Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

learn more… | top users | synonyms

1
vote
2answers
3k views

Expected value of $\ln X$ if $X$ is $\Gamma(a,b)$ distributed.

I'm new here and hope you can help. It's really late here in South Africa, maybe my mind just doesn't want to function now! But I need to figure out how to get a closed form expression hopefully for ...
4
votes
1answer
453 views

Integral with exp and erf

I found an integral calculated from what I understand with “differentation under the integration sign” method. $$ ...
5
votes
0answers
158 views

Are there asymptotic expressions for multiple zetas $\small \zeta(s),\zeta(s,s),\zeta(s,s,s),\ldots$ where $\small s=1+\delta, \delta\to 0$?

Playing around with elementary symmetric functions I tried to generalize that to infinite series and arrived at the well known concept of MZV ("multiple zeta values"). At the moment I'm only ...
1
vote
1answer
138 views

Does division of polynomials give an increasing function?

How can I show that \begin{equation} f(a)=\frac{\sum_{i=1}^{k^*-1} \left(\begin{array}{c} K \\ i \\ \end{array} \right) \left(-1-\frac{1}{ar}\right)^i+1}{\sum_{i=1}^{k^*-1} \left(\begin{array}{c} K \\ ...
3
votes
1answer
341 views

An example of divergent series with the Lerch function

I am often working with divergent series all around being this the bread and butter for a theoretical physicist. Thanks to the excellent work of Hardy these have lost their mystical Aurea and so, they ...
4
votes
1answer
4k views

Taylor Expansion of Error Function

I am working on a question that involves finding the Taylor expansion of the error function. The question is stated as follows The error function is defined by $\mathrm{erf}(x):=\frac ...
6
votes
1answer
434 views

Was the definition of $\mathrm{erf}$ changed at some point?

I have seen two competing definitions of the error function. When I was an undergrad, Spiegel's Mathematical Handbook of formulas and tables (mine is the 1968 edition) was the definitive authority, ...
4
votes
1answer
971 views

Limit for gamma function

How can I prove that $$\displaystyle \Gamma(z)=\lim_{n \to \infty} \displaystyle \int_0^n \left( 1-\frac{t}{n}\right)^n t^{z-1}\ \text{d} t\;=\displaystyle \int_0^{\infty} e^{-t} t^{z-1}\ \text{d} ...
6
votes
2answers
1k views

Asymptotic expansion of integral involving modified Bessel-function

I would like to obtain the asymptotic expression for $\alpha \to \infty$ of the following integral $$I(\alpha)=\int_0^\infty\!dx\,x (1 - \cos[2\alpha K_0(x)]) = \int_0^\infty\!dx\, 2x \sin^2[\alpha ...
1
vote
1answer
170 views

A uniqueness proposition involving Erf, the error function

This is a MathOverflow cross-post (currently no answer there) and a generalization of a previous MathOverflow question, "Reducing system of equations involving Erf, Error Function". Consider the ...
6
votes
0answers
147 views

relationship between solution of quintic in terms of $_{4}F_{3}$ hypergeometric function and theta functions

There is one approach (Bring radical/method of differential resolvents) to the general solution to the quintic that gives the solution for a particular root $v\in\{v_{1},v_{2},v_{3},v_{4},v_{5}\}$ in ...
2
votes
1answer
373 views

How does $\int_1 ^x \cos(2\pi/t) dt$ have complex values for real values of $x$?

This question is closely related to one I just asked here. I believe that it is just different enough to warrant another question; please let me know if it does not. In the question mentioned above, ...
3
votes
1answer
189 views

Deriving the form of the Exponential Integral from a given integral

The Wikipedia entry on Asymptotic Expansion outlines a detailed example, where it refers to the fact that the integral \begin{equation} \int_0^\infty \frac{e^{-w/t}}{1-w} \, dw \end{equation} ...
11
votes
4answers
387 views

Evaluate $\int_0^1 \frac{\ln(1+bx)}{1+x} dx $

What is $ \displaystyle\int_0^1 \frac{\ln(1+bx)}{1+x} dx $? I call it $f(b)$ and differentiate with respect to be $b,$ a bit of partial fractions and the $x$ integral can be done. Then I ...
2
votes
1answer
258 views

What does $ \langle Y_{lm} | Y _{\lambda\mu} \rangle = \delta_{l\lambda} \delta _{m\mu} $ mean?

In Rotation Matrices for Real Spherical Harmonics. Direct Determination by Recursion, I can almost completely understand the recurrence relations described, but for one part. The $Y^l_m$ function is ...
1
vote
0answers
152 views

Proving or disproving that if $\Gamma(a)+\Gamma(b)= 121\,645\,106\,635\,852\,800$ both $a$ and $b$ are integers.

I found some formula about special function very complicated, so I am curious how you people solve this by hand. $$\Gamma(a)+\Gamma(b)= 121\,645\,106\,635\,852\,800$$ but $a$ and $b$ are very ...
2
votes
2answers
6k views

Definition of Sinc function

I just want to make clear of the definition of sinc(x). I know there is a normalized and unnormalized definition for the sinc function. If we have unnormalized sinc then we have: ...
4
votes
2answers
118 views

Limits of a function involving $\mathrm{cn}(x,k)$

Given $$f(x) = \frac{1 - \mathrm{cn}(x,k)}{{\sqrt3}(1+\mathrm{cn}(x,k)) - 1 + \mathrm{cn}(x,k)}$$ what would be $$\lim_{x\to 0} f(x)$$ and $$\lim_{x\to\infty} f(x)$$ when ...
3
votes
1answer
999 views

Differentiation of generating function of Hermite's polynomials

The generating function of Hermite's polynomials is given by $G(x,t)=e^{2xt-t^2}$ for $x, t \in \mathbf{R}$. It is known that $\displaystyle G(x,t)=\sum_{n=0}^\infty H_n(x) \frac{t^n}{n!}$ for $x, t ...
8
votes
2answers
2k views

An explanation of spherical harmonics?

Could somebody please explain spherical harmonics in a simpler manner than it is demonstrated on various websites (like the Wikipedia page which simply overflows my buffer with symbols). I've tried ...
1
vote
1answer
289 views

What is the fractional derivative of the function $\pi \cot (\pi x)$?

What is the fractional derivative of the function $\pi \cot (\pi x)$? I derived the following expression: $(\pi \cot (\pi q))^{(p)}=-\frac{\zeta'(p+1,q)+(\psi(-p)+\gamma ) \zeta (p+1,q)}{\Gamma ...
2
votes
1answer
267 views

how to evaluate $\int_0^1\sin(\frac{1}{x})dx$?

How can I evaluate the integral of $$\int_0^1\sin\left(\frac{1}{x}\right)dx.$$ Maybe it needs the cosine integral to evaluate it, but I cannot understand it very well. Thanks a lot.
4
votes
2answers
209 views

Could you give a application of a special function on number theory or analysis?

With the best effort i have ever taken, i couldn't find a application of a special function on number theory or analysis on the internet. By the way, why is the applications of special functions in ...
2
votes
2answers
716 views

Euler Gamma function $\Gamma(z)$ on $\mathbb{C}$

I'm working on an exercise about the Gamma Function from Euler. First, $\Gamma (z)= \int_0^\infty e^{-t}t^{z-1}dt$. Now, if we consider the "similar" function $\int_{\frac{1}{n}}^\infty ...
1
vote
0answers
210 views

analytic continuation of an integral involving the mittag-leffler function

i have posted this question on MO, and i didn't get an answer . we have the following integral : $$I(s)=\int_{0}^{\infty} \frac{s}{2x}\left(E_{s/2}((\pi x)^{s/2})-1\right)\omega(x)dx -\lim_{z \to 1 ...
6
votes
4answers
285 views

Gamma identity $\lim_{n\to \infty}n^{p+1}\int_{0}^{1}e^{-nx}\ln(1+x^{p})dx=\Gamma(p+1)$

I ran across what appears to be another Gamma identity. Show that $$\lim_{n\to \infty}n^{p+1}\int_{0}^{1}e^{-nx}\ln(1+x^{p}) \,\mathrm dx=\Gamma(p+1)=p!$$ I tried several different subs and ...
5
votes
2answers
1k views

Convergence of $\Gamma(p)$ for $0<p\leq 1$ and divergence for $p \leq0$.

Can someone show me a proof or any clear resource about convergence of gamma function for values of $p$ less than zero. If possible I need proofs using integration by parts. My problem evaluating ...
2
votes
1answer
170 views

Closed form of integral of $\operatorname{erfc} \log t$

Is there any closed form expression for the following integral? $$ \int\limits_t^\infty \left(1- \operatorname{erf}(\log x) \right )dx $$ or equivalently: $$ \int\limits_t^\infty ...
11
votes
2answers
1k views

Integral with spherical symmetry over cube

Is it possible to calculate the integral $$I = \int_{-1}^1 \mathrm dx \int_{-1}^1 \mathrm dy \int_{-1}^1 \mathrm dz \frac{1}{x^2 + y^2 + z^2}$$ analytically? I tried using spherical coordinates $$I ...
2
votes
1answer
344 views

How to show integral of different order Hankel transformed functions are equal?

Say I have a function $p_v(r) \in L^2(\mathbb{R})$ given by $$p_v(r) = \int_0^\infty P(k) J_v(rk)\,k\,dk$$ From mucking around in MATLAB it seems the following is true: $$\int_{r=0}^\infty ...
1
vote
2answers
252 views

Bounds on integral $x^{-a} \int_{1}^x y^{a-1} \exp(-y a) dy$

Consider the function $$ I(a,x) = x^{-a} \int_{1}^x y^{a-1} \exp(-y a) dy $$ where $x \geq 1$, and $a \geq 0$. I am not really interested in the parameter $x$, so define $$ I(a) = \sup_{x \geq 1} ...
2
votes
0answers
494 views
7
votes
1answer
1k views

Variations on the Stirling's formula for $\Gamma(z)$

I am currently reading some material that makes heavy usage of Hypergeometric functions, and there is one particular point about applying Stirling's approximation to various terms consisting of ...
6
votes
2answers
435 views

Inequality involving the regularized gamma function

Prove that $$Q(x,\ln 2) := \frac{\int_{\ln 2}^{\infty} t^{x-1} e^{-t} dt}{\int_{0}^{\infty} t^{x-1} e^{-t} dt} \geqslant 1 - 2^{-x}$$ for all $x\geqslant 1$. ($Q$ is the regularized gamma function.) ...
4
votes
2answers
150 views

'Error term' in zeta function [duplicate]

Possible Duplicate: What is the expression of $n$ that equals to $\sum_{i=1}^n \frac{1}{i^2}$? Asymptotic formulas for the n-th harmonic number are well-known: $$ \sum_{k=1}^n\frac1n=\log ...
11
votes
1answer
460 views

Integral of digamma function

I was attempting to evaluate a series $$\sum_{n=1}^\infty \frac{1}{n} \ln\left(1+\frac{1}{n}\right)$$ Since $$\frac{1}{n}\ln\left(1+\frac{1}{n}\right)=\int_0^1 \frac{1}{n(n+t)}dt,$$ I rewrote it as ...
1
vote
0answers
259 views

How is the Riemann-Siegel formula applied?

What is the application of the Riemann-Siegel formula: $$ \zeta(s) = \sum_{n=1}^N\frac{1}{n^s} + \gamma(1-s)\sum_{n=1}^M\frac{1}{n^{1-s}} + R(s) , $$ where $ \displaystyle\gamma(s) = ...
1
vote
1answer
175 views

How to evaluate $\sum J_0(\alpha n) z^{-n}$ in closed form?

I need to evaluate $\sum_{n = -\infty}^{\infty} J_0(\alpha n) z^{-n}$ in closed form, where $z$ is complex variable and $J_0()$ is the zeroth order Bessel function of the first kind. How do I evaluate ...
12
votes
2answers
621 views

Integrating $\frac{x^k }{1+\cosh(x)}$

In the course of solving a certain problem, I've had to evaluate integrals of the form: $$\int_0^\infty \frac{x^k}{1+\cosh(x)} \mathrm{d}x $$ for several values of k. I've noticed that that, for k a ...
1
vote
1answer
84 views

Show the convergence of a series

Given the series: $$S(\lambda,\Phi)=\sum_{n=0}^{\infty}{J_n}(\lambda)e^{in\Phi}$$ where the $J_n(\lambda)$ is the Bessel function of order $n$ I have some difficulty to give a proof of its ...
1
vote
1answer
72 views

Nonlinear equation containing parametric integral

I have the following equation: $$I(k,x)=\int_0^xJ_k(\tau^k)d\tau=\alpha$$ where $\alpha$ is a given constant $\alpha\in \mathbb{R}$ and $k$ integer with $k\gt 0$. $J_k(x)$ is the Bessel function of ...
13
votes
1answer
3k views

Quotient of gamma functions?

I'm sorry if this is a simple question, but this page on Wolfram Research states that it follows from Stirling's formula that: $$ \frac{\Gamma(x+\beta)}{\Gamma(x)} \approx x^\beta $$ for large $x$, ...
0
votes
2answers
854 views

Looking for function of bell-like curve that peaks quickly.

I'm writing a little Sage/Python script that would graph the cumulative effects of taking a particular medication at different time intervals / doses. Right now, I'm using the following equation: ...
4
votes
1answer
473 views

An addition property of Weierstrass $\wp$

I want to show $$ \left( \begin{array}{ccccc} &1 &\wp(v) &\wp'(v) \\ &1 &\wp(w) &\wp'(w) \\ &1 &\wp(v+w) &-\wp'(v+w) \end{array} \right)=0 $$ ...
5
votes
1answer
161 views

how to evaluate a definite integral (looks almost like nonintegral moments of a Gaussian)

I'd like to show the following equality (at least Mathematica claims it is an equality): \begin{multline*} \int_0^\infty x^p \exp(-(ax - b)^2)\, dx = \frac{1}{2} e^{-b^2} a^{-p-1} \left(\Gamma ...
3
votes
1answer
432 views

Concave functions on discrete domain

We are given a positive, non-decreasing function $f$ defined on natural numbers with $f(0) = 0$. $f$ has a submodularity-like property: $f(x+y) \leq f(x) + f(y) $ for all natural numbers $x$ and ...
2
votes
1answer
197 views

integration question about dilogarithm

I want to show that $$\operatorname{Li}_2(z)=z\int_{0}^{\infty}\frac{x}{e^x-z}dx$$ It is the integral of the Bose–Einstein distribution in dilogarithm case. Thank you!
3
votes
1answer
235 views

Dilogarithm asymptotics for an exponential parameter.

So this question is about this dilogarithm function. Assume the argument $z$ is real then I want to show the formula $$\operatorname{Li}_2(e^{-z})=\frac{\pi^2}{6} + z\log z -z+O(z^2) $$ as $z$ ...
2
votes
1answer
260 views

Question on Inverse Pochhammer Symbol

I struggled quite a while without success, so I highly appreciate if anybody can help me, proving that: $$ \frac{1}{\left(1+1/n \right)^{(M)}}=\sum_{k=0}^M \frac{(-1)^k}{k!(M-k)!(nk+1)}, $$ where ...
8
votes
1answer
354 views

Does the series of squares of Legendre polynomials converge?

I am a physicist working on an electrostatic problem and this series popped up: $\sum^{\infty}_{l=0} (P_l(x))^2$ where $P_l$ is the $l$-th Legendre polynomial. Computing this numerically I think the ...