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

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9
votes
0answers
358 views

New generalization of Riemann Zeta?

I am interested in the following generalization of the Riemann Zeta function: $$ \zeta_M(s,c) = \sum_{n=1}^\infty \left(\frac{n^2}{c^2} + \frac{c^2}{n^2}\right)^{-s} $$ This is most closely related ...
8
votes
1answer
353 views

Prove $f(x)=\int\frac{e^x}{x}\mathrm dx$ is not an elementary function

How do I prove that the exponential integral $$f(x)=\int \frac{e^x}{x}\mathrm dx$$ is not an elementary function? Also, what are the general methods and tricks to prove that an integral or solution ...
12
votes
3answers
340 views

How this integral $ \int_0^z\frac{1-e^x}{x} dx$ is connected to the Gamma function and Euler constant?

This is my first question in this forum; I hope it is an appropriate question. The Wolframalpha website tells me that $$ \int\nolimits_0^z\frac{1-e^x}{x} dx = \log (-z)+\Gamma(0, -z)+\gamma\quad ...
6
votes
2answers
491 views

Beta Function — finding a lower bound based on parameters

I would like to show that $$ 1-\frac{1}{c}Beta\left(c+1,\frac{1}{c}\right) \geq \frac{1}{c+1}.$$ for all $c \geq 2$. I have plotted it out for $c$ up through 200, and it seems to hold. Does anyone ...
1
vote
1answer
87 views

What are these theta functions appearing in Sloane's database

Looking at Sloane's database, I found a neat formula for the lambda-invariant. Let $q:\tau \mapsto \exp(\pi i \tau)$ on the complex upper-half plane. Then $$\lambda(q) = 16q\;\prod_{k>0} \frac{(1 ...
8
votes
1answer
1k views

On the growth of the Jacobi theta function

So, I ran into this exercise from Stein & Shakarchi. CA, Chapter 5: Show that if $\tau$ is fixed with positive imaginary part, then the Jacobi theta function $$\theta(z | r) = ...
1
vote
0answers
51 views

Further simplifiable?

I have an equation $(xy')'+kxy=xf(x)$ where $k$ is not an eigenvalue; $y(x)$ and $f(x)$ are subjected to boundary conditions (i) bounded as $x\to 0$; (ii) $y(1)=0=f(1)$ I want to get the ...
3
votes
1answer
91 views

Integrating a differential equation?

How does $(xJ_0'(x))'+xJ_0(x)=0\implies\int\nolimits_0^1 x J_0(ax)J_0(bx) dx={bJ_0(a)J_0'(b)-aJ_0(b)J_0'(a)\over{a^2-b^2}}?$ Thanks. Perhaps int by parts? But how do I get the RHS form?
1
vote
0answers
310 views

Nested Integral of exponential function with trigonometric identities

Is there any possibility to simplify the following integral or any function that is equivalent to the following integral? $$ ...
6
votes
1answer
883 views

Inverse of the polylogarithm

The polylogarithm can be defined using the power series $$ \operatorname{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s}. $$ Contiguous polylogs have the ladder operators $$ \operatorname{Li}_{s+1}(z) ...
13
votes
4answers
559 views

Intriguing polynomials coming from a combinatorial physics problem

For real $0<q<1$, integer $n >0 $ and integer $k\ge 0$, define $$[k, n]_q \equiv -\sum_{m=1}^{n} q^{m(k+1)} (q^{-n}; q)_m = -\sum_{m=1}^{n} q^{m(k+1)} \prod_{l=0}^{m-1} (1-q^{l-n})$$ ...
1
vote
0answers
88 views

Computing the PDF of a product of the sum of 2 Nakagmi-m R.V.s with a Normal R.V

I really have two questions: One is about computing a PDF and the second is about how to sum a series involving $K_v(x)$ that the PDF in question seems to contain. I have come across the following ...
2
votes
1answer
149 views

Solving for $y$ in $y = x \ln(y)$

I want to solve $y = x \ln(y)$ for $y$ in terms of $x$. Wolfram Alpha kindly produces this plot with the solution, $y = -x W(-\frac{1}{x})$, where $W$ is the Lambert function. However, that only ...
8
votes
2answers
598 views

Is there a combinatorial way to see the link between the beta and gamma functions?

The Wikipedia page on the beta function gives a simple formula for it in terms of the gamma function. Using that and the fact that $\Gamma(n+1)=n!$, I can prove the following formula: $$ ...
7
votes
1answer
829 views

the limit of the ratio of two $\Gamma(x)$ functions

I am interested in the quantity $$ a_{n} = \sqrt{n/2} \frac{\Gamma((n-1)/2)}{\Gamma(n/2)}$$ (this is the geometric bias of the non-central t-distribution with $n$ d.f.) After some plotting, my hunch ...
4
votes
2answers
544 views

Digamma function integral

Does anyone how to get a finite value to this integral ? $ \int\nolimits_{0}^{\infty} dx \frac{ \Psi (1/4+ix/2) +\Psi (1/4-ix/2)}{x^{2}+1/4} $ i have tried residue theorem but i got nonsenses :( can ...
15
votes
2answers
1k views

Logarithmic derivative of Riemann Zeta function

Given the logarithmic derivative of the zeta function $\dfrac{\zeta^\prime (s)}{\zeta(s)}$ how does it behave near $s=1$? I mean if for $s=1$ the Laurent series for the logarithmic derivative becomes ...
4
votes
0answers
218 views

please help with the a gamma function since i don't even have the idea?

How to prove: $$\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} \frac{\Gamma(\alpha_1+x)}{\beta_1^{\alpha_1+x}}\, \frac{\Gamma(\alpha_2-x)}{\beta_2^{\alpha_2-x}}\, ...
3
votes
1answer
302 views

Conceptually, what is the difference between the Beta function and the Beta distribution?

I have read the Wikipedia pages on the Beta function and the Beta distribution, but I'm still not sure I have a good intuition for what's going on. I'm am hoping someone will be kind enough to ...
5
votes
1answer
147 views

orthonormal polynomials

Here is the question: Suppose $P_0, P_1, P_2, \dots$ are polynomials orthonormal with respect to the inner product $$(f,g)=\int_a^b f(x)g(x)W(x)dx,$$ where $W(x) > 0$ is a weight function and ...
8
votes
2answers
166 views

Is there a neat way to show $\int_{-1}^1 \frac{ U_n(z) U_n(z)}{\sqrt{1-z^2}} \mathrm{d} z = \pi (n+1)$

In answering a question on math.SE, I attempted to find integral of Fejér kernel by using $$ K_n(t) = \frac{1}{n} U_{n-1}^2\left( \cos \frac{t}{2} \right) $$ where $U_n(z)$ stands for the ...
3
votes
1answer
191 views

Solving for a variable under $\Gamma(z)$

My friend came to see me regarding some calculation in probability where he would like to know if it is possible to solve for a variable analytically under the gamma function. By this I mean say we ...
3
votes
1answer
1k 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 ...
2
votes
0answers
107 views

On a certain lemma concerning Incomplete Hypergeometric Functions of Fox type

I am reposting this from MO, since, I guess, my question might have been too elementary for MO and did not receive any reaction at all for the past 30 hours. Here is the link. Here is my post: I am ...
13
votes
5answers
1k views

Proof that $x \Phi(x) + \Phi'(x) \geq 0$ $\forall x$, where $\Phi$ is the normal CDF

As title. Can anyone supply a simple proof that $$x \Phi(x) + \Phi'(x) \geq 0 \quad \forall x\in\mathbb{R}$$ where $\Phi$ is the standard normal CDF, i.e. $$\Phi(x) = \int_{-\infty}^x ...
7
votes
2answers
468 views

An integral about Bessel function

Is there somebody who knows the solution for the integral $$\int_0^\infty\frac{J^3_1(ax)J_0(bx)}{x^2} dx$$ where $a>0,b>0$ and $J(\cdot)$ the bessel function of the first kind with integer ...
3
votes
1answer
863 views

What is a cardinal basis spline?

Wikipedia says: the normalized cardinal B-splines tend to the Gaussian function and writes them as "Bk". Meanwhile, cnx.org Signal Reconstruction says: The basis splines Bn are shown ... ...
3
votes
2answers
899 views

Hankel function in terms of planewaves

It is well know that planewaves are a complete basis for solutions to the wave equation. Let us assume a 2D space, and at fixed temporal frequency, the equation reduces to the Helmholtz equation. In ...
10
votes
2answers
604 views

Why is this sum equal to the Logarithmic Integral?

I am using this sum: $$\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}\left((-1)^{k-1} (n-1) + \sum_{j=1}^{k-1}\frac{(-1)^{j+k-1}n (\log n)^j}{j!}\right)$$ Empirically, this is precisely equal to ...
1
vote
1answer
157 views

What are the $\ker$, $\mathrm{kei}$ functions?

In a book titled 'Ordinary Differential Equations and Useful Polynomials', under the chapter 'Bessel's function', the author has introduced four new functions $\mathrm{ber}$, $\mathrm{bei}$, $\ker$, ...
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 ...
10
votes
2answers
2k views

Integral of product of two error functions (erf)

In the course of my research I came across the following integral: $$\int\nolimits_{-\infty}^{\infty}\operatorname{erf}(a+x)\operatorname{erf}(a-x)dx$$ where ...
6
votes
1answer
720 views

series including infinite sum

I am looking for the approximation of the following function: $$\rho(a,b)=1-e^{-(a+b)}\sum_{m=1}^{\infty}\left(\sqrt{\frac{b}{a}}\right)^m I_m(2\sqrt{ab})$$ where $I_m(x)$ is the modified Bessel ...
2
votes
1answer
428 views

Identity for complete elliptic integral of the second kind

Why does $|x-1| \; E\left(-\dfrac{4x}{(x-1)^2}\right) = |x+1| \; E\left(\dfrac{4x}{(x+1)^2}\right)$, where $E(m)$ is the complete elliptic integral of the second kind, with parameter $m$?
9
votes
3answers
489 views

infinite series of the form $\sum\limits_{k=1}^{\infty}\frac{1}{a^{k}+1}$

Is there a method for evaluating infinite series of the form: $\displaystyle\sum_{k=1}^{\infty}\frac{1}{a^{k}+1}, \;\ a\in \mathbb{N}$?. For instance, say $a=2$ and we have ...
5
votes
0answers
193 views

Beta integral and Chu-Vandermonde identity

Chu vandermonde identity states that ${s+t \choose n}=\sum_{k=0}^n {s \choose k}{t \choose n-k} $ Now how to prove that this identity is a discrete form of beta integral? i see as a starting ...
1
vote
0answers
221 views

Improper integrals over the square and modulus square of an associated Legendre function

I am trying to evaluate integrals of the type $$\int dz\, P^{\,\mu}_\nu(z)^2 \qquad \mathrm{and} \qquad \int dz\, \left|P^{\,\mu}_\nu(z)\right|^2$$ where $P^\mu_\nu$ are associated Legendre functions. ...
3
votes
1answer
119 views

Simplifying an Integral

I'm looking to simplify the integral $$\int\nolimits_0^{\infty}\dfrac{(t^b+1)^n}{(1+t)^{nb+2}} dt$$. (This arises out of the sum of a bunch of Beta functions, ie $\displaystyle\sum_{i=0}^{n} ...
2
votes
1answer
252 views

Asymptotic order of $\frac{\mathrm{erfi}(\sqrt{x})}{\exp(x)\sqrt{x}}$

I need to approximate this expression in order to sum it. Asymptotically I obtain $\frac1{\sqrt{\pi}x}+\frac1{2\sqrt{\pi} x^2} + O\left(\frac1{x^3}\right)$. Although this looks fine there is the ...
6
votes
2answers
292 views

Integral form of $\Gamma (x)$

While trying to represent the poles and analytic continuation of $\Gamma (x)$ , the author uses the following equality: ...
7
votes
3answers
226 views

Understanding a specific meromorphic function that comes from a statistical physics research problem

Dear Math Stackexchange, I'm a physics researcher working on a problem in quantum statistical physics. I've encountered the following function which I do not recognize (neither does Mathematica): ...
1
vote
1answer
854 views

Properties of Inverse Lambert W function

Starting with the two-branched Lambert W function (from Wikipedia): Suppose we just flip it like this: Is there a single power series for this $y=W^{-1}(x)$?
5
votes
2answers
605 views

On the absolute integrability of Bessel functions

Reading "How do you integrate a Bessel function", it didn't seem like it was an easy task. Thinking more about Bessel functions, speficially $J_0(x)$, it occurred that it looked a lot like the sinc ...
2
votes
1answer
539 views

partial sum of Basel problem related to series involving Beta function

I ran across a series and got to wondering how this is so. We are all familiar with the famous $\displaystyle\sum_{k=1}^{\infty}\frac{1}{k^{2}}=\frac{{\pi}^{2}}{6}$ But, how can we show: ...
3
votes
0answers
269 views

2 dimensional Fourier transform integral

I'm trying to calculate the two dimensional Fourier integral $$\iint \mathrm d\vec{R} \; (x^2 + y^2) \; e^{-2 \sqrt{ x^2 + y^2 + z^2}} \; e^{i\vec{K}\vec{R}} \;,$$ with $\vec{R}=(x,y)$. Switching to ...
1
vote
2answers
2k views

Easy approximation of the incomplete beta function $\text{B}_x(a,b)$

I need to calculate $\text{B}_x(a,b)$ on the cheap, without too many coefficients and loops. For the complete $\text{B}(a,b)$, I can use $\Gamma(a)\Gamma(b)/\Gamma(a+b)$, and Stirling's approximation ...
5
votes
2answers
196 views

Lerch-$\small \zeta(\varphi,0,-n)$ of integer *n* purely real and imaginary ($\small \zeta_\varphi (-n)^2 $ is real) for $\small n \ge 2$?

Are the Lerch-$\zeta(\varphi,0,-n) $ of integer n (for shortness I use the notation of my earlier question $\small \zeta_\varphi(-n)$) periodically purely real and imaginary: $\zeta_\varphi (-n)^2 $ ...
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 ...
4
votes
1answer
508 views

Approximating Lambert W for input below 0

As a small part of a much bigger project, I need to be able to approximate the numerical output of the Lambert W function. I have found decent approximations (good up to at least 4 decimal places), ...
14
votes
0answers
564 views

Integral involving Complete Elliptic Integral of the First Kind K(k)

I have run into an integral involving the complete elliptic integral, which can be put into the following form after changing integration variables to the modulus: ...