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

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5
votes
3answers
3k views

Inverse of $y=xe^x$

I feel like finding the inverse of $y=xe^x$ should have an easy answer but can't find it.
9
votes
2answers
310 views

$\frac{\mathrm d^2 \log(\Gamma (z))}{\mathrm dz^2} = \sum\limits_{n = 0}^{\infty} \frac{1}{(z+n)^2}$

How do I show $$\frac{\mathrm d^2 \log(\Gamma(z))}{\mathrm dz^2} = \sum_{n = 0}^{\infty} \frac{1}{(z+n)^2}$$? $\Gamma(z)$ is the gamma function.
2
votes
3answers
315 views

Proving $\lim_{n \to \infty} 2^{2n-1} \sqrt{n} \frac{ \Gamma(n)^{2}}{\Gamma(2n)} = \sqrt{\pi}$

How does one prove this identity: $$\lim_{n \to \infty} 2^{2n-1} \sqrt{n} \frac{ \Gamma(n)^{2}}{\Gamma(2n)} = \sqrt{\pi}$$ Taken from Gamelin : Complex Analysis
5
votes
1answer
183 views

Is this a known special function?

Is this a known special function: $$\int_0^1 a^p(1-a)^{1-p}\,b^{1-p}\,(1-b)^p dp\qquad ?$$ I am really only interested in maximizing this over $(a,b)$ in $[0,1] \times [0,1]$, so a pointer to a nice ...
3
votes
1answer
278 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 $E(\pi/2,m)=E(m)$? ...
4
votes
1answer
594 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 ...
1
vote
1answer
469 views

Separate elliptic integrals into real and imaginary parts

This is somewhat of a follow-up question to ‘Elliptic integrals with parameter outside 0<m<1’. I have an equation that I'm attempting to simplify that has terms that look something like this: ...
0
votes
2answers
594 views

What is the Series Expansion of the following function?

I was wondering what the expansion series of the function $$ f(x) = -\frac{1}{x^3} \cdot \frac{1}{\Gamma(x) \cdot \Gamma(-(\exp(\frac{2}{3}\pi\cdot i))x) \cdot \Gamma(-(\exp(\frac{4}{3}\pi \cdot ...
2
votes
1answer
671 views

Elliptic integrals with parameter outside $0<m<1$

I'm attempting to implement an equation (for calculating magnetic forces between coils, eqs (22–24) in the linked paper) that requires the use of elliptic integrals. Unfortunately these equations ...
5
votes
1answer
214 views

Generalization of cos: is this function known?

Consider a function $f_1$ defined by $f_1(x)=1-x+o(x)$ and $f_1(2x)=f_1(x)^2 + 0$. It's simple to find that $f_1(x)=e^{-x}$ (for example by writing series near $x=0$). Consider a function $f_2$ ...
1
vote
1answer
168 views

Inequality concerning the Gamma function

For each $n>0$, how do we prove that $$\Gamma'(n+1)> \log{n} \cdot \Gamma(n+1)$$ I had spent about half an hour on this question, but just could find any way of proceeding for the solution. ...
24
votes
4answers
2k views

Proving a known zero of the Riemann Zeta has real part exactly 1/2

Much effort has been expended on a famous unsolved problem about the Riemann Zeta function $\zeta(s)$. Not surprisingly, it's called the Riemann hypothesis, which asserts: $$ \zeta(s) = 0 ...
1
vote
2answers
144 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
2answers
1k views

On deriving the arclength of a hyperbola

In my attempts to derive the closed form for the arclength of the hyperbola, I wound up with the following integral: $$\int\frac{\sqrt{1-m\;\sin^2 u}}{\sin^2 u}\mathrm{d}u$$ I am aware that such ...
12
votes
1answer
506 views

Proving $\sqrt{1-x^2}\ge \operatorname{erf}(\sqrt{-\log x})$

Can anyone see a nice way to prove the following for $0\le x \le 1$? $$\sqrt{1-x^2}\ge \operatorname{erf}(\sqrt{-\log x})$$ $\operatorname{erf}$ is defined as $$\operatorname{erf}(z) = ...
8
votes
3answers
578 views

Are there addition formulas for the Riemann Zeta function?

In particular for two real numbers $a$ and $b$, I'd like to know if there are formulas for $\zeta (a+b)$ and $\zeta (a-b)$ as a function of $\zeta (a)$ and $\zeta (b)$. The closest I could find ...
3
votes
2answers
319 views

Implicit function $y = e^{(y-1)/x}$

I'd like to know if the function $ y = f(x) : [0,1] \rightarrow [0,1]$ defined implicitly by the transcendental equation $$\displaystyle y = e^{(y-1)/x}$$ is "well known" (name, properties) or is ...
11
votes
1answer
1k views

Iterative refinement algorithm for computing exp(x) with arbitrary precision

I'm working on a multiple-precision library. I'd like to make it possible for users to ask for higher precision answers for results already computed at a fixed precision. My $\mathrm{sqrt}(x)$ can ...
3
votes
1answer
275 views

A singularity of hypergeometric functions

Do generalized hypergeometric functions $${}_p F_q(a_1,\ldots,a_p; b_1, \ldots,b_q; z) $$ with $p = q+1$ always possess a singularity at $z=1$, independent of the their parameters $a_1,\ldots,a_p$ ...
1
vote
2answers
392 views

Legendre functions in number theory

I have heard that Legendre functions are important in number theory. Can any one tell me how? The Legendre function of the first kind $P_s$ is defined by \begin{eqnarray*}P_s(x) =& ...
3
votes
1answer
538 views

The partial fraction expansion of $\frac{1}{x^n - 1}$

If $n$ is an integer, is there a nice way to write the partial fraction expansion of $\frac{1}{x^n - 1}$? I figure that if $\zeta$ is the $n$-th root of unity, then for some coefficients $a_0, a_1, ...
2
votes
2answers
190 views

Proving identity $\displaystyle\sum_{j\geq 1}[(j+t)^{-1}-j^{-1}]=\displaystyle\sum_{k\geq 1}\zeta (k+1)(-t)^{k}$

Motivation: In S.J. Patterson's An introduction to the theory of the Riemann Zeta-Function it is proved (p.132) that $\displaystyle -\Gamma ^{\prime }(t)/\Gamma (t)=\gamma +t^{-1}+\underset{j\geq ...
2
votes
1answer
276 views

Deriving Eulers Addition Theorem for Elliptic Integrals

In the book Elliptic Curves - McKean & Moll we are given the outline for a proof of Eulers addition theorem: The (projective) quartic $\mathbf y^2 = (1-\mathbf x^2)(1-k^2 \mathbf x^2)$ has ...
11
votes
1answer
1k views

Integral Representation of Infinite series

Let's take a look at the following integrals : 1) $\displaystyle \int\limits_{0}^{1} \frac{\log{x}}{1+x} \ dx = -\frac{\pi^{2}}{12} = -\frac 1 2 \sum\limits_{n=1}^{\infty} \frac{1}{n^2}= -\frac 1 2 ...
1
vote
1answer
345 views

Gamma integrals

Is anything known about these integrals? Textbook suggestions are welcome $f(n,p)=\int_{x=-0.5}^p \frac{n!}{x!(n-x)!} dx$ $n>0, p\le n+0.5$ For instance, as n grows $2^n-f(n,n+1/2)$ seems to be ...
16
votes
2answers
3k views

Beta function derivation

How do I derive the Beta function using the definition of the beta function as the normalizing constant of the Beta distribution and only common sense random experiments? I'm pretty sure this is ...
3
votes
3answers
795 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 ...
5
votes
2answers
874 views

Two ways of defining the gamma function $\Gamma (x)$. How to show they are equivalent?

In the Portuguese book Análise Matemática (Mathematical Analysis) by C. Sarrico, it is proved there exist $$\displaystyle\lim_{n\rightarrow +\infty}f_{n}(x)=\lim_{n\rightarrow +\infty }e^{\log ...
7
votes
1answer
240 views

Is the Gamma function superadditive?

A function $f$ is superadditive if $f(x) + f(y) \le f(x+y)$. The question is: Does a real number $a$ exists such that for all real numbers with $x, y\ \ge \ a $ $$ \Gamma(x) + \Gamma(y) \le ...
8
votes
2answers
2k views

How to evaluate Riemann Zeta function

How do I evaluate this function for given $s$? $$\zeta(s) = \sum_{n=1}^\infty \frac1{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$$
9
votes
2answers
869 views

Minimum of the Gamma Function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$?

The $\Gamma (x)$ function has just one minimum for $x>0$ . This result uses some properties of the gamma function: $\Gamma ^{\prime \prime }(x)>0$ and $\Gamma (x)>0$ for all $x>0$ $\Gamma (1)=\Gamma ...
4
votes
2answers
599 views

Question Relating Gamma Function to Riemann Zeta function evaluated at integers

I was just reading a paper of Ramanujan entitled " On question 330 of Professor Sanjana" when i got stuck up with a Proposition which i am unable to answer. The proposition is if $ \displaystyle ...
88
votes
9answers
5k views

Why is Euler's Gamma function the “best” extension of the factorial function to the reals?

There are lots (an infinitude) of smooth functions that coincide with f(n)=n! on the integers. Is there a simple reason why Euler's Gamma function $\Gamma (z) = \int_0^\infty t^{z-1} e^t dt$ is ...
8
votes
4answers
3k views

How to accurately calculate the error function erf(x) with a computer?

I am looking for an accurate algorithm to calculate the error function I have tried using [this formula] (http://stackoverflow.com/a/457805) (Handbook of Mathematical Functions, formula ...