scale " ] ) estfit. weibull < - fitdist ( r, " weibull " ) # added to account for the estimated parameters as. numeric ( ks. test ( r, " pweibull ", shape = estfit. weibull $ estimate [ " shape " ], scale = estfi


##t. weibull $ estimate [ " scale " ] ) $ statistic ) } ) the ecdf of the simulated ks - statistics looks as follows : plot ( ecdf ( stats ), las = 1, main = " ks - test statistic simulation ( cdf ) ", col = " darkorange ", lwd = 1. 7 ) grid ( ) finally, our $ p $ - value using the simulated null distribution of the ks - statistics is : fit < - logspline ( stats ) 1 - plogspline ( ks. test ( x, " pweibull ", shape = fit. weibull $ estimate [ " shape " ], scale = fit. weibull $ estimate [ " scale " ] ) $ statistic, fit ) [ 1 ] 0. 4889511 this confirms our graphical conclusion that the sample is compatible with a weibull distribution. as explained here, we can use bootstrapping to add pointwise confidence intervals to the estimated weibull pdf or cdf : xs < - seq ( 10, 65, len = 500 ) true. weibull < - rweibull ( 1e6, shape = fit. weibull $ estimate [ " shape " ], scale = fit. weibull $ estimate [ " scale " ] ) boot. pdf < - sapply ( 1 : 1000, function ( i ) { xi < - sample ( x, size = length ( x ), replace = true ) mle. est < - suppresswarnings ( fitdist ( xi, distr = " weibull " ) ) dweibull ( xs, shape = mle. est $ estimate [ " shape " ], scale = mle. est $ estimate [ " scale " ] ) } ) boot. cdf < - sapply ( 1 : 1000, function ( i ) { xi < - sample ( x, size = length ( x ), replace = true ) mle. est < - suppresswarnings ( fitdist ( xi, distr = " weibull " ) ) pweibull ( xs, shape = mle. est $ estimate [ " shape " ], scale = mle. est $ estimate [ " scale " ] ) } ) # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -


- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - # plot pdf # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - par ( bg = " white ", las = 1, cex = 1. 2 ) plot ( xs, boot. pdf [, 1 ], type = " l ", col = rgb (. 6,. 6,. 6,. 1 ), ylim = range ( boot. pdf ), xlab = " x ", ylab = " probability density " ) for ( i in 2 : ncol ( boot. pdf ) ) lines ( xs, boot. pdf [, i ], col = rgb (. 6,. 6,. 6,. 1 ) ) # add pointwise confidence bands quants < - apply ( boot. pdf, 1, quantile, c ( 0. 025, 0. 5, 0. 975 ) ) min. point < - apply ( boot. pdf, 1, min, na. rm = true ) max. point < - apply ( boot. pdf, 1, max, na. rm = true ) lines ( xs, quants [ 1, ], col = " red ", lwd = 1. 5, lty = 2 ) lines ( xs, quants [ 3, ], col = " red ", lwd = 1. 5, lty = 2 ) lines ( xs, quants [ 2, ], col = " darkred ", lwd = 2 ) # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - # plot cdf # - - - - - - - - - - -


- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - par ( bg = " white ", las = 1, cex = 1. 2 ) plot ( xs, boot. cdf [, 1 ], type = " l ", col = rgb (. 6,. 6,. 6,. 1 ), ylim = range ( boot. cdf ), xlab = " x ", ylab = " f ( x ) " ) for ( i in 2 : ncol ( boot. cdf ) ) lines ( xs, boot. cdf [, i ], col = rgb (. 6,. 6,. 6,. 1 ) ) # add pointwise confidence bands quants < - apply ( boot. cdf, 1, quantile, c ( 0. 025, 0. 5, 0. 975 ) ) min. point < - apply ( boot. cdf, 1, min, na. rm = true ) max. point < - apply ( boot. cdf, 1, max, na. rm = true ) lines ( xs, quants [ 1, ], col = " red ", lwd = 1. 5, lty = 2 ) lines ( xs, quants [ 3, ], col = " red ", lwd = 1. 5, lty = 2 ) lines ( xs, quants [ 2, ], col = " darkred ", lwd = 2 ) # lines ( xs, min. point, col = " purple " ) # lines ( xs, max. point, col = " purple " ) automatic distribution fitting with gamlss the gamlss package for r offers the ability to try many different distributions and select the " best " according to the gaic ( the generalized akaike information criterion ). the main function is fitdist. an important option in this function is the type of the distributions that are tried. for example, setting type = " realline " will try all implemented distributions defined on the whole real line whereas type = " realsplus " will only try distributions defined on the real positive line. another important option is the parameter $ k


$, which is the penalty for the gaic. in the example below, i set the parameter $ k = 2 $ which means that the " best " distribution is selected according to the classic aic. you can set $ k $ to anything you like, such as $ \ log ( n ) $ for the bic. library ( gamlss ) library ( gamlss. dist ) library ( gamlss. add ) x < - c ( 37. 50, 46. 79, 48. 30, 46. 04, 43. 40, 39. 25, 38. 49, 49. 51, 40. 38, 36. 98, 40. 00, 38. 49, 37. 74, 47. 92, 44. 53, 44. 91, 44. 91, 40. 00, 41. 51, 47. 92, 36. 98, 43. 40, 42. 26, 41. 89, 38. 87, 43. 02, 39. 25, 40. 38, 42. 64, 36. 98, 44. 15, 44. 91, 43. 40, 49. 81, 38. 87, 40. 00, 52. 45, 53. 13, 47. 92, 52. 45, 44. 91, 29. 54, 27. 13, 35. 60, 45. 34, 43. 37, 54. 15, 42. 77, 42. 88, 44. 26, 27. 14, 39. 31, 24. 80, 16. 62, 30. 30, 36. 39, 28. 60, 28. 53, 35. 84, 31. 10, 34. 55, 52. 65, 48. 81, 43. 42, 52. 49, 38. 00, 38. 65, 34. 54, 37. 70, 38. 11, 43. 05, 29. 95, 32. 48, 24. 63, 35. 33, 41. 34 ) fit < - fitdist ( x, k = 2, type = " realplus ", trace = false, try. gamlss = true ) summary ( fit ) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * family : c ( " wei2 ",


" weibull type 2 " ) call : gamlssml ( formula = y, family = dist [ i ], data = sys. parent ( ) ) fitting method : " nlminb " coefficient ( s ) : estimate std. error t value pr ( > | t | ) eta. mu - 24. 3468041 2. 2141197 - 10. 9962 < 2. 22e - 16 * * * eta. sigma 1. 8661380 0. 0892799 20. 9021 < 2. 22e - 16 * * * according to the aic, the weibull distribution ( more specifically wei2, a special parametrization of it ) fits the data best. the exact parameterization of the distribution wei2 is detailed in this document on page 279. let's inspect the fit by looking at the residuals in a worm plot ( basically a de - trended q - q - plot ) : we expect the residuals to be close to the middle horizontal line and 95 % of them to lie between the upper and lower dotted curves, which act as 95 % pointwise confidence intervals. in this case, the worm plot looks fine to me indicating that the weibull distribution is an adequate fit.


understanding $ p $ - value suppose, that you want to test the hypothesis that the average height of male students at your university is $ 5 $ ft $ 7 $ inches. you collect heights of $ 100 $ students selected at random and compute the sample mean ( say it turns out to be $ 5 $ ft $ 9 $ inches ). using an appropriate formula / statistical routine you compute the $ p $ - value for your hypothesis and say it turns out to be $ 0. 06 $. in order to interpret $ p = 0. 06 $ appropriately, we should keep several things in mind : the first step under classical hypothesis testing is the assumption that the hypothesis under consideration is true. ( in our context, we assume that the true average height is $ 5 $ ft $ 7 $ inches. ) imagine doing the following calculation : compute the probability that the sample mean is greater than $ 5 $ ft $ 9 $ inches assuming that our hypothesis is in fact correct ( see point 1 ). in other words, we want to know $ $ \ mathrm { p } ( \ mathrm { sample \ : mean } \ ge 5 \ : \ mathrm { ft } \ : 9 \ : \ mathrm { inches } \ : | \ : \ mathrm { true \ : value } = 5 \ : \ mathrm { ft } \ : 7 \ : \ mathrm { inches } ). $ $ the calculation in step 2 is what is called the $ p $ - value. therefore, a $ p $ - value of $ 0. 06 $ would mean that if we were to repeat our experiment many, many times ( each time we select $ 100 $ students at random and compute the sample mean ) then $ 6 $ times out of $ 100 $ we can expect to see a sample mean greater than or equal to $ 5 $ ft $ 9 $ inches. given the above understanding, should we still retain our assumption that our hypothesis is true ( see step 1 )? well, a $ p = 0. 06 $ indicates that one of two things have happened : ( a ) either our hypothesis is correct and an extremely unlikely event has occurred ( e. g., all $ 100 $ students are student athletes ) or ( b ) our assumption is incorrect and the sample we have obtained is not that unusual. the traditional way to choose between ( a ) and ( b ) is to choose an arbitrary cut - off for $ p $. we choose ( a ) if $ p > 0. 05 $ and ( b )


if $ p < 0. 05 $.


it's called chip - on - board. the die is glued to the pcb and wires are bonded from it to pads. the pulsonix pcb software i use has it as an optional extra. the main benefit is reduced cost, since you don't have to pay for a package.


proof by contradiction is often used to show that a language is not regular : let $ p $ a property true for all regular languages, if your specific language does not verify $ p $, then it's not regular. the following properties can be used : the pumping lemma, as exemplified in dave's answer ; closure properties of regular languages ( set operations, concatenation, kleene star, mirror, homomorphisms ) ; a regular language has a finite number of prefix equivalence class, myhill – nerode theorem. to prove that a language $ l $ is not regular using closure properties, the technique is to combine $ l $ with regular languages by operations that preserve regularity in order to obtain a language known to be not regular, e. g., the archetypical language $ i = \ { a ^ n b ^ n \ mid n \ in \ mathbb { n } \ } $. for instance, let $ l = \ { a ^ p b ^ q \ mid p \ neq q \ } $. assume $ l $ is regular, as regular languages are closed under complementation so is $ l $'s complement $ l ^ c $. now take the intersection of $ l ^ c $ and $ a ^ \ star b ^ \ star $ which is regular, we obtain $ i $ which is not regular. the myhill – nerode theorem can be used to prove that $ i $ is not regular. for $ p \ geq 0 $, $ i / a ^ p = \ { a ^ { r } b ^ rb ^ p \ mid r \ in \ mathbb { n } \ } = i. \ { b ^ p \ } $. all classes are different and there is a countable infinity of such classes. as a regular language must have a finite number of classes $ i $ is not regular.


summary : i find a formula for the diameter of a bubble large enough to support one human and plug in known values to get $ d = 400 \, { \ rm m } $. i'll have a quantitative stab at the answer to the question of how large an air bubble has to be for the carbon dioxide concentration to be in a breathable steady state, whilst a human is continuously producing carbon dioxide inside the bubble. fick's law of diffusion is that the flux of a quantity through a surface ( amount per unit time per unit area ) is proportional to the concentration gradient at that surface, $ $ \ vec { j } = - d \ nabla \ phi, $ $ where $ \ phi $ is concentration and $ d $ is the diffusivity of the species. we want to find the net flux out of the bubble at the surface, or $ \ vec { j } = - d _ { \ text { surface } } \ nabla \ phi $. $ d _ { \ text { surface } } $ is going to be some funny combination of the diffusivity of $ co _ 2 $ in air and in water, but since the coefficient in water is so much lower, really diffusion is going to be dominated by this coefficient : it can't diffuse rapidly out of the surface and very slowly immediately outside the surface, because the concentration would then pile up in a thin layer immediately outside until it was high enough to start diffusing back in again. so i'm going to assume $ d _ { \ text { surface } } = d _ { \ text { water } } $ here. to estimate $ \ nabla \ phi $, we can first assume $ \ phi ( \ text { surface } ) = \ phi ( \ text { inside } ) $, fixing $ \ phi ( \ text { inside } ) $ from the maximum nonlethal concentration of co2 in air and the molar density of air ( $ = p / rt $ ) ; then assuming the bubble is a sphere of radius $ a $, because in a steady state the concentration outside is a harmonic function, we can find $ $ \ phi ( r ) = \ phi ( \ text { far } ) + \ frac { ( \ phi ( \ text { inside } ) - \ phi ( \ text { far } ) ) a } { r }, $ $ where $ \ phi ( \ text { far } ) $ is the concentration far from the bubble,


assumed to be constant. then $ $ \ nabla \ phi ( a ) = - \ frac { ( \ phi ( \ text { inside } ) - \ phi ( \ text { far } ) ) a } { a ^ 2 } = - \ frac { \ phi ( \ text { inside } ) - \ phi ( \ text { far } ) } { a } $ $ yielding $ $ j = d \ frac { \ phi ( \ text { inside } ) - \ phi ( \ text { far } ) } { a }. $ $ next we integrate this over the surface of the bubble to get the net amount leaving the bubble, and set this $ = $ the amount at which carbon dioxide is exhaled by the human, $ \ dot { n } $. since for the above simplifications $ j $ is constant over the surface ( area $ a $ ), this is just $ ja $. so we have $ $ \ dot { n } = d _ { \ text { water } } a \ frac { \ phi ( \ text { inside } ) - \ phi ( \ text { far } ) } { a } = d _ { \ text { water } } 4 \ pi a ( \ phi ( \ text { inside } ) - \ phi ( \ text { far } ) ). $ $ finally assuming $ \ phi ( \ text { far } ) = 0 $ for convenience, and rearranging for diameter $ d = 2a $ $ $ d = \ frac { \ dot { n } } { 2 \ pi d _ { \ text { water } } \ phi ( \ text { inside } ) } $ $ and substituting $ d = 1. 6 \ times 10 ^ { - 9 } \, { \ rm m } ^ 2 \, { \ rm s } ^ { - 1 } $ ( from wiki ) $ \ phi \ approx 1. 2 \, { \ rm mol } \, { \ rm m } ^ { - 3 } $ ( from osha maximum safe level of 3 % at stp ) $ \ dot { n } = 4 \ times 10 ^ { - 6 } \, { \ rm m } ^ 3 \, { \ rm s } ^ { - 1 } = 4. 8 \ times 10 ^ { - 6 } \, { \ rm mol } \, { \ rm s } ^ { - 1 } $ ( from $ \ % { \ rm co


} _ 2 \ approx 4 \ % $, lung capacity $ \ approx 500 \, { \ rm ml } $ and breath rate $ \ approx \ frac { 1 } { 5 } \, { \ rm s } ^ { - 1 } $ ) i get $ d \ approx 400 \, { \ rm m } $. it's interesting to note that this is independent of pressure : i've neglected pressure dependence of $ d $ and human resilience to carbon dioxide, and the maximum safe concentration of carbon dioxide is independent of pressure, just derived from measurements at stp. finally, a bubble this large will probably rapidly break up due to buoyancy and plateau - rayleigh instabilities.


i would also recommend to take a look at pomegranate, a nice python package for probabilistic graphical models. it includes solvers for hmms and much more. under the hood it uses cythonised code, so it's also quite fast.


this effect is known as inharmonicity, and it is important for precision piano tuning. ideally, waves on a string satisfy the wave equation $ $ v ^ 2 \ frac { \ partial ^ 2 y } { \ partial x ^ 2 } = \ frac { \ partial ^ 2 y } { \ partial t ^ 2 }. $ $ the left - hand side is from the tension in the string acting as a restoring force. the solutions are of the form $ \ sin ( kx - \ omega t ) $, where $ \ omega = kv $. applying fixed boundary conditions, the allowed values of the wavenumber $ k $ are integer multiples of the lowest possible wavenumber, which implies that the allowed frequencies are integer multiplies of the fundamental frequency. this predicts evenly spaced harmonics. however, piano strings are made of thick wire. if you bend a thick wire, there's an extra restoring force in addition to the wire's tension, because the inside of the bend is compressed while the outside is stretched. one can show that this modifies the wave equation to $ $ v ^ 2 \ frac { \ partial ^ 2 y } { \ partial x ^ 2 } - a \ frac { \ partial ^ 4 y } { \ partial x ^ 4 } = \ frac { \ partial ^ 2 y } { \ partial t ^ 2 }. $ $ upon taking a fourier transform, we have the nonlinear dispersion relation $ $ \ omega = kv \ sqrt { 1 + ( a / v ^ 2 ) k ^ 2 } $ $ which " stretches " evenly spaced values of $ k $ into nonuniformly spaced values of $ \ omega $. higher harmonics are further apart. we can write this equation in terms of the harmonic frequencies $ f _ n $ as $ $ f _ n \ propto n \ sqrt { 1 + bn ^ 2 } $ $ which should yield a good fit to your data. note that the frequencies have no dependence on the amplitude, as you noted, and this is because our modified wave equation is still linear in $ y $. this effect must be taken into account when tuning a piano, since we perceive two notes to be in tune when their harmonics overlap. this results in stretched tuning, where the intervals between the fundamental frequencies of different keys are slightly larger than one would expect. that is, a piano whose fundamental frequencies really were tuned to simple ratios would sound out of tune!


your model of what you do mentally is incorrect. in fact, you operate in two steps : eliminate all points that are too far, in $ o ( 1 ) $ time. measure the $ m $ points that are about as close, in $ \ theta ( m ) $ time. if you've played games like petanque ( bowls ) or curling, this should be familiar — you don't need to examine the objects that are very far from the target, but you may need to measure the closest contenders. to illustrate this point, which green dot is closest to the red dot? ( only by a little over 1 pixel, but there is one that's closest. ) to make things easier, the dots have even been color - coded by distance. this picture contains $ m = 10 $ points which are nearly on a circle, and $ n \ gg 10 $ green points in total. step 1 lets you eliminate all but about $ m $ points, but step 2 requires checking each of the $ m $ points. there is no a priori bound for $ m $. a physical observation lets you shrink the problem size from the whole set of $ n $ points to a restricted candidate set of $ m $ points. this step is not a computation step as commonly understood, because it is based on a continuous process. continuous processes are not subject to the usual intuitions about computational complexity and in particular to asymptotic analysis. now, you may ask, why can't a continuous process completely solve the problem? how does it come to these $ m $ points, why can't we refine the process to get $ m = 1 $? the answer is that i cheated a bit : i presented a set of points which is generated to consists of $ m $ almost - closest points and $ n - m $ points which are further. in general, determining which points lie within a precise boundary requires a precise observation which has to be performed point by point. a coarse process of elimination lets you exclude many obvious non - candidates, but merely deciding which candidates are left requires enumerating them. you can model this system in a discrete, computational world. assume that the points are represented in a data structure that sorts them into cells on a grid, i. e. the point $ ( x, y ) $ is stored in a list for the cell $ ( \ lfloor x \ rfloor, \ lfloor y \ rfloor ) $. if you're looking


for the points that are closest to $ ( x _ 0, y _ 0 ) $ and the cell that contains this point contains at most one other point, then it is sufficient to check the containing cell and the 8 neighboring cells. the total number of points in these 9 cells is $ m $. this model respects some key properties of the human model : $ m $ is potentially unbounded — a degenerate worse case of e. g. points lying almost on a circle is always possible. the practical efficiency depends on having selected a scale that matches the data ( e. g. you'll save nothing if your dots are on a piece of paper and your cells are 1 km wide ).


for more than two, the moving knife is a nice solution. somebody takes a knife and moves it slowly across the sandwich. any player may say " cut ". at that moment, the sandwich is cut and the piece given to the one who said " cut ". as he has said that is an acceptable piece, he believes he has at least $ \ frac 1n $ of the sandwich. the rest have asserted ( by not saying " cut " ) that is it at most $ \ frac 1n $ of the sandwich, so the average available is now at least their share. recurse.


to add to rightskewed answer : while it is true that : gencode is an additive set of annotation ( the manual one done by havana and an automated one done by ensembl ), the annotation ( gtf ) files are quite similar for a few exceptions involving the x chromosome and y par and additional remarks in the gencode file ( see more at faq - gencode ). what are the actual differences between different annotation databases? they are a few differences, but the main one for me ( and it could be stupid ) is that refseq is developed by the american ncbi and the ensembl is mainly developed by the european embl - ebi. often, labs or people will just start using what is the best known to them ( because of a course or workshop ) or because they start working with one of the databases with one specific tool and keep with it later. my lab, for reasons still unknown to me, prefers ensembl annotations ( we're working with transcript / exon expression estimation ), while some software ship with refseq annotations. your lab might be mostly european based people or they might also have read papers like the one from frankish et al. comparison of gencode and refseq gene annotation and the impact of reference geneset on variant effect prediction. bmc genomics 2015 ; 16 ( suppl 8 ) : s2 - doi : 10. 1186 / 1471 - 2164 - 16 - s8 - s2 from the frankish et al. paper paper : the gencode comprehensive transcripts contain more exons, have greater genomic coverage and capture many more variants than refseq in both genome and exome datasets, while the gencode basic set shows a higher degree of concordance with refseq and has fewer unique features. as for : are there significant differences between them today, or are they, for all intents and purposes, interchangeable ( e. g., are exon coordinates between refseq and ensembl annotations interchangeable )? no. i don't think they are great differences between them as that the global picture should stay the same ( although you will see different results if you are interested in a small set of genes ). however, they are not directly interchangeable. particularly as there are many versions of ensembl and refseq based on different genome annotations ( and those won '


t be interchangeable between themselves either in most cases ). however, you can easily translate most [ 1 ] of your refseq ids to ensembl ids and vice - versa with tools as for example ( there are devoted libraries / api as well like biocondutor : biomart [ 1 ] most as sometimes, they might be annotated in one of the database but haven't ( yet ) an equivalent in the other. edit in fine, even if people tends to keep to what they are used to ( and that the annotations are constantly expanded and corrected ) depending on the research subject one might be interested in using one database over another : from zhao s, zhang b. a comprehensive evaluation of ensembl, refseq, and ucsc annotations in the context of rna - seq read mapping and gene quantification. bmc genomics. 2015 ; 16 : 97. paper : when choosing an annotation database, researchers should keep in mind that no database is perfect and some gene annotations might be inaccurate or entirely wrong. [.. ] wu et al. [ 27 ] suggested that when conducting research that emphasizes reproducible and robust gene expression estimates, a less complex genome annotation, such as refgene, might be preferred. when conducting more exploratory research, a more complex genome annotation, such as ensembl, should be chosen. [.. ] [ 27 ] wu p - y, phan jh, wang md. assessing the impact of human genome annotation choice on rna - seq expression estimates. bmc bioinformatics. 2013 ; 14 ( suppl 11 ) : s8. doi : 10. 1186 / 1471 - 2105 - 14 - s11 - s8.


i wrote this handout as a complement to oppenheim and willsky. please take a look at table 4. 1 on page 14, reproduced below. ( click for larger image. ) i wrote that table specifically to answer questions such as yours. note the similarities and differences among the four operations : " series " : periodic in time, discrete in frequency " transform " : aperiodic in time, continuous in frequency " continuous time " : continuous in time, aperiodic in frequency " discrete time " : discrete in time, periodic in frequency i hope you find these notes helpful! please feel free to distribute as you wish.


it is a theorem of liouville, reproven later with purely algebraic methods, that for rational functions $ f $ and $ g $, $ g $ non - constant, the antiderivative of $ $ f ( x ) \ exp ( g ( x ) ) \, \ mathrm dx $ $ can be expressed in terms of elementary functions if and only if there exists some rational function $ h $ such that it is a solution of $ $ f = h'+ hg'$ $ $ e ^ { x ^ 2 } $ is another classic example of such a function with no elementary antiderivative. i don't know how much math you've had, but some of this paper might be comprehensible in its broad strokes : liouville's original paper : liouville, j. " suite du memoire sur la classification des transcendantes, et sur l'impossibilite d'exprimer les racines de certaines equations en fonction finie explicite des coefficients. " j. math. pure appl. 3, 523 - 546, 1838. michael spivak's book on calculus also has a section with a discussion of this.


i'll try to give a succinct answer to some of your questions. please bear in mind that this is not strictly my field of research, so some of my info may be outdated / incorrect. there are many tools that are specifically designed to formally prove properties of java and c + +. however i need to make a small digression here : what does it mean to prove correctness of a program? the java type checker proves a formal property of a java program, namely that certain errors, like adding a float and an int, can never occur! i imagine you are interested in much stronger properties, namely that your program can never enter into an unwanted state, or that the output of a certain function conforms to a certain mathematical specification. in short, there is a wide gradient of what " proving a program correct " can mean, from simple security properties to a full proof that the program fulfills a detailed specification. now i'm going to assume that you are interested in proving strong properties about your programs. if you are interested in security properties ( your program can not reach a certain state ), then in general it seems the best approach is model checking. however if you wish to fully specify the behavior of a java program, your best bet is to use a specification language for that language, for instance jml. there are such languages for specifying the behavior of c programs, for instance acsl, but i don't know about c + +. once you have your specifications, you need to prove that the program conforms to that specification. for this you need a tool that has a formal understanding of both your specification and the operational semantics of your language ( java or c + + ) in order to express the adequacy theorem, namely that the execution of the program respects the specification. this tool should also allow you to formulate or generate the proof of that theorem. now both of these tasks ( specifying and proving ) are quite difficult, so they are often separated in two : one tool that parses the code, the specification and generates the adequacy theorem. as frank mentioned, krakatoa is an example of such a tool. one tool that proves the theorem ( s ), automatically or interactively. coq interacts with krakatoa in this manner, and there are some powerful automated tools like z3 which can also be used. one ( minor ) point : there are some theorems which are much too hard to be proven with automated methods


, and automatic theorem provers are known to occasionally have soundness bugs which make them less trustworthy. this is an area where coq shines in comparison ( but it is not automatic! ). if you want to generate ocaml code, then definitely write in coq ( gallina ) first, then extract the code. however, coq is terrible at generating c + + or java, if it is even possible. can the above tools handle threading and performance issues? probably not, performance and threading concerns are best handled by specifically designed tools, as they are particularly hard problems. i'm not sure i have any tools to recommend here, though martin hofmann's polyni project seems interesting. in conclusion : formal verification of " real world " java and c + + programs is a large and well - developed field, and coq is suitable for parts of that task. you can find a high - level overview here for example.


a modified implementation of vivek's answer. peddy is a python package that samples an input. vcf at ~ 25000 sites and projects onto a principal component space built on 2504 thousand genome samples. the author has extensive documentation of the tool's features and a link to the preprint. i downloaded the. vcf and. vcf. tbi for the na12878 sample from genome in a bottle's ftp here. then, created a custom. ped file, na12878. ped, with the contents : na12878 hg001 0 0 2 0 at the command line : $ peddy - - plot - - prefix myvcf hg001 _ grch37 _ giab _ highconf _ cg - illfb - illgatkhc - ion - 10x - solid _ chrom1 - x _ v. 3. 3. 2 _ highconf _ pgandrtgphasetransfer. vcf. gz na12878. ped the output files all have the prefix myvcf., here's myvcf. pca _ check. png


( note : the full desciption is a bit complex, and has several subtleties which i prefered to ignore. the following is merely the high - level ideas for the qtm model ) when defining a quantum turing machine ( qtm ), one would like to have a simple model, similar to the classical tm ( that is, a finite state machine plus an infinite tape ), but allow the new model the advantage of quantum mechanics. similarly to the classical model, qtm has : $ q = \ { q _ 0, q _ 1,.. \ } $ - a finite set of states. let $ q _ 0 $ be an initial state. $ \ sigma = \ { \ sigma _ 0, \ sigma _ 1,... \ } $, $ \ gamma = \ { \ gamma _ 0,.. \ } $ - set of input / working alphabet an infinite tape and a single " head ". however, when defining the transition function, one should recall that any quantum computation must be reversible. recall that a configuration of tm is the tuple $ c = ( q, t, i ) $ denoting that the tm is at state $ q \ in q $, the tape contains $ t \ in \ gamma ^ * $ and the head points to the $ i $ th cell of the tape. since, at any given time, the tape consist only a finite amount of non - blank cells, we define the ( quantum ) state of the qtm as a unit vector in the hilbert space $ \ mathcal { h } $ generated by the configuration space $ q \ times \ sigma ^ * \ times \ mathrm { z } $. the specific configuration $ c = ( q, t, i ) $ is represented as the state $ $ | c \ rangle = | q \ rangle | t \ rangle | i \ rangle. $ $ ( remark : therefore, every cell in the tape isa $ \ gamma $ - dimensional hilbert space. ) the qtm is initialized to the state $ | \ psi ( 0 ) \ rangle = | q _ 0 \ rangle | t _ 0 \ rangle | 1 \ rangle $, where $ t _ 0 \ in \ gamma ^ * $ is concatenation of the input $ x \ in \ sigma ^ * $ with many " blanks " as needed ( there is a subtlety here to determine the maximal length, but i ignore it ). at


each time step, the state of the qtm evolves according to some unitary $ u $ $ $ | \ psi ( i + 1 ) \ rangle = u | \ psi ( i ) \ rangle $ $ note that the state at any time $ n $ is given by $ | \ psi ( n ) \ rangle = u ^ n | \ psi ( 0 ) \ rangle $. $ u $ can be any unitary that " changes " the tape only where the head is located and moves the head one step to the right or left. that is, $ \ langle q ', t ', i'| u | q, t, i \ rangle $ is zero unless $ i'= i \ pm 1 $ and $ t'$ differs from $ t $ only at position $ i $. at the end of the computation ( when the qtm reaches a state $ q _ f $ ) the tape is being measured ( using, say, the computational basis ). the interesting thing to notice, is that each " step " the qtm's state is a superposition of possible configurations, which gives the qtm the " quantum " advantage. the answer is based on masanao ozawa, on the halting problem for quantum turing machines. see also david deutsch, quantum theory, the church - turing principle and the universal quantum computer.


real signals are " mirrored " in the real and negative halves of the fourier transform because of the nature of the fourier transform. the fourier transform is defined as the following - $ $ h ( f ) = \ int h ( t ) e ^ { - j2 \ pi ft } dt $ $ basically it correlates the signal with a bunch of complex sinusoids, each with its own frequency. so what do those complex sinusoids look like? the picture below illustrates one complex sinusoid. the " corkscrew " is the rotating complex sinusoid in time, while the two sinusoids that follow it are the extracted real and imaginary components of the complex sinusoid. the astute reader will note that the real and imaginary components are the exact same, only they are out of phase with each other by 90 degrees ( $ \ frac { \ pi } { 2 } $ ). because they are 90 degrees out of phase they are orthogonal and can " catch " any component of the signal at that frequency. the relationship between the exponential and the cosine / sine is given by euler's formula - $ $ e ^ { jx } = \ cos ( x ) + j \ cdot \ sin ( x ) $ $ this allows us to modify the fourier transform as follows - $ $ h ( f ) = \ int h ( t ) e ^ { - j2 \ pi ft } dt \ \ = \ int h ( t ) ( \ cos ( 2 \ pi ft ) - j \ cdot \ sin ( 2 \ pi ft ) ) dt $ $ at the negative frequencies the fourier transform becomes the following - $ $ h ( - f ) = \ int h ( t ) ( \ cos ( 2 \ pi ( - f ) t ) - j \ sin ( 2 \ pi ( - f ) t ) ) dt \ \ = \ int h ( t ) ( \ cos ( 2 \ pi ft ) + j \ cdot \ sin ( 2 \ pi ft ) ) dt $ $ comparing the negative frequency version with the positive frequency version shows that the cosine is the same while the sine is inverted. they are still 90 degrees out of phase with each other, though, allowing them to catch any signal component at that ( negative ) frequency. because both the positive and negative frequency sinusoids are 90 degrees out of phase and have the same magnitude, they will both respond to real signals in the same way. or rather


, the magnitude of their response will be the same, but the correlation phase will be different. edit : specifically, the negative frequency correlation is the conjugate of the positive frequency correlation ( due to the inverted imaginary sine component ) for real signals. in mathematical terms, this is, as dilip pointed out, the following $ h ( - f ) = [ h ( f ) ] ^ * $ another way to think about it : imaginary components are just that.. imaginary! they are a tool, which allows the employ of an extra plane to view things on and makes much of digital ( and analog ) signal processing possible, if not much easier than using differential equations! but we can't break the logical laws of nature, we can't do anything'real'with the imaginary content $ ^ \ dagger $ and so it must effectively cancel itself out before returning to reality. how does this look in the fourier transform of a time based signal ( complex frequency domain )? if we add / sum the positive and negative frequency components of the signal the imaginary parts cancel, this is what we mean by saying the positive and negative elements are conjugate to each - other. notice that when an ft is taken of a time - signal there exists these conjugate signals, with the'real'part of each sharing the magnitude, half in the positive domain, half in the negative, so in effect adding the conjugates together removes the imaginary content and provides the real content only. $ ^ \ dagger $ meaning we can't create a voltage that is $ 5i $ volts. obviously, we can use imaginary numbers to represent real - world signals that are two - vector - valued, such as circularly polarized em waves.


here ’ s one to get things started. let $ u $ be a non - empty open subset of $ \ bbb r $. for $ x, y \ in u $ define $ x \ sim y $ iff $ \ big [ \ min \ { x, y \ }, \ max \ { x, y \ } \ big ] \ subseteq u $. it ’ s easily checked that $ \ sim $ is an equivalence relation on $ u $ whose equivalence classes are pairwise disjoint open intervals in $ \ bbb r $. ( the term interval here includes unbounded intervals, i. e., rays. ) let $ \ mathscr { i } $ be the set of $ \ sim $ - classes. clearly $ u = \ bigcup _ { i \ in \ mathscr { i } } i $. for each $ i \ in \ mathscr { i } $ choose a rational $ q _ i \ in i $ ; the map $ \ mathscr { i } \ to \ bbb q : i \ mapsto q _ i $ is injective, so $ \ mathscr { i } $ is countable. a variant of the same basic idea is to let $ \ mathscr { i } $ be the set of open intervals that are subsets of $ u $. for $ i, j \ in \ mathscr { i } $ define $ i \ sim j $ iff there are $ i _ 0 = i, i _ 1, \ dots, i _ n = j \ in \ mathscr { i } $ such that $ i _ k \ cap i _ { k + 1 } \ ne \ varnothing $ for $ k = 0, \ dots, n - 1 $. then $ \ sim $ is an equivalence relation on $ \ mathscr { i } $. for $ i \ in \ mathscr { i } $ let $ [ i ] $ be the $ \ sim $ - class of $ i $. then $ \ left \ { \ bigcup [ i ] : i \ in \ mathscr { i } \ right \ } $ is a decomposition of $ u $ into pairwise disjoint open intervals. both of these arguments generalize to any lots ( = linearly ordered topological space ), i. e., any linearly ordered set $ \ langle x, \ le \ rangle $


with the topology generated by the subbase of open rays $ ( \ leftarrow, x ) $ and $ ( x, \ to ) $ : if $ u $ is a non - empty open subset of $ x $, then $ u $ is the union of a family of pairwise disjoint open order - convex sets. ( a set $ c \ subseteq x $ is order - convex if whenever $ u, v \ in c $ and $ u < x < v $, then $ x \ in c $. intervals and rays are always order - convex, and the converse is true if the order is complete. ) it does take a little work to verify that these sets are open. alternatively, $ u $ is the union of a family of pairwise disjoint intervals, where the intervals are allowed to have endpoints in the completion of the linear order. in general the family need not be countable, of course.


( background : i have a chronic pain condition, and it is extremely painful for me to do any sort of repetitive fine motor activity, including writing and typing. i earned an undergraduate degree in math with quite little handwriting, and i'm currently a graduate student. ) unfortunately, i've never been able to find any one solution or approach to replace the work that many people do with pencil and paper, but there are lots of partial solutions that work for different situations and types of mathematics. here are some of the ones that i use and which, based on your description, might be applicable. getting really proficient at mental math. i think you've already done a lot of this, so all i'm going to say is that i can do a lot more math in my head than i would've believed was possible pre - disability. it just takes a lot of practice. of course, it doesn't work for everything ; some calculations just require remembering too many things at once. speech - to - text. if you have clear speech, then a speech - to - text system is potentially very helpful. ( i use dragon naturallyspeaking. ) maybe you already use one for general writing and computer use. if so, then you have almost certainly noticed that they are not designed for mathematics ( or, for that matter, computer science ), so some additional software is necessary in order to do math. i use a system based on natlatex to dictate all of my formal mathematics, including anything in that i'm going to turn in for my coursework. basically, natlatex defines a speakable form of many common latex commands, including everything you need for most mathematical expressions. using a custom vocabulary in dragon, i can dictate a plain text file containing this natlatex source. i then use in scripts from the natlatex project to transform my dictated text into actual latex source, which i can then compile into nicely typeset mathematics using a standard latex compiler. ( actually, i use a batch file to automate the process. ) just as a note, i have made several modifications to natlatex in order to optimize it for mathematics ( the original author was a physicist ) and to adjust for changes in latex. feel free to contact me if you want a copy of the modified scripts. i do eventually intend to post them somewhere online, but i need to spend some time updating the documentation first ( and that's really hard to justify


spending time on it while i'm preparing for comprehensive exams! ). the big advantage to a dictation system like this is that you get to go through the process of formally presenting your work, which is really helpful for checking understanding and practicing proof writing. disadvantages include a steep learning curve and not being able to see your work ( typeset, at least ) in real time. you also have very limited choices in text editors because dragon will only " play nice " with a couple of them. programming languages and computer algebra systems. when you actually need to do calculations, plot functions, etc., it is hard to beat a good computer algebra system. there are lots of choices out there, and i think the choice of which one to use ultimately comes down to personal preference and perhaps compatibility with whatever assistive technology you use. of course, you still face the problem of how to handle getting input to that computer algebra system. typing with one finger on an on - screen keyboard sounds rather slow and tedious. here are a few alternatives you might want to look into. if you have head and neck movement, a head pointer is one option. i sometimes use one that actually marketed as a gaming device called trackir. ( gaming peripherals are much less expensive than assistive technology peripherals! ) you can use this to type on an on - screen keyboard or to interact with the input panels found in many computer algebra systems. for mouse click, you could use a switch with your finger or a dwell / hover click. eye tracking is a technology that has recently become much more affordable due to relatively new consumer - level devices marketed for gaming applications. just a couple months ago, i got an tobii eyex eye tracker, and it has been great! i use it with a simple mouse emulation script and dasher to write code in sage. dasher is really cool, by the way, and quite good for text entry with low - precision input devices like eye trackers. it's also much faster than most on - screen keyboards. just as a warning, the eyex is not intended as an assistive device so you do have to do a some software configuration in order to make it do what you want it to do. but you're studying computer science, so i don't think that should be a problem for you. ( freepie, optikey, and dasher are all good pieces of free software to look up and consider for use with the eyex. ) the big advantage


to computer algebra systems is that you can do all of the numerical calculations and tedious symbolic manipulations without needing to ever write down a single number. the disadvantage is that hiding all of the details can sometimes hinder your conceptual understanding. those are the main strategies i use. i hope something here can help you out, too. edit : there has been a request for some examples of what natlatex can do, so here are a couple of different examples pulled files on my hard drive. discrete math example : natlatex input ( dictated with dragon ) given a poset " ( p, precedes ) ", a collection of linear extensions " { calligraphy r } equals left curly brace precedes sub one, precedes sub two, low dots, precedes sub k right curly brace " is called a ` ` realizer'' of " p " if " precedes equals intersection of sub { i equals one } to the k precedes sub k ", where each relation " precedes sub i " is interpreted as a set of ordered pairs and " intersection of " is set intersection. equivalently, " { calligraphy r } " is a realizer of " p " if, for all " p, q in p ", " p precedes q " if and only if " p precedes sub i q " for all " one less than or equal to i less than or equal to k ". latex output given a poset \ ( ( p, \ prec ) \ ), a collection of linear extensions \ ( { \ mathcal r } = \ { \ prec _ 1, \ prec _ 2, \ ldots, \ prec _ k \ } \ ) is called a ` ` realizer'' of \ ( p \ ) if \ ( \ prec = \ bigcap _ { i = 1 } ^ k \ prec _ k \ ), where each relation \ ( \ prec _ i \ ) is interpreted as a set of ordered pairs and \ ( \ bigcap \ ) is set intersection. equivalently, \ ( { \ mathcal r } \ ) is a realizer of \ ( p \ ) if, for all \ ( p, q \ in p \ ), \ ( p \ prec q \ ) if and only if \ ( p \ prec _ i q \ ) for all \ ( 1 \ leq i \ leq k \ ). analysis example : nat


##latex input ( dictated with dragon ) begin theorem [ monotone convergence theorem ] let " left curly brace f sub n right curly brace sub { n equals one } to the infinity " be a sequence of nonnegative measurable functions with " f sub one less than or equal to f sub two less than or equal to low dots less than or equal to f sub n less than or equal to f sub { n + 1 } less than or equal to low dots " and " limit of sub n f sub n equals f " ( pointwise ). then, " f " is measurable and @ begin { equation } limit of sub { n right arrow infinity } integral f sub n d greek mu equals integral limit of sub { n right arrow infinity } f sub n d greek mu equals integral f d greek mu @ end { equation } end theorem latex output \ begin { theorem } [ monotone convergence theorem ] let \ ( \ { f _ n \ } _ { n = 1 } ^ \ infty \ ) be a sequence of nonnegative measurable functions with \ ( f _ 1 \ leq f _ 2 \ leq \ ldots \ leq f _ n \ leq f _ { n + 1 } \ leq \ ldots \ ) and \ ( \ lim _ n f _ n = f \ ) ( pointwise ). then, \ ( f \ ) is measurable and \ begin { equation } \ lim _ { n \ rightarrow \ infty } \ int f _ n d \ mu = \ int \ lim _ { n \ rightarrow \ infty } f _ n d \ mu = \ int f d \ mu \ end { equation } \ end { theorem }


simula 67 is generally considered the first object - oriented language and predates smalltalk by a number of years. it also used the this keyword for the same concept, which can be seen in this book chapter extract : class linker ; begin ref ( linker ) next, sex, employment ; text id ; procedure add _ to _ list ( lhead ) ; name lhead ; ref ( linker ) lhead ; begin next : - lhead ; lhead : - this linker end.. of.. add.. to.. list ; procedure onto _ lists ( gender, occupation ) ; name gender, occupation ; ref ( linker ) gender, occupation ; begin sex : - gender ; employment : - occupation ; gender : - occupation : - this linker end.. of.. onto.. lists ; inimage ; id : - copy ( sysin. image ) ; inimage ; end - - of - - linker ;


it's a very general statement, but it's not always true. i'll explain why it's often true, and give a counter - example at the end. your majority component b and the impurity ( let's call it a ) form a binary system. in most cases, such binary mixtures exhibit a solid – liquid phase diagram as follows : ( image taken from these lecture notes ). this binary phase diagram has pure a on the left, pure b on the right. a and b form, somewhere, a eutectic. it is the point here at concentration e and temperature y. because the existence of a eutectic point is guaranteed for any a / b binary system, and because the eutectic corresponds to a lower temperature, your liquidus curve decreases with increasing impurity concentration, and the impurity thus lowers the melting point. however, not all binary mixtures form a eutectic. in the words of wikipedia : not all binary alloys have a eutectic point ; for example, in the silver - gold system the melt temperature ( liquidus ) and freeze temperature ( solidus ) both increase monotonically as the mix changes from pure silver to pure gold. the corresponding phase diagram is as follows :


waves always travel. even standing waves can always be interpreted as two traveling waves that are moving in opposite directions ( more on that below ). keeping the idea that waves must travel in mind, here's what happens whenever you figure out a way to build a region in which the energy of such a moving wave cancels out fully : if you look closely, you will find that you have created a mirror, and that the missing energy has simply bounced off the region you created. examples include opals, peacock feathers, and ordinary light mirrors. the first two reflect specific frequencies of light because repeating internal structures create a physical regions in which that frequency of light cannot travel - that is, a region in which near - total energy cancellation occurs. an optical mirror uses electrons at the top of their fermi seas to cancel out light over a much broader range of frequencies. in all three examples the light bounces off the region, with only a little of its energy being absorbed ( converted to heat ). a skip rope ( or perhaps a garden hose ) provides a more accessible example. first, lay out the rope or hose along its length, then give it quick, sharp clockwise motion. you get a helical wave that travels quickly away from you like a moving corkscrew. no standing wave, that! you put a friend at the other end, but she does not want your wave hitting her. so what does she do? first she tries sending a clockwise wave at you too, but that seems to backfire. your wave if anything seems to hit harder and faster. so she tries a counterclockwise motion instead. that seems to work much better. it halts the forward progress of the wave you launched at her, converting it instead to a loop. that loop still has lots of energy, but at least now it stays in one place. it has become a standing wave, in this case a classic skip - rope loop, or maybe two or more loops if you are good at skip rope. what happened is that she used a canceling motion to keep your wave from hitting her. but curiously, her cancelling motion also created a wave, one that is twisted in the opposite way ( counterclockwise ) and moving towards you, just as your clockwise wave moved towards her. as it turns out, the motion you are already doing cancels her wave too, sending it right back at her. the wave is now trapped between your two cancelling actions. the sum of the two waves, which now looks sinusoidal


instead of helical, has the same energy as your two individual helical waves added together. i should note that you really only need one person driving the wave, since any sufficiently solid anchor for one end of the rope will also prevent the wave from entering it, and so end up reflecting that wave just as your friend did using a more active approach. physical media such as peacock features and fermi sea electrons also use a passive approach to reflection, with the same result : the energy is forbidden by cancellation from entering into some region of space. so, while this is by no means a complete explanation, i hope it provides some " feel " for what complete energy cancellation really means : it's more about keeping waves out. thinking of cancellation as the art of building wave mirrors provides a different and less paradoxical - sounding perspective on a wide variety of phenomena that alter, cancel, or redirect waves.


to clarify a bit. the p - value is uniformly distributed when the null hypothesis is true and all other assumptions are met. the reason for this is really the definition of alpha as the probability of a type i error. we want the probability of rejecting a true null hypothesis to be alpha, we reject when the observed $ \ text { p - value } < \ alpha $, the only way this happens for any value of alpha is when the p - value comes from a uniform distribution. the whole point of using the correct distribution ( normal, t, f, chisq, etc. ) is to transform from the test statistic to a uniform p - value. if the null hypothesis is false then the distribution of the p - value will ( hopefully ) be more weighted towards 0. the pvalue. norm. sim and pvalue. binom. sim functions in the teachingdemos package for r will simulate several data sets, compute the p - values and plot them to demonstrate this idea. also see : murdoch, d, tsai, y, and adcock, j ( 2008 ). p - values are random variables. the american statistician, 62, 242 - 245. for some more details. edit : since people are still reading this answer and commenting, i thought that i would address @ whuber's comment. it is true that when using a composite null hypothesis like $ \ mu _ 1 \ leq \ mu _ 2 $ that the p - values will only be uniformly distributed when the 2 means are exactly equal and will not be a uniform if $ \ mu _ 1 $ is any value that is less than $ \ mu _ 2 $. this can easily be seen using the pvalue. norm. sim function and setting it to do a one sided test and simulating with the simulation and hypothesized means different ( but in the direction to make the null true ). as far as statistical theory goes, this does not matter. consider if i claimed that i am taller than every member of your family, one way to test this claim would be to compare my height to the height of each member of your family one at a time. another option would be to find the member of your family that is the tallest and compare their height with mine. if i am taller than that one person then i am taller than the rest as well and my claim is true, if i am not taller than that one person then my claim is false. testing a composite null can be


seen as a similar process, rather than testing all the possible combinations where $ \ mu _ 1 \ leq \ mu _ 2 $ we can test just the equality part because if we can reject that $ \ mu _ 1 = \ mu _ 2 $ in favour of $ \ mu _ 1 > \ mu _ 2 $ then we know that we can also reject all the possibilities of $ \ mu _ 1 < \ mu _ 2 $. if we look at the distribution of p - values for cases where $ \ mu _ 1 < \ mu _ 2 $ then the distribution will not be perfectly uniform but will have more values closer to 1 than to 0 meaning that the probability of a type i error will be less than the selected $ \ alpha $ value making it a conservative test. the uniform becomes the limiting distribution as $ \ mu _ 1 $ gets closer to $ \ mu _ 2 $ ( the people who are more current on the stat - theory terms could probably state this better in terms of distributional supremum or something like that ). so by constructing our test assuming the equal part of the null even when the null is composite, then we are designing our test to have a probability of a type i error that is at most $ \ alpha $ for any conditions where the null is true.


great question, and one about which there has historically been a lot of speculation, and there is currently a lot of misinformation. i will first address the two answers given by other users, which are both incorrect but have been historically suggested by scientists. then i will try to explain the current understanding ( which is not simple or complete ). my answer is derived directly from the literature, and in particular from mable ( 2004 ), which in turn is part of the 2004 special issue of the biological journal of the linnean society tackling the subject. the'sex'answer... in 1925 hj muller addressed this question in a famous paper, " why polyploidy is rarer in animals than in plants " ( muller, 1925 ). muller briefly described the phenomenon that polyploidy was frequently observed in plants, but rarely in animals. the explanation, he said, was simple ( and is approximate to that described in matthew piziak's answer ) : animals usually have two sexes which are differentiated by means of a process involving the diploid mechanism of segregation and combination whereas plants - at least the higher plants - are usually hermaphroditic. muller then elaborated with three explanations of the mechanism : he assumed that triploidy was usually the intermediate step in chromosome duplication. this would cause problems, because if most animals'sex was determined by the ratios of chromosomes ( as in drosophila ), triploidy would lead to sterility. in the rare cases when a tetraploid was accidentally created, it would have to breed with diploids, and this would result in a ( presumably sterile ) triploid. if, by chance, two tetraploids were to arise and mate, they would be at a disadvantage because, he said, they would be randomly allocated sex chromosomes and this would lead to a higher proportion of non - viable offspring, and thus the polyploid line would be outcompeted by the diploid. unfortunately, whilst the first two points are valid facts about polyploids, the third point is incorrect. a major flaw with muller's explanation is that it only applies to animals with chromosomal ratio - based sex determination, which we have since discovered is actually relatively few animals. in 1925 there was comparatively little systematic study of life, so we really didn't know what proportion of plant or animal taxa showed polyploidy. muller's answer doesn't explain why most animals, e. g. those with


y - dominant sex determination, exhibit relatively little polyploidy. another line of evidence disproving muller's answer is that, in fact, polyploidy is very common among dioecious plants ( those with separate male and female plants ; e. g. westergaard, 1958 ), while muller's theory predicts that prevalence in this group should be as low as in animals. the'complexity'answer... another answer with some historical clout is the one given by daniel standage in his answer, and has been given by various scientists over the years ( e. g. stebbins, 1950 ). this answer states that animals are more complex than plants, so complex that their molecular machinery is much more finely balanced and is disturbed by having multiple genome copies. this answer has been soundly rejected ( e. g. by orr, 1990 ) on the basis of two key facts. firstly, whilst polyploidy is unusual in animals, it does occur. various animals with hermaphroditic or parthenogenetic modes of reproduction frequently show polyploidy. there are also examples of mammalian polyploidy ( e. g. gallardo et al., 2004 ). in addition, polyploidy can be artificially induced in a wide range of animal species, with no deleterious effects ( in fact it often causes something akin to hybrid vigour ; jackson, 1976 ). it's also worth noting here that since the 1960s susumo ohno ( e. g. ohno et al. 1968 ; ohno 1970 ; ohno 1999 ) has been proposing that vertebrate evolution involved multiple whole - genome duplication events ( in addition to smaller duplications ). there is now significant evidence to support this idea, reviewed in furlong & holland ( 2004 ). if true, it further highlights that animals being more complex ( itself a large, and in my view false, assumption ) does not preclude polyploidy. the modern synthesis... and so to the present day. as reviewed in mable ( 2004 ), it is now thought that : polyploidy is an important evolutionary mechanism which was and is probably responsible for a great deal of biological diversity. polyploidy arises easily in both animals and plants, but reproductive strategies might prevent it from propagating in certain circumstances, rather than any reduction in fitness resulting from the genome duplication. polyploidy may be more prevalent in animals


than previously expected, and the imbalance in data arises from the fact that cytogenetics ( i. e. chromosome counting ) of large populations of wild specimens is a very common practise in botany, and very uncommon in zoology. in addition, there are now several new suspected factors involved in ploidy which are currently being investigated : polyploidy is more common in species from high latitudes ( temperate climates ) and high altitudes ( soltis & soltis, 1999 ). polyploidy frequently occurs by the production of unreduced gametes ( through meiotic non - disjunction ), and it has been shown that unreduced gametes are produced with higher frequency in response to environmental fluctuations. this predicts that polyploidy should be more likely to occur in the first place in fluctuating environments ( which are more common at higher latitudes and altitudes ). triploid individuals, the most likely initial result of a genome duplication event, in animals and plants often die before reaching sexual maturity, or have low fertility. however, if triploid individuals do reproduce, there is a chance of even - ploid ( fertile ) individuals resulting. this probability is increased if the species produces large numbers of both male and female gametes, or has some mechanism of bypassing the triploid individual stage. this may largely explain why many species with'alternative'sexual modes ( apomictic, automictic, unisexual, or gynogenetic ) show polyploidy, as they can keep replicating tetraploids, thus increasing the chance that eventually a sexual encounter with another tetraploid will create a new polyploid line. in this way, non - sexual species may be a crucial evolutionary intermediate in generating sexual polyploid species. species with external fertilisation are more likely to establish polyploid lines - a greater proportion of gametes are involved in fertilisation events and therefore two tetraploid gametes are more likely to meet. finally, polyploidy is more likely to occur in species with assortative mixing. that is, when a tetraploid gamete is formed, if the genome duplication somehow affects the individual so as to make it more likely that it will be fertilised by another tetraploid, then it is more likely that a polyploid line will be established. thus it may be partly down to evolutionary chance as to how easily a species '


reproductive traits are affected. for example in plants, tetraploids often have larger flowers or other organs, and thus are preferentially attractive to pollinators. in frogs, genome duplication leads to changes in the vocal apparatus which can lead to immediate reproductive isolation of polyploids. references furlong, r. f. & holland, p. w. h. ( 2004 ) polyploidy in vertebrate ancestry : ohno and beyond. biological journal of the linnean society. 82 ( 4 ), 425 – 430. gallardo, m. h., kausel, g., jimenez, a., bacquet, c., gonzalez, c., figueroa, j., kohler, n. & ojeda, r. ( 2004 ) whole - genome duplications in south american desert rodents ( octodontidae ). biological journal of the linnean society. 82 ( 4 ), 443 – 451. jackson, r. c. ( 1976 ) evolution and systematic significance of polyploidy. annual review of ecology and systematics. 7209 – 234. mable, b. k. ( 2004 ) ‘ why polyploidy is rarer in animals than in plants ’ : myths and mechanisms. biological journal of the linnean society. 82 ( 4 ), 453 – 466. muller, h. j. ( 1925 ) why polyploidy is rarer in animals than in plants. the american naturalist. 59 ( 663 ), 346 – 353. ohno, s. ( 1970 ) evolution by gene duplication. ohno, s. ( 1999 ) gene duplication and the uniqueness of vertebrate genomes circa 1970 – 1999. seminars in cell & developmental biology. 10 ( 5 ), 517 – 522. ohno, s., wolf, u. & atkin, n. b. ( 1968 ) evolution from fish to mammals by gene duplication. hereditas. 59 ( 1 ), 169 – 187. orr, h. a. ( 1990 ) ‘ why polyploidy is rarer in animals than in plants ’ revisited. the american naturalist. 136 ( 6 ), 759 – 770. soltis, d. e. & soltis, p. s. ( 1999 ) polyploidy : recurrent formation and genome evolution. trends in ecology & evolution. 14 ( 9 ), 348 – 352. ste


##bbins, c. l. ( 1950 ) variation and evolution in plants. westergaard, m. ( 1958 ) the mechanism of sex determination in dioecious flowering plants. in : advances in genetics. academic press. pp. 217 – 281. ( i'll come back and add links to the references later )


disclaimer : different people view this differently. i side with lakatos : logic is a tool. proofs are a way to verify one's intuition ( and in many cases to improve one's intuition ) and it is a tool to check the consistency of theories in a process of refining the axioms. the fact that every proof boils down to a tautology is true but irrelevant to mathematics. here is an isomorphic question to the question you posed : a painting is just blobs of paint of different colour on canvas. so, are we to deduce from this fact that the art of painting is reduced to just placing paint on canvas? technically, the answer is yes. but the painter does much more than that. in fact, it is clear that while the painter must possess quite a large amount of skill in placing paint on canvas, this skill is the least relevant ( while absolutely necessary ) for the creative process of painting. so it is with mathematics. being able to prove is essential, but is the least relevant skill for doing mathematics. in mathematics we don't deduce things from axioms. rather we try to capture a certain idea by introducing axioms, check which theorems follow from the axioms and compare these results against the idea we are trying to capture. if the results agree we are happy. if the results disagree, we change the axioms. the ideas we try to capture transcend the deductive system. the deductive system is there to help us find consequences from the axioms, but it does not tell us how to gauge the validity of results against the idea we try to capture, nor how to adjust the axioms. this is my personal point of view of what mathematics is ( or at least what a sizable portion of it is ). it is very close to what physics is. physics is not just some theories about matter and its interactions with stuff. rather it is trying to model reality. so does mathematics, it's just not entirely clear which reality it is trying to model.


unfortunately, nothing in the bonding situation in carbon monoxide is easily explained, especially not the dipole moment. according to the electronegativities of the elements, you would expect the partial positive charge to be at the carbon and a partial negative charge at oxygen. however, this is not the case, which can only be explained by molecular orbital theory. a complete analysis of this can be found in gernot frenking, christoph loschen, andreas krapp, stefan fau, and steven h. strauss, j. comp. chem., 2007, 28 ( 1 ), 117 - 126. ( i believe it is available free of charge. ) responsible for the dipole moment is the highest occupied molecular orbital, a $ \ pmb { \ sigma } $ orbital, which has its largest coefficient at the carbon atom. in first order approximation, this orbital can be considered the lone pair of carbon. all other valence orbitals are more strongly polarised towards the oxygen. the orbital that can in first order approximation be considered as the oxygen lone pair has almost only s character and therefore contributes only little to the dipole moment. \ begin { align } \ ce { { } ^ { \ ominus } \! : c # o : ^ { \ oplus } } & & \ text { dipole : } ~ | \ mathbf { q } | = 0. 11 ~ \ mathrm { d } & & \ text { direction : } ~ \ longleftarrow \ end { align } i have reproduced the mo scheme of carbon monoxide for you below. please note, that the blue / orange coloured orbitals are virtual ( unoccupied ) orbitals, which should be taken with a grain of salt. there are two possible decomposition schemes to explain the bonding, both of them involve donor - acceptor interactions. the term " dative bonding " should be avoided here, it is better to use it only for bonds, that consist purely of donor - acceptor interactions, as for example in $ \ ce { h3n \ bond { - > } bh3 } $. below, the two decomposition schemes are reproduced from figure 6 ( b & c ) in the linked paper. please note, that this decomposition does not include hybridised orbitals. the left decomposition is a better description, since it retains the $ c _ { \ infty { } v } $ symmetry of the molecule. we can see a donor - acceptor $ \ sigma $


bond and two electrons sharing $ \ pi $ bonds. in the right configuration we assume an electron sharing $ \ sigma $ bond, an electron sharing $ \ pi $ bond and a donor - acceptor $ \ pi $ bond. it is very important to understand, that the concept of a dative bond, that you are trying to employ here is only right by coincidence. the reason that the dipole moment is oriented towards the carbon is only to find in the weakly bonding homo, the lone pair of carbon.


here is an approach. we give some preliminary results. the poly - hurwitz zeta function the poly - hurwitz zeta function may initially be defined by the series $ $ \ begin { align } \ displaystyle \ zeta ( s \ mid a, b ) : = \ sum _ { n = 1 } ^ { + \ infty } \ frac { 1 } { ( n + a ) ^ { s } ( n + b ) }, \ quad \ re a > - 1, \, \ re b > - 1, \, \ re s > 0. \ tag1 \ end { align } $ $ this special function is a natural extension of the hurwitz zeta function initially defined as $ $ \ zeta ( s, a ) = \ sum _ { n = 0 } ^ { \ infty } \ frac { 1 } { ( n + a ) ^ s }, \ quad \ re a > 0, \ re s > 1, \ tag2 $ $ which is a natural extension itself of the riemann zeta function initially defined as $ $ \ zeta ( s ) = \ sum _ { n = 1 } ^ { \ infty } \ frac { 1 } { n ^ s }, \ quad \ re s > 1. \ tag3 $ $ the poly - hurwitz function appears in different places with different notations, one may find it here : [ masri, p. 2 and p. 15 ( 2004 ) ], [ murty, p. 17 ( 2006 ) ], [ sinha, p. 45 ( 2002 ) ]. in this answer we are dealing with a simplified version of a general poly - hurwitz function. the series in $ ( 1 ) $ converges absolutely for $ \ displaystyle \ re s > 0 $. moreover, the convergence of the series is uniform on every half - plane $ $ \ displaystyle h _ { \ delta } = \ left \ { s \ in \ mathbb { c }, \ re s \ geq \ delta \ right \ }, \, \ delta \ in \ mathbb { r }, \, \ delta > 0, $ $ therefore the poly - hurwitz zeta function $ \ displaystyle \ zeta ( \ cdot \ mid a, b ) $ is analytic on the half - plane $ \ displaystyle \ re s > 0 $. let $ a $, $ b $ and $ s $ be complex numbers such


that $ \ re a > - 1, \, \ re b > - 1, \, \ re s > 0 $. one may observe that $ $ \ begin { align } \ zeta ( s \ mid a, b ) & = \ sum _ { n = 1 } ^ { + \ infty } \ frac { 1 } { ( n + a ) ^ { s } ( n + b ) } \ \ & = \ sum _ { n = 1 } ^ { + \ infty } \ frac { ( n + b ) + ( a - b ) } { ( n + a ) ^ { s + 1 } ( n + b ) } \ \ & = \ sum _ { n = 1 } ^ { + \ infty } \ frac { 1 } { ( n + a ) ^ { s + 1 } } + ( a - b ) \ sum _ { n = 1 } ^ { + \ infty } \ frac { 1 } { ( n + a ) ^ { s + 1 } ( n + b ) } \ tag4 \ end { align } $ $ giving the functional identity $ $ \ begin { align } \ zeta ( s \ mid a, b ) = \ zeta ( s + 1, a + 1 ) + ( a - b ) \ zeta ( s + 1 \ mid a, b ) \ tag5 \ end { align } $ $ where $ \ displaystyle \ zeta ( \ cdot, \ cdot ) $ is the standard hurwitz zeta function. from $ ( 5 ) $, we obtain by induction, for $ n = 1, 2, 3, \ ldots $, $ $ \ begin { align } \ zeta ( s \ mid a, b ) = \ sum _ { k = 1 } ^ { n } ( a - b ) ^ { k - 1 } \ zeta ( s + k, a + 1 ) + ( a - b ) ^ n \ zeta ( s + n \ mid a, b ). \ tag6 \ end { align } $ $ we use $ ( 6 ) $ to extend $ \ displaystyle \ zeta ( \ cdot \ mid a, b ) $ to a meromorphic function on each open set $ \ re s > - n $, $ n \ geq 1 $. since the hurwitz zeta function is analytic on the whole complex plane except for a simple pole at $ 1 $ with residue $


1 $, then from $ ( 6 ) $ the poly - hurwitz zeta function $ \ displaystyle \ zeta ( \ cdot \ mid a, b ) $ is analytic on the whole complex plane except for a simple pole at $ 0 $ with residue $ 1 $. the poly - stieltjes constants in 1885 stieltjes has found that the laurent series expansion around $ 1 $ of the riemann zeta function $ $ \ zeta ( 1 + s ) = \ frac { 1 } { s } + \ sum _ { k = 0 } ^ { \ infty } \ frac { ( - 1 ) ^ { k } } { k! } \ gamma _ k s ^ k, \ quad s \ neq 0, \ tag7 $ $ is such that the coefficients of the regular part of the expansion are given by $ $ \ begin { align } \ gamma _ k & = \ lim _ { n \ to \ infty } \ left ( \ sum _ { n = 1 } ^ n \ frac { \ log ^ k n } { n } - \ frac { \ log ^ { k + 1 } \! n } { k + 1 } \ right ). \ end { align } \ tag8 $ $ euler was the first to define a constant of this form ( 1734 ) $ $ \ begin { align } \ gamma & = \ lim _ { n \ to \ infty } \ left ( 1 + \ frac12 + \ frac13 + \ cdots + \ frac1n - \ log n \ right ) = 0. 577215 \ ldots. \ end { align } $ $ the constants $ \ displaystyle \ gamma _ k $ are called the stieltjes constants and due to the fact that $ \ displaystyle \ gamma _ 0 = \ gamma $ they are sometimes called the generalized euler's constants. similarly, wilton ( 1927 ) and berndt ( 1972 ) established that the laurent series expansion in the neighbourhood of $ 1 $ of the hurwitz zeta function $ $ \ begin { align } \ zeta ( 1 + s, a ) = \ frac1s + \ sum _ { k = 0 } ^ { \ infty } \ frac { ( - 1 ) ^ { k } } { k! } \ gamma _ { k } ( a ) \ : s ^ { k }, \ quad


\ re a > 0, \, s \ neq 0, \ tag { 9 } \ end { align } $ $ is such that the coefficients of the regular part of the expansion are given by $ $ \ begin { align } \ gamma _ k ( a ) & = \ lim _ { n \ to \ infty } \ left ( \ sum _ { n = 0 } ^ n \ frac { \ log ^ k ( n + a ) } { n + a } - \ frac { \ log ^ { k + 1 } ( n + a ) } { k + 1 } \ right ), \ quad \ re a > 0, \ end { align } \ tag { 10 } $ $ with $ \ displaystyle \ gamma _ { 0 } ( a ) = - \ psi ( a ) = - \ gamma'( a ) / \ gamma ( a ) $. the coefficients $ \ gamma _ k ( a ) $ are called the generalized stieltjes constants. we have seen from $ ( 6 ) $ that the poly - hurwitz zeta function admits a laurent series expansion around $ 0 $. let's denote by $ \ displaystyle \ gamma _ k ( a, b ) $ the coefficients of the regular part of $ \ displaystyle \ zeta ( \ cdot \ mid a, b ) $ around $ 0 $. i will call these coefficients the poly - stieltjes constants. do we have an analog of $ ( 10 ) $ for $ \ displaystyle \ gamma _ k ( a, b ) $? the following result is new. theorem 1. let $ a, b $ be complex numbers such that $ \ re a > - 1, \, \ re b > - 1 $. consider the poly - hurwitz zeta function $ $ \ begin { align } \ zeta ( s \ mid a, b ) : = \ sum _ { n = 1 } ^ { + \ infty } \ frac { 1 } { ( n + a ) ^ { s } ( n + b ) }, \ quad \ re s > 0. \ tag { 11 } \ end { align } $ $ then the meromorphic extension of $ \ displaystyle \ zeta ( \ cdot \ mid a, b ) $ admits the following laurent series expansion around $ 0 $, $ $ \ zeta ( s \ mid a, b ) = \ frac { 1 } { s } + \ sum


_ { k = 0 } ^ { + \ infty } \ frac { ( - 1 ) ^ { k } } { k! } \ gamma _ k ( a, b ) s ^ k, \ quad s \ neq 0, \ tag { 12 } $ $ where the poly - stieltjes constants $ \ displaystyle \ gamma _ k ( a, b ) $ are given by $ $ \ begin { align } \ gamma _ k ( a, b ) & = \ lim _ { n \ to + \ infty } \ left ( \ sum _ { n = 1 } ^ n \ frac { \ log ^ k ( n + a ) } { n + b } - \ frac { \ log ^ { k + 1 } \! n } { k + 1 } \ right ) \ end { align } \ tag { 13 } $ $ with $ $ \ gamma _ { 0 } ( a, b ) = - \ psi ( b + 1 ) = - \ gamma'( b + 1 ) / \ gamma ( b + 1 ). \ tag { 14 } $ $ proof. let $ a, b $ be complex numbers such that $ \ re a > - 1, \, \ re b > - 1 $. we first assume $ \ re s > 0 $. observing that, for each $ n \ geq 1 $, $ $ \ left | \ sum _ { k = 0 } ^ { \ infty } \ frac { \ log ^ k ( n + a ) } { n + b } \ frac { ( - 1 ) ^ { k } } { k! } s ^ k \ right | \ leq \ sum _ { k = 0 } ^ { \ infty } \ left | \ frac { \ log ^ k ( n + a ) } { n + b } \ right | \ frac { | s | ^ k } { k! } < \ infty $ $ and that $ $ \ sum _ { n = 1 } ^ { \ infty } \ left | \ sum _ { k = 0 } ^ { \ infty } \ frac { \ log ^ k ( n + a ) } { n + b } \ frac { ( - 1 ) ^ { k } } { k! } s ^ k \ right | = \ sum _ { n = 1 } ^ { \ infty }


\ left | \ frac1 { ( n + a ) ^ s ( n + b ) } \ right | = \ sum _ { n = 1 } ^ { \ infty } \ frac1 { | n + a | ^ { \ re s } | n + b | } < \ infty, $ $ we obtain $ $ \ begin { align } & \ sum _ { k = 0 } ^ { \ infty } \ frac { ( - 1 ) ^ { k } } { k! } \ lim _ { n \ to + \ infty } \ left ( \ sum _ { n = 1 } ^ n \ frac { \ log ^ k ( n + a ) } { n + b } - \ frac { \ log ^ { k + 1 } \! n } { k + 1 } \ right ) s ^ k \ \ \ \ & = \ lim _ { n \ to + \ infty } \ sum _ { k = 0 } ^ { \ infty } \ frac { ( - 1 ) ^ { k } } { k! } \ left ( \ sum _ { n = 1 } ^ n \ frac { \ log ^ k ( n + a ) } { n + b } - \ frac { \ log ^ { k + 1 } \! n } { k + 1 } \ right ) s ^ k \ \ \ \ & = \ lim _ { n \ to + \ infty } \ sum _ { k = 0 } ^ { \ infty } \ left ( \ sum _ { n = 1 } ^ n \ frac { ( - 1 ) ^ { k } } { k! } \ frac { \ log ^ k ( n + a ) } { n + b } s ^ k - \ frac { ( - 1 ) ^ { k } } { k! } \ frac { \ log ^ { k + 1 } \! n } { k + 1 } s ^ k \ right ) \ \ \ \ & = \ lim _ { n \ to + \ infty } \ left ( \ sum _ { n = 1 } ^ n \ sum _ { k = 0 } ^ { \ infty } \ frac { ( - 1 ) ^ { k } } { k! } \ frac { \ log ^ k ( n + a ) } { n + b } s ^


k - \ sum _ { k = 0 } ^ { \ infty } \ frac { ( - 1 ) ^ { k } } { k! } \ frac { \ log ^ { k + 1 } \! n } { k + 1 } s ^ k \ right ) \ \ \ \ & = \ lim _ { n \ to + \ infty } \ left ( \ sum _ { n = 1 } ^ n \ frac1 { ( n + a ) ^ s ( n + b ) } + \ frac1 { n ^ s } - \ frac1s \ right ) \ \ \ \ & = \ zeta ( s \ mid a, b ) - \ frac1 { s } \ end { align } $ $ as desired. then, using $ ( 6 ) $, we extend the preceding identity by analytic continuation to all $ s \ neq 0 $. to prove $ ( 14 ) $, we start from a standard series representation of the digamma function ( see abram. & steg. p. 258 6. 3. 16 ) : $ $ \ begin { align } - \ psi ( b + 1 ) & = \ gamma - \ sum _ { n = 1 } ^ { \ infty } \ left ( \ frac1n - \ frac1 { b + n } \ right ) \ \ & = \ lim _ { n \ to + \ infty } \ left ( \ gamma - \ sum _ { n = 1 } ^ n \ left ( \ frac1n - \ frac1 { b + n } \ right ) \ right ) \ \ & = \ lim _ { n \ to + \ infty } \ left ( \ left ( \ sum _ { n = 1 } ^ n \ frac1 { b + n } - \ ln n \ right ) - \ left ( \ sum _ { n = 1 } ^ n \ frac1n - \ ln n - \ gamma \ right ) \ right ) \ \ & = \ lim _ { n \ to + \ infty } \ left ( \ sum _ { n = 1 } ^ n \ frac1 { b + n } - \ ln n \ right ) \ \ \ \ & = \ gamma _ 0 ( a, b ) \ end { align } $ $ using $ ( 13 ) $. $ \ qquad \ qquad


\ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ box $ one of the consequences of theorem 1 is the new possibility to express some series in terms of the poly - stieltjes constants. theorem 2. let $ a, b, c $ be complex numbers such that $ \ re a > - 1, \, \ re b > - 1, \, \ re c > - 1 $. then $ $ \ begin { align } ( b - a ) \ sum _ { n = 1 } ^ { + \ infty } \ frac { \ log ( n + c ) } { ( n + a ) ( n + b ) } = \ gamma _ 1 ( c, a ) - \ gamma _ 1 ( c, b ), \ tag { 15 } \ end { align } $ $ similarly $ $ \ begin { align } \ sum _ { n = 1 } ^ { + \ infty } \ frac1 { n + b } \ left ( { \ log ( n + a ) - \ log ( n + c ) } \ right ) = \ gamma _ 1 ( a, b ) - \ gamma _ 1 ( c, b ), \ tag { 16 } \ end { align } $ $ with the poly - stieltjes constant $ $ \ gamma _ 1 ( a, b ) = \ lim _ { n \ to + \ infty } \ left ( \ sum _ { n = 1 } ^ n \ frac { \ log ( n + a ) } { n + b } - \ frac { \ log ^ 2 \! n } 2 \ right ). $ $ proof. let $ a, b, c $ be complex numbers such that $ \ re a > - 1, \, \ re b > - 1, \, \ re c > - 1 $. we have $ $ ( b - a ) \ frac { \ log ( n + c ) } { ( n + a ) ( n + b ) } = \ frac { \ log ( n + c ) } { n + a } - \ frac { \ log ( n + c ) } { n +


b } $ $ giving, for $ n \ geq1 $, $ $ \ begin { align } ( b - a ) & \ sum _ { n = 1 } ^ n \ frac { \ log ( n + c ) } { ( n + a ) ( n + b ) } = \ \ \ \ & \ left ( \ sum _ { n = 1 } ^ n \ frac { \ log ( n + c ) } { n + a } - \ frac { \ log ^ 2 \! n } 2 \ right ) - \ left ( \ sum _ { n = 1 } ^ n \ frac { \ log ( n + c ) } { n + b } - \ frac { \ log ^ 2 \! n } 2 \ right ) \ tag { 17 } \ end { align } $ $ letting $ n \ to \ infty $ and using $ ( 13 ) $ gives $ ( 15 ) $. we have, for $ n \ geq1 $, $ $ \ begin { align } & \ sum _ { n = 1 } ^ n \ frac1 { n + b } \ left ( { \ log ( n + a ) - \ log ( n + c ) } \ right ) \ \ \ \ & = \ left ( \ sum _ { n = 1 } ^ n \ frac { \ log ( n + a ) } { n + b } - \ frac { \ log ^ 2 \! n } 2 \ right ) - \ left ( \ sum _ { n = 1 } ^ n \ frac { \ log ( n + c ) } { n + b } - \ frac { \ log ^ 2 \! n } 2 \ right ) \ tag { 18 } \ end { align } $ $ letting $ n \ to \ infty $ and using $ ( 13 ) $ gives $ ( 16 ) $. $ \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ box $ juantheron's integral let ’ s first give a numerical evaluation of juantheron ’ s integral. i would like to thank jonathan borwein and david


h. bailey who obtained the result below to $ 1000 $ digits, in just 3. 9 seconds run time, using david's new mpfun - mpfr software, along with the tanh - sinh quadrature program included with the mpfun - mpfr package. they also tried the integral with david's mpfun - fort package, which has a completely different underlying multiprecision system, and they obtained the same result below. finally, they computed the integral with mathematica $ 11. 0 $ ; it agreed with the result below, although it required about 10 times longer to run. proposition 1. we have $ $ \ begin { align } \ int _ { 0 } ^ { \ large \ frac { \ pi } 2 } \! & \ frac1 { ( 1 + x ^ 2 ) ( 1 + \ tan x ) } \ mathrm dx \ \ \ \ = 0. & 59738180945180348461311323509087376430643859042555 \ \ & 67307703207161550311033249824121789098990404474443 \ \ & 73300942847961727020952797366230453350097928752529 \ \ & 62099371263365268445580755896768905606293308536674 \ \ & 89639352215352393870280616186538538722285601087082 \ \ & 81730013060929540132583577240799018025603130403772 \ \ & 83596189879605956759516344861849456740112012597646 \ \ & 30195536341071109827787231788650530475635336662512 \ \ & 50757672078320586388500276160658476344052492489409 \ \ & 64026178233152015087197531


##148322444147655936720008 \ tag { 19 } \ \ & 40650450631581050321100329502169853063154902765446 \ \ & 58804861176982696627707544105655815406116180984371 \ \ & 54148587721902800400109013880620460529382772599713 \ \ & 06874977209651994186527207589425408866256042399213 \ \ & 80515694164361264997143539392018681691584285790381 \ \ & 65536517701019826846772718498479534803417547866296 \ \ & 23842162877309354675086691711521468623807334908897 \ \ & 71491673168051054009130049879837629516862688171756 \ \ & 13790927986073268994254629238035029442300668334396 \ \ & 901581838911515359223628586133156893962372426055 \ cdots \ end { align } $ $ david h. bailey confirmed that mathematica $ 11. 0 $, in spite of the great numerical precision, could not find a closed - form of the integral. the next result proves that the op integral admits a closed form in terms of the poly - stieltjes constants. proposition 2. we have $ $ \ begin { align } \ int _ { 0 } ^ { \ large \ frac { \ pi } 2 } \! \! \ frac1 { ( 1 + x ^ 2 ) ( 1 + \ tan x ) } \ mathrm dx & = \ frac { ( e ^ 2 + 1 ) ^ 2 } { 2 ( e ^ 4 + 1 ) } \ arctan \


! \ frac { \ pi } { 2 } - \ frac { e ^ 4 - 1 } { 4 ( e ^ 4 + 1 ) } \ log \ left ( 1 + \ frac { \ pi ^ 2 } { 4 } \ right ) \ \ \ \ & + \ frac { 64 \ pi ^ 2 \ log 3 } { ( \ pi ^ 2 + 16 ) ( 9 \ pi ^ 2 + 16 ) } \ \ \ \ & + \ frac { \ im } { 2 \ pi } \ gamma _ 1 \! \ left ( \! \ frac34, \ frac34 + \ frac { i } { \ pi } \! \ right ) - \ frac { \ im } { 2 \ pi } \ gamma _ 1 \! \ left ( \! \ frac14, \ frac34 + \ frac { i } { \ pi } \! \ right ) \ tag { 20 } \ \ \ \ & + \ frac { \ im } { 2 \ pi } \ gamma _ 1 \! \ left ( \! \ frac34, \ frac14 - \ frac { i } { \ pi } \! \ right ) - \ frac { \ im } { 2 \ pi } \ gamma _ 1 \! \ left ( \! \ frac14, \ frac14 - \ frac { i } { \ pi } \! \ right ) \ end { align } $ $ with the poly - stieltjes constant $ $ \ gamma _ 1 ( a, b ) = \ lim _ { n \ to + \ infty } \ left ( \ sum _ { n = 1 } ^ n \ frac { \ log ( n + a ) } { n + b } - \ frac { \ log ^ 2 \! n } 2 \ right ). $ $ proof. we proceed in three steps. step 1. one may write $ $ \ require { cancel } \ begin { align } & \ int _ { 0 } ^ { \ large \ frac { \ pi } { 2 } } \ frac { 1 } { ( 1 + x ^ 2 ) ( 1 + \ tan x ) } \ mathrm dx \ \ & = \ int _ { 0 } ^ { \ large \ frac { \ pi } { 2 } } \ frac { \ cos x } { ( 1 + x ^ 2 )


( \ cos x + \ sin x ) } \ mathrm dx \ \ & = \ frac12 \ int _ { 0 } ^ { \ large \ frac { \ pi } { 2 } } \ frac { ( \ cos x + \ sin x ) + ( \ cos x - \ sin x ) } { ( 1 + x ^ 2 ) ( \ cos x + \ sin x ) } \ mathrm dx \ \ & = \ frac12 \ int _ { 0 } ^ { \ large \ frac { \ pi } { 2 } } \! \ frac { 1 } { 1 + x ^ 2 } \ mathrm dx + \ frac12 \ int _ { 0 } ^ { \ large \ frac { \ pi } { 2 } } \ frac { 1 } { ( 1 + x ^ 2 ) } \ frac { ( \ cos x - \ sin x ) } { ( \ cos x + \ sin x ) } \ mathrm dx \ \ & = \ frac12 \ arctan \! \ frac { \ pi } { 2 } + \ frac12 \ int _ { 0 } ^ { \ large \ frac { \ pi } { 2 } } \ frac { 1 } { 1 + x ^ 2 } \ tan ( x - \ pi / 4 ) \ : \ mathrm dx \ \ & = \ frac12 \ arctan \! \ frac { \ pi } { 2 } + \ frac12 \ int _ { - \ large \ frac { \ pi } { 4 } } ^ { \ large \ frac { \ pi } 4 } \ frac { 1 } { 1 + ( x + \ pi / 4 ) ^ 2 } \ tan x \ : \ mathrm dx \ \ & = \ frac12 \ arctan \! \ frac { \ pi } { 2 } + \ frac12 \ int _ 0 ^ { \ large \ frac { \ pi } 4 } \ left ( \ frac1 { 1 + ( x + \ pi / 4 ) ^ 2 } - \ frac1 { 1 + ( x - \ pi / 4 ) ^ 2 } \ right ) \ tan x \ : \ mathrm dx \ \ & = \ frac12 \ arctan \! \ frac { \ pi } { 2 } - \ frac { \


im } 2 \! \ int _ { 0 } ^ { \ large \ frac { \ pi } { 4 } } \! \! \ left ( \! \ frac1 { x + \ pi / 4 + i } + \ frac1 { x - \ pi / 4 - i } \! \ right ) \ tan x \ : \ mathrm dx \ tag { 21 } \ end { align } $ $ let ’ s evaluate the latter integral. step 2. one may recall that the tangent function, as a meromorphic function, can be expressed as an infinite sum of rational functions : $ $ \ tan x = \ sum _ { n = 0 } ^ { + \ infty } \ frac { 2x } { \ pi ^ 2 ( n + 1 / 2 ) ^ 2 - x ^ 2 }, \ quad x \ neq \ pm \ pi / 2, \ pm 3 \ pi / 2, \ pm 5 \ pi / 2, \ ldots. \ tag { 22 } $ $ we have the inequality $ $ \ sup _ { x \ in [ 0, \ pi / 4 ] } \ left | \ frac { 2x } { \ pi ^ 2 ( n + 1 / 2 ) ^ 2 - x ^ 2 } \ right | \ leq \ frac1 { ( n + 1 / 2 ) ^ 2 }, \ quad n = 0, 1, 2, \ ldots, \ tag { 23 } $ $ the convergence in $ ( 22 ) $ is then uniform on $ [ 0, \ pi / 4 ] $. thus, plugging $ ( 22 ) $ into $ ( 21 ) $, we are allowed to integrate $ ( 21 ) $ termwise. each term, via a partial fraction decomposition, is evaluated to obtain $ $ \ begin { align } \ int _ { 0 } ^ { \ large \ frac { \ pi } 4 } \! & \ left ( \! \ frac1 { x + \ pi / 4 + i } + \ frac1 { x - \ pi / 4 - i } \! \ right ) \ frac { 2x } { \ pi ^ 2 ( n + 1 / 2 ) ^ 2 - x ^ 2 } \ : \ mathrm dx \ \ & = \ frac { 2 \ tau } { \ pi ^ 2 ( n + 1 / 2 ) ^ 2 - \ tau ^ 2 } \ log \


left ( \ frac { 4 \ tau - \ pi } { 4 \ tau + \ pi } \ right ) \ \ & + \ frac1 { \ pi } \ frac1 { ( n + 1 / 2 + \ tau / \ pi ) } \ left ( \ log \! \ left ( n + \ frac34 \ right ) - \ log \! \ left ( n + \ frac14 \ right ) \ right ) \ \ & + \ frac1 { \ pi } \ frac1 { ( n + 1 / 2 - \ tau / \ pi ) } \ left ( \ log \! \ left ( n + \ frac34 \ right ) - \ log \! \ left ( n + \ frac14 \ right ) \ right ) \ end { align } \ tag { 24 } $ $ where for the sake of convenience we have set $ \ tau : = \ pi / 4 + i $. step 3. we sum $ ( 24 ) $ from $ n = 0 $ to $ \ infty $ obtaining $ $ \ begin { align } \ int _ { 0 } ^ { \ large \ frac { \ pi } { 4 } } \! & \ left ( \! \ frac1 { x + \ pi / 4 + i } + \ frac1 { x - \ pi / 4 - i } \! \ right ) \ tan x \ : \ mathrm dx \ \ & = \ sum _ { n = 0 } ^ { \ infty } \ frac { 2 \ tau } { \ pi ^ 2 ( n + 1 / 2 ) ^ 2 - \ tau ^ 2 } \ log \ left ( \ frac { 4 \ tau - \ pi } { 4 \ tau + \ pi } \ right ) \ \ & + \ frac1 { \ pi } \ sum _ { n = 0 } ^ { \ infty } \ frac1 { ( n + 1 / 2 + \ tau / \ pi ) } \ left ( \ log \! \ left ( n + \ frac34 \ right ) - \ log \! \ left ( n + \ frac14 \ right ) \ right ) \ \ & + \ frac1 { \ pi } \ sum _ { n = 0 } ^ { \ infty } \ frac1 { ( n + 1 / 2 - \ tau / \ pi ) } \ left ( \ log


\! \ left ( n + \ frac34 \ right ) - \ log \! \ left ( n + \ frac14 \ right ) \ right ), \ tag { 25 } \ end { align } $ $ then, singling out the first terms in the two last series and using theorem $ 2 $ $ ( 16 ) $, we get $ $ \ begin { align } \ int _ { 0 } ^ { \ large \ frac { \ pi } { 4 } } \! & \ left ( \! \ frac1 { x + \ pi / 4 + i } + \ frac1 { x - \ pi / 4 - i } \! \ right ) \ tan x \ : \ mathrm dx \ \ & = \ tan \ tau \ log \ left ( \ frac { 4 \ tau - \ pi } { 4 \ tau + \ pi } \ right ) + \ frac { 4 \ pi } { \ pi ^ 2 - 4 \ tau ^ 2 } \ log 3 \ \ & + \ frac1 { \ pi } \ gamma _ 1 \! \ left ( \! \ frac34, \ frac12 + \ frac { \ tau } { \ pi } \! \ right ) - \ frac1 { \ pi } \ gamma _ 1 \! \ left ( \! \ frac14, \ frac12 + \ frac { \ tau } { \ pi } \! \ right ) \ tag { 26 } \ \ & + \ frac1 { \ pi } \ gamma _ 1 \! \ left ( \! \ frac34, \ frac12 - \ frac { \ tau } { \ pi } \! \ right ) - \ frac1 { \ pi } \ gamma _ 1 \! \ left ( \! \ frac14, \ frac12 - \ frac { \ tau } { \ pi } \! \ right ) \ end { align } $ $ and the substitution $ \ tau = \ pi / 4 + i $ gives the desired result. $ \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \


box $ achille hui's conjecture is true. achille hui has announced in the comments that the op integral is equal to $ $ \ begin { align } \ frac { \ arctan ( \ frac { \ pi } { 2 } ) - t \ log \ sqrt { 1 + \ frac { \ pi ^ 2 } { 4 } } } { 1 + t ^ 2 } + \ frac { \ pi ^ 2 } 4 \ sum _ { n = 0 } ^ { \ infty } \ frac { ( 2n + 1 ) \ left ( \ log \ left ( n + \ frac34 \ right ) - \ log \ left ( n + \ frac14 \ right ) \ right ) } { \ left ( 1 + \ pi ^ 2 \ left ( n + \ frac14 \ right ) ^ 2 \ right ) \ left ( 1 + \ pi ^ 2 \ left ( n + \ frac34 \ right ) ^ 2 \ right ) } \ tag { 27 } \ end { align } $ $ with $ \ displaystyle t : = \ tanh ( 1 ) $. the first term in $ ( 27 ) $, with a little algebra is seen to be equal to the sum of the first two terms on the right hand side of $ ( 20 ) $. we establish the veracity of the conjecture using proposition $ 2 $ and using the next result. proposition 3. we have $ $ \ begin { align } & \ frac { \ pi ^ 2 } 4 \ sum _ { n = 0 } ^ { \ infty } \ frac { ( 2n + 1 ) \ left ( \ log \ left ( n + \ frac34 \ right ) - \ log \ left ( n + \ frac14 \ right ) \ right ) } { \ left ( 1 + \ pi ^ 2 \ left ( n + \ frac14 \ right ) ^ 2 \ right ) \ left ( 1 + \ pi ^ 2 \ left ( n + \ frac34 \ right ) ^ 2 \ right ) } \ \ \ \ & = \ frac { 64 \ pi ^ 2 \ log 3 } { ( \ pi ^ 2 + 16 ) ( 9 \ pi ^ 2 + 16 ) } \ \ \ \ & + \ frac { \ im } { 2 \ pi } \ gamma _ 1 \! \ left ( \! \ frac34, \ frac34 +


\ frac { i } { \ pi } \! \ right ) - \ frac { \ im } { 2 \ pi } \ gamma _ 1 \! \ left ( \! \ frac14, \ frac34 + \ frac { i } { \ pi } \! \ right ) \ tag { 28 } \ \ \ \ & + \ frac { \ im } { 2 \ pi } \ gamma _ 1 \! \ left ( \! \ frac34, \ frac14 - \ frac { i } { \ pi } \! \ right ) - \ frac { \ im } { 2 \ pi } \ gamma _ 1 \! \ left ( \! \ frac14, \ frac14 - \ frac { i } { \ pi } \! \ right ) \ end { align } $ $ with the poly - stieltjes constant $ $ \ gamma _ 1 ( a, b ) = \ lim _ { n \ to + \ infty } \ left ( \ sum _ { n = 1 } ^ n \ frac { \ log ( n + a ) } { n + b } - \ frac { \ log ^ 2 \! n } 2 \ right ). $ $ proof. observe that the first term of the series on the left hand side of $ ( 28 ) $, given by $ n = 0 $, is just equal to $ $ \ frac { 64 \ pi ^ 2 \ log 3 } { ( \ pi ^ 2 + 16 ) ( 9 \ pi ^ 2 + 16 ) }. $ $ by a partial fraction decomposition, one may check that $ $ \ begin { align } \ frac { \ pi ^ 2 } 4 & \ frac { ( 2n + 1 ) } { \ left ( 1 + \ pi ^ 2 \ left ( n + \ frac14 \ right ) ^ 2 \ right ) \ left ( 1 + \ pi ^ 2 \ left ( n + \ frac34 \ right ) ^ 2 \ right ) } = \ frac { \ im } { 2 \ pi } \ left ( \! \ frac1 { n + \ frac34 + \ frac { i } { \ pi } } - \ frac1 { n + \ frac14 + \ frac { i } { \ pi } } \! \ right ) \ tag { 29 } \ end { align } $ $ then,


multiplying $ ( 29 ) $ by $ \ left ( \ log \! \ left ( n + \ frac34 \ right ) - \ log \! \ left ( n + \ frac14 \ right ) \ right ) $ and summing from $ n = 1 $ to $ \ infty $ we get, using theorem $ 2 $ $ ( 16 ) $, the result $ ( 28 ) $. $ \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ box $