Questions tagged [special-functions]

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

11
votes
4answers
564 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 ...
3
votes
0answers
229 views

Solid angle spanned by disc/rewriting expression with elliptic integrals

The solid angle spanned by a disc of unit radius, as observed from a point $(r,z)$ at a distance $z>0$ above a point in the disc plane with at distance $r>0$ to the center, can be expressed as $...
5
votes
1answer
384 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
0answers
63 views

Deriving functions for empiracal distributions -very applied mathatics

First I am a new user of this site. Second my math background is very limited, although I do have a lot of experience in applied statistics. Component or piece part failures on high value parts($1000 ...
5
votes
1answer
2k views

what does the “L” in “L-function” stand for?

I haven't been able to find a reference that tells what word (if a word) the L is short for.
7
votes
2answers
393 views

What is Eulerian?

I encountered an interesting function which is called "Eulerian" by the Wolfram's MathWorld: $$\phi(q)=\prod_{k=1}^{\infty} (1-q^{k})$$ It is interesting because it seems that roots of any ...
23
votes
6answers
17k views

Algorithm to compute Gamma function

The question is simple. I would like to implement the Gamma function in my calculator written in C; however, I have not been able to find an easy way to programmatically compute an approximation to ...
16
votes
1answer
5k views

Is there an Inverse Gamma $\Gamma^{-1} (z) $ function?

Since $\Gamma$ is not one to one over the complex domain, Is it possible to define some principal values ( analogues to Principal Roots for the Root function ) so we can have a $\Gamma^{-1} (z)$ (...
27
votes
3answers
2k views

Why isn't the gamma function defined so that $\Gamma(n) = n! $?

As a physics student, I have occasionally run across the gamma function $$\Gamma(n) \equiv \int_0^{\infty}t^{n-1}e^{-t} \textrm{d}t = (n-1)!$$ when we want to generalize the concept of a factorial. ...
7
votes
3answers
3k views

Integral Representations of Hermite Polynomial?

One of my former students asked me how to go from one presentation of the Hermite Polynomial to another. And I'm embarassed to say, I've been trying and failing miserably. (I'm guessing this is a ...
3
votes
0answers
477 views

How can I integrate a Bessel function divided by a “shifted” value?

Sorry to ask yet another "how do I do this integral" question! But I've really been having a hard time with this one. $$\int_0^\infty \frac{J_0(x)}{|C^2 - x^2|}\mathrm{d}x$$ I've been through a lot ...
11
votes
3answers
22k views

How do you integrate a Bessel function? I don't want to memorize answers or use a computer, is this possible?

I am attempting to integrate a Bessel function of the first kind multiplied by a linear term: $\int xJ_n(x)\mathrm dx$ The textbooks I have open in front of me are not useful (Boas, Arfken, various ...
3
votes
2answers
448 views

Periodicity of EllipticPi

We are trying to implement transformations to evaluate the incomplete integral of the third kind $\Pi(n;\phi|m)$ for arbitrary inputs, and I can't find any references for how to calculate this ...
30
votes
4answers
4k views

When is an elliptic integral expressible in terms of elementary functions?

After seeing this recent question asking how to calculate the following integral $$ \int \frac{1 + x^2}{(1 - x^2) \sqrt{1 + x^4}} \, dx $$ and some of the comments that suggested that it was an ...
2
votes
1answer
1k 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 ...
8
votes
1answer
846 views

Stirling-type formula for the logarithmic derivative of the Gamma function

How may one go about proving $\displaystyle\frac{\Gamma'(s)}{\Gamma(s)}=O(\log|s|)$, (away from the poles) directly? By a direct proof, I mean not to go through the usual Stirling formula with ...
11
votes
7answers
1k views

A gamma function inequality

I would like to prove $$\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)} \le \frac{1}{\sqrt{n}}$$ for all natural $n \ge 1$. The inequality does seem to be true numerically, but the proof eludes me.
5
votes
1answer
719 views

How can I prove this identity involving the digamma function?

I'm trying to prove an identity involving the digamma function $\psi(z)$, but I can't seem to figure out a way to do it. Can anyone help me out? The identity is $$\psi\left(\frac{m}{2} + iy\right) + \...
4
votes
2answers
2k views

Domain of the Gamma function

I need to find the domain of the Gamma function, that is to say all $z \in \mathbb{C}$, for which the integral: $$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \mathrm dt$$ converges. I started by ...
17
votes
3answers
6k views

What is a special function?

When I read some issues here I see from time to time incorrect references to the field special functions, it might e.g. be a discussion around Dirac's $\delta$-function which is tagged (special-...
3
votes
2answers
428 views

Is it problem of Mathematica or my own?

The following is a plot comparing Exp[Derivative[1,0][Zeta][0,x]+1/2Log[2 Pi]] and Gamma[x]: In theory the blue and the red ...
6
votes
1answer
2k views

Some questions about the gamma function

Show that $\Gamma(y) = \int_0^{\infty}{e^{-x}x^{y-1}\,dx}$ is finite for $y>0$ both as an improper Riemann integral and as a Lebesgue integral. Show $\Gamma'(y) = \int_0^{\infty}{e^{-x}x^{y-1}\...
10
votes
1answer
390 views

density of roots of a family of polynomials: $(1-x^2)^{v+n}$

My research has brought me to the following, very general problem. Given a fixed, but arbitrary, natural number, $\displaystyle v$, consider the following family of polynomials: The $\displaystyle (...
22
votes
5answers
12k views

Roots of Legendre Polynomial

I was wondering if the following properties of the Legendre polynomials are true in general. They hold for the first ten or fifteen polynomials. Are the roots always simple (i.e., multiplicity $1$)? ...
5
votes
3answers
1k views

Why can't erf be expressed in terms of elementary functions?

I have seen this claim on Wikipedia and other places. Which branch of mathematics does this result come from?
11
votes
4answers
12k 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.
10
votes
2answers
344 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
363 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
205 views

Is this a known special function?

Is this a known special function: $$\int\nolimits_0^1 a^p(1-a)^{1-p}\\,b^{1-p}\\,(1-b)^p dp\qquad ?$$ I am really only interested in maximizing this over $(a,b)$ in $[0,1] \times [0,1]$, so a ...
4
votes
1answer
289 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 $E(\pi/2,m)=E(m)$? ...
5
votes
1answer
1k 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
861 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
678 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 i))...
3
votes
1answer
1k 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
239 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
184 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. ...
33
votes
4answers
4k 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
185 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
6
votes
2answers
3k 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 ...
13
votes
1answer
638 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) = \frac{2}{\...
8
votes
3answers
631 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
350 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 ...
12
votes
1answer
2k 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 ...
4
votes
1answer
415 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
561 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) =& \frac{1}{2\pi}\...
5
votes
1answer
937 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
206 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 1}{\...
2
votes
1answer
347 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 four ...
14
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 \...
4
votes
1answer
408 views

Gamma integrals

Is anything known about these integrals? Textbook suggestions are welcome \begin{equation*} f(n,p)=\int_{x=-0.5}^p \frac{n!}{x!(n-x)!} dx, \end{equation*} $n>0, p\le n+0.5$. For instance, as $n$ ...