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

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2
votes
1answer
250 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 ...
1
vote
0answers
297 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? $$ ...
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 ...
8
votes
2answers
557 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: $$ ...
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 ...
7
votes
1answer
776 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 ...
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
217 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
293 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 ...
7
votes
1answer
322 views

Is there a gamma-like function for the q-factorial?

I'm looking at quantum calculus and just trying to understand what is going with this subject. Looking at the q-factorial made me wonder if this function could take all real or even complex numbers in ...
5
votes
1answer
144 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
578 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 ...
2
votes
0answers
106 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 ...
3
votes
2answers
869 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 ...
5
votes
1answer
721 views

Efficiently calculating the logarithmic integral with complex argument

My number theory library of choice doesn't implement the logarithmic integral for complex values. I thought that I might take a crack at coding it, but I thought I'd ask here first for algorithmic ...
6
votes
1answer
697 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 ...
1
vote
1answer
154 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 $\mathrm{ber}$, $\mathrm{bei}$, $\ker$, ...
9
votes
3answers
476 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
191 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
215 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. ...
7
votes
3answers
224 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
799 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 $y=W^{-1}(x)$?
3
votes
1answer
284 views

Inconsistent naming of elliptic integrals

This may be a question whose answer is lost in the mists of time, but why is the elliptical integral of the first kind denoted as $F(\pi/2,m)=K(m)$ when that of the second kind has $E(\pi/2,m)=E(m)$? ...
1
vote
1answer
80 views

Nonhomogeneous equation involving logarithm

This is probably a lame question, but what is the general approach to solving $\log z +z \sigma +1=0$ for $z$? Wolfram Alpha obtains a Lambert W-function, but I don't quite see how.
5
votes
1answer
364 views

Where are this kind of series used, $\vartheta_{4}(0,e^{\alpha \cdot z})$?

In my recent explorations I stumbled upon the following series $$ \vartheta_{4}(0,e^{\alpha \cdot z})=1+2\sum_{k=1}^{\infty} (-1)^{k}\cdot e^{\alpha \cdot z\cdot k^{2}} ; \alpha \in \mathbb{R}, z ...
2
votes
1answer
508 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: ...
12
votes
1answer
458 views

elliptic generalizations of Euler's trick

So Euler employed the following identity $$\sin(z) = z \prod_{n=1}^{\infty} \left[1-\left(\frac{z}{n\pi}\right)^{2}\right]$$ to evaluate $\zeta(2n)$, for $n\in\mathbb{N}$ I'm curious if there's been ...
3
votes
0answers
267 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 ...
4
votes
2answers
201 views

How to verify integral with hypergeometric function

Trying to evaluate the following integral, Mathematica returns this result: $$ \int \frac{e^{-\tau \omega}}{1+e^{-\beta \omega}} d \omega = \frac{e^{(\beta - \tau) \omega} \cdot {}_2F_1(1, ...
4
votes
1answer
485 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), ...
5
votes
2answers
194 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 $ ...
9
votes
2answers
2k views

Addition theorems for elliptic functions: is there a painless way?

The Weierstrass $\wp$ function satisfies the addition formula $$\wp(z+Z)+\wp(z)+\wp(Z) = \left(\frac{\wp'(z)-\wp'(Z)}{\wp(z)-\wp(Z)}\right)^2.$$ Of course, this is just the $x$-coordinate of ...
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 ...
3
votes
1answer
323 views

Periodic Zeta Function Functional Equation

Recall that the periodic zeta function has the Dirichlet series $$F(\lambda,s)= \sum_{n=1}^\infty \frac{e^{2\pi i n\lambda}}{n^s}.$$ This defines an analytic function for $\Re s>0$ and has a ...
1
vote
2answers
153 views

On functions similar to Hurwitz zeta function

Denoted as $\zeta(s,a)$ for a > 0 Where do I find topics on the Hurwitz zeta function for a < 0? Any links or resources would be appreciated. (Please dont mention wiki or mathworld) Thanks
5
votes
0answers
528 views

Tricky integral $\int_{a}^{b}\frac{\gamma d \gamma}{\gamma + \phi_{1}(\mu)-e^{-\frac{\phi_{2}(\mu)}{\gamma}}}$

in this integral $a=\psi_{1}(\mu), \ b=\psi_{2} (\mu)$. I expanded the function in Taylor series (3 terms) around ($\gamma= \frac{b}{2}$), numerically (for varioud values of $\mu$, and other constants ...
4
votes
1answer
302 views

Asymptotic approximation for confluent hypergeometric function

I have the following nasty expression that I would like to expand in powers of $\frac{1}{N}$: \begin{align} \frac{2^{\frac{3}{2}} 3^{\frac{1}{2}} \Biggl[ \sqrt{u} \cdot ...
2
votes
1answer
517 views

Is Riemann Zeta Function symmetrical about the real axis?

From wikipedia, http://en.wikipedia.org/wiki/Riemann_zeta_function "Furthermore, the fact that $\zeta(s) = \zeta(s^*)^*$ for all complex s ≠ 1 ($s^*$ indicating complex conjugation) implies that the ...
4
votes
3answers
913 views

Hint on how to prove $\zeta ( 2) =\pi ^{2}/6$ using the complex Fourier series of $f(x)=x$

I know how to prove $\zeta (2)=\pi ^{2}/6$ by using the trigonometric Fourier series expansion of $x^{2}/4$. How can one prove the same result using the complex Fourier series of $f(x)=x$ for $0\leq ...
4
votes
2answers
1k views

Tables of Hypergeometric Functions

I'm looking for a book, set of tables, or other reference which contains a comprehensive list of hypergeometric identities; that is, something which allows a hypergeometric fucntion to be expressed in ...
15
votes
3answers
6k views

Analytic continuation- Easy explanation?

Today, as I was flipping through my copy of Higher Algebra by Barnard and Child, I came across a theorem which said, The series $$ 1+\frac{1}{2^p} +\frac{1}{3^p}+...$$ diverges for $p\leq 1$ and ...
3
votes
1answer
106 views

How to show $ \frac{Q_{k}(1-z^2)-zP_{k}(1-z^2)}{Q_{k}(1-z^2)+zP_{k}(1-z^2)}=\left(\frac{1-z}{1+z}\right)^{2k+1} $ analytically?

Let \begin{eqnarray*} P_{k}(z)={_2F_1}(-k,\frac{1}{2}-k; -2k; z), \ \label{e0} Q_{k}(z)={_2F_1}(-k,-\frac{1}{2}-k; -2k; z) , \end{eqnarray*} where $k\ge 1$ is an integer. How to show ...
6
votes
1answer
1k views

How to decompose displaced Hermite-Gauss function into higher order HGs?

The Hermite-Gauss functions appear commonly in physics. These functions are formed from the product of a Hermite polynomial and a Gaussian: $$ u_n(x) = \left(\frac{2}{\pi w_0^2}\right)^{1/4} ...
15
votes
3answers
502 views

The graph of $x^{n}+y^{n}=r^{n}$ for sufficiently large $n$

The graph of the function $x^{n}+y^{n}=r^{n}$ for certain large values of $n$ looks suspiciously like a square. See this page from wolframalpha. Have any results been proven regarding this ...
4
votes
1answer
206 views

Closed form for some integrals related to the complementary error function

While studying the use of the trapezoidal rule for numerically evaluating the complementary error function $\mathrm{erfc}(z)$, the following integrals showed up when I was trying to derive expressions ...
1
vote
2answers
2k views

How can I solve this equation (contains error function)?

Edited out incorrect formula Can someone please solve this equation for x? I have no idea what to do with the $\mathrm{erf}$ (error function). Edit: Hm, it did not work correctly... here is the ...
10
votes
3answers
461 views

Erroneous numerical approximations of $\zeta\left(\frac{1}{2}\right)$?

By definition of the Riemann Zeta Function, $$\zeta\left(\frac{1}{2}\right) = \sum_{n=1}^\infty \frac{1}{\sqrt{n}}.$$ Since $\forall n \geq 1 : \frac{1}{\sqrt{n}} \geq \frac{1}{n}$, we have that for ...
34
votes
2answers
7k views

Why is the error function defined as it is?

$\newcommand{\erf}{\operatorname{erf}}$ This may be a very naïve question, but here goes. The error function $\erf$ is defined by $$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt.$$ Of ...