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

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4
votes
1answer
165 views

Definite integral with $\mathrm{Si}$ in integrand

Does the function $$f(t) = \int_0^{\sqrt{3}} (x^2-1) \;\mathrm{Si}((x^2-1)\,t)\; \mathrm{d}x$$ have a representation in terms of elementary functions of $t$ for real, positive $t$? Here, ...
1
vote
1answer
161 views

How to approximate $\text{li}(z)$ numerically?

I'm trying to implement a function to calculate $\pi(x)$ via Riemann's formula: $$ \pi(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan ...
2
votes
1answer
912 views

Electric Potential of an off axis charge (Legendre Generating Function)

An insulated disk, uniform surface charge density $\sigma$, of radius R is laid on the xy plane. Deduce the electric potential $V(z)$ along the z-axis. Next ...
2
votes
1answer
1k views

Normalization of the Bessel function

I would greatly appreciate assistance with the following problem. show: $$\int _0 ^\infty J_n(x)dx = 1; \forall n \in \mathbb{N}^+$$ for $J_o,$ use $$\mathscr{L}{J_o(at)} = \int _0 ^\infty ...
1
vote
1answer
218 views

Fractional part, periodic function

I don't know how to solve this problem: Let $f$ be a continuous real function such that $\{f(x)\} = f(\{x\})$ for each $x$ ($\{x\}$ is the fractional part of number x) Prove that then $f$ or ...
3
votes
3answers
114 views

Can this function be expressed in terms of other well-known functions?

Consider the function $$f(a) = \int^1_0 \frac {t-1}{t^a-1}dt$$ Can this function be expressed in terms of 'well-known' functions for integer values of $a$? I know that it can be relatively simply ...
9
votes
1answer
526 views

Evaluation of an integral involving the Lambert W function

Wikipedia claims that $$\int_0^\infty W\left(\frac{1}{x^2}\right) \,\text dx=\sqrt{2\pi}$$ and a numerical computation seems to confirm this. How can this result be proven?
1
vote
1answer
314 views

asymptotic behavior of the real part of the Riemann zeta function for $0<\sigma<1$

consider the zeta function $\zeta(\sigma+it)$ for $\sigma>1$ : $$\zeta(\sigma+it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma+it}}$$ And: $$\zeta(\sigma-it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma-it}}$$ ...
0
votes
1answer
42 views

Why is this function homogenous to the specified degree?

I have this function $$ w(q) = (1 - \alpha)q^nBk^\alpha + c $$ The paper I'm reading says that w is homogenous of degree $$ n/(1-\alpha) $$ and so small differences in q cause large differences ...
2
votes
0answers
196 views

Problem with understanding first (and second) derivative of a two-sided infinite series

For the function $$f(x)=b^x-1 = x_1 \qquad g(x)=\log(1+x)/\log(b) $$ and its iterative notation $$ x_0=x \qquad x_h=f(x_{h-1})=g(x_{h+1}) \qquad x_{-1}=g(x_0) $$ with b from the interval $1 \lt b \lt ...
3
votes
2answers
55 views

$\frac{\mathrm{d} g(x)}{\mathrm{d}x}=h(x)$ and $\frac{\mathrm{d} h(x)}{\mathrm{d}x}=g(x)$ where $h(x)\neq g(x)$

Is there any other solution to : $$\frac{\mathrm{d} g(x)}{\mathrm{d}x}=h(x)$$ $$\frac{\mathrm{d} h(x)}{\mathrm{d}x}=g(x)$$ other than $h(x)=g(x)=e^x$? By varying $\alpha,\beta$ in $$\frac{\mathrm{d} ...
3
votes
0answers
239 views

Saddle point and stationary point approximation of the Airy equation

Happy New Year to you all. Let $$\tag 1 J(N)=\int_a^b e^{Nf(x)}dx$$ where $N\in\mathbb R$ and $N>>1$ and $f(x)$ has a global maximum at $x=x_0$ with Taylor expansion $$f(x) \approx ...
1
vote
1answer
171 views

Is a Macdonald function a Bessel function with imaginary argument??

I mean that $$ K_{a} (x)= CJ_{a}(ix).$$ Here $C$ is a complex number, and $a$ is real. So is the Macdonald function a Bessel function in disguise (or proportional) of complex argument??
0
votes
0answers
73 views

Summing equally spaced samples of a periodic function

I'm a little stuck at the moment and wondered if someone could point me in the direction of the theory I need to read. I have a $2\pi$-periodic function, $f:\mathbb{R}\rightarrow\mathbb{R}$ which I ...
18
votes
2answers
2k views

The Gamma function and the Pi function

I have been studying differential equation, in particular special functions. Euler's Gamma function, and Gauss's Pi function are essentially the same, differing only by an offset of one unit. for ...
2
votes
1answer
96 views

About the values of the $\Gamma$ function

The $\Gamma$ function is defined by $$\Gamma(z)=\int_{0}^{+\infty}t^{z-1}e^{-t}dt$$ where $z$ is a complex number. We know that if $z$ is real then the values of $\Gamma$ are also real. I am ...
3
votes
0answers
144 views

A Thue-Morse Zeta function ( Generalized Riemann Zeta function and new GRH )

Consider $t_n$ as the Thue-Morse sequence. Let $m$ be a positive integer and $s$ a complex number. Odiuos Number Now consider the sequence of functions below $f(1,s)=1+2^{-s}+3^{-s}+4^{-s}+...$ ...
6
votes
2answers
462 views

Prove that $\int_0^1 \psi{(x) \sin(2 n \pi x)} \space\mathrm{dx}=-\frac{\pi}{2}$

Prove that $$\int_0^1 \psi{(x) \sin(2 n \pi x)} \space\mathrm{dx}=-\frac{\pi}{2}, \space n\ge1$$ where $\psi(x)$ - digamma function
2
votes
1answer
997 views

error function (erf) with better precision

Currently I'm using this C++ routine to approximate the error function ...
4
votes
2answers
553 views

Solving the integral of a Modified Bessel function of the second kind

I would like to find the answer for the following integral $$\int x\ln(x)K_0(x) dx $$ where $K_0(x)$ is the modified Bessel function of the second kind and $\ln(x)$ is the natural-log. Do you have ...
7
votes
1answer
177 views

Finding x in $\frac{\,_2F_1(\frac{1}{5},\frac{4}{5},\,1,\,1-x)}{\,_2F_1(\frac{1}{5},\frac{4}{5},\,1,\,x)} = \sqrt{n}$

I was trying to find a closed-form for $0<x<1$ in, $$\frac{\,_2F_1(\frac{1}{m},\,1-\frac{1}{m},\,1,\,1-x)}{\,_2F_1(\frac{1}{m},\,1-\frac{1}{m},\,1,\,x)} = \sqrt{n}$$ where $\,_2F_1(a,b,c,z)$ ...
2
votes
0answers
76 views

asymptotics of $ J_{iu} (ia)$ for a Bessel function

Let $J_{iu}(ia)$ be the Bessel function of imaginary order. ($a$ is a real number (positive or negative) and $u$ is also real.) In the limit $u \to \infty $ how does the function $J_{iu} (ia)$ ...
3
votes
2answers
68 views

Showing integrability (Riemann)

I was trying to show whether or not the function: $f: [0,1 ] \rightarrow \mathbb{R}$ $f(x)= \frac {1}{n}$ for $x = \frac {1}{n}$ $(n \in \mathbb{N})$ and $f(x) = 1$ if the condition isn't ...
2
votes
1answer
79 views

What is the “name” of this function?

There is a function I met in complex analysis. $$f(\lambda) = \int \limits_{-\infty}^{\infty}\frac{e^{i\lambda x}}{\sqrt{1 + x^{2n}}}dx$$
10
votes
2answers
771 views

An inequality from the handbook of mathematical functions (by Abramowitz and Stegun)

Prove that $$\frac{1}{x+\sqrt{x^2+2}}<e^{x^2}\int\limits_x^{\infty}e^{-t^2} \, \text dt \le\frac{1}{x+\sqrt{x^2+\displaystyle\tfrac{4}{\pi}}}, \space (x\ge 0)$$
0
votes
1answer
485 views

Identity concerning $e^{ia\sin{x}}$ as a series of bessel functions

Prove the following identity: \begin{equation} e^{ia\sin{x}}=\sum_{-\infty}^{+\infty}J_k(a) e^{ikx}, \end{equation} where $a$ is a real constant and $J_k$ is the Bessel function of the first type of ...
1
vote
0answers
105 views

Does this series converge (squares of associated Legendre polynomials)?

Consider the following series (where $l,\,m\in\mathrm{Z}\,$): $S = \displaystyle\sum^{\infty}_{l\,=\,2} \frac{2l+1}{(l-1)(l+2)(1+l^2)}\sum^{l}_{m\,=\,-l}\frac{(l-m)!}{(l+m)!}\Big(P^m_l(x)\,\Big)^2$, ...
0
votes
2answers
202 views

$e^x-x-4$equating with zero

I want to find out the values of x where the $f(x) = e^x-x-4$ will equal zero. My problem by solving this myself is that I cannot use logarithm natural (ln) because I have a normal x: $f(x) = e^x - ...
1
vote
2answers
574 views

Scale modified Bessel functions to then unscale later

So I have some variables $\,x_{1},\, x_{2},\, \nu\, =\, 12.654,\, 13.487,\, 0\,$ and the following function: $\dfrac{(x_{1}\cdot(-BesselK(\nu,x_{1}\cdot125))\cdot ...
1
vote
1answer
62 views

Bounds on geometric sum

Consider the sum $\sum_{x=1}^{\infty} \frac{\log{x}}{z^x}$. We can assume that $z\geq1$ (and is real). Mathematica gives this sum as -PolyLog^(1, 0)[0,1/z] ...
2
votes
1answer
178 views

Inequality for Gamma functions

Let $k, n ,m \in N$ and such that $0\leq k \leq n \leq m$. When the following ineuality is true? $$ ...
4
votes
2answers
152 views

Advice on an integral involving the error function

I'd like to calculate the following integral: $$\int^{\infty}_{0} \mathrm{erf}\left(\frac{\alpha}{\sqrt{1+x}} - \frac{\sqrt{1+x}}{\beta}\right) \exp\left(-\frac{x}{\gamma}\right)\, dx,$$ where ...
3
votes
1answer
425 views

Problem with the Dirichlet Eta Function

I was doing a bit of self-study of sequences, and I considered $$\sum_{n=1}^{\infty}\frac {(-1)^n \ln(n)}{n} $$ which I then found out is ${\eta}'(1)$, the derivative of the Dirichlet Eta Function ...
2
votes
2answers
1k views

Quotient of Gamma functions

I am trying to find a clever way to compute the quotient of two gamma functions whose inputs differ by some integer. In other words, for some real value $x$ and an integer $n < x$, I want to find a ...
1
vote
1answer
78 views

reference needed for Gamma function

Please help me to find a reference (book) for the following upper bound of Gamma function For $x \geq 1$ $$ \Gamma(x)\leq x^{x-1}. $$ Thank you.
1
vote
2answers
4k views

Meaning of function with circle and cross

I've seen this function M2 = tmp ⊕ Pi. What does the circle with cross do?
2
votes
1answer
682 views

Conformal mapping from triangle to upper half plane in terms of Weierstrass $\wp$

I'm trying to explicitly compute a conformal map $f:\Delta \rightarrow \mathbb{H}$ where $\Delta$ is a triangle and $\mathbb{H}$ is the upper half plane, in terms of the Weierstrass $\wp$ function. I ...
2
votes
1answer
378 views

Show that the series representation of the Bessel function works

For the following series representation of the Bessel function: $$w = J_n = \sum_{k=0}^{\infty} \frac{(-1)^k z^{n+2k}}{k!(n+k)!2^{n+2k}}.$$ I want to show that w is indeed the Bessel function, such ...
0
votes
1answer
162 views

Riemann's Zeta function [duplicate]

Possible Duplicate: Riemann Zeta Function and Analytic Continuation Calculating the Zeroes of the Riemann-Zeta function It is stated that Riemann's Zeta function has zeros at negative ...
2
votes
3answers
256 views

Adding imaginary number to exponential of Euler Gamma function

This is gamma function: $\Gamma (n) = \int_0^\infty x^{n-1}e^{-x}\,dx$ What will be Result if I add Imaginary Number to Exponential of Euler Gamma Function? $$? = \int_0^\infty x^{n-1}e^{-ix}\,dx$$ ...
1
vote
1answer
57 views

weird bessel zero question

given 'a' and 'b' fixed i define the function $$ f(t)= bJ_{2t}(a) $$ here $ J_{n} $ is a Bessel function but in this cases i would be interested in getting the solutions (?? are there any ? ) for ...
3
votes
0answers
277 views

Questions about the Fourier expansion of $e^{iz\cot(x)}$

By analogy with Jacobi–Anger expansion, one expects that $e^{iz\cot(x)}$ has a Fourier expansion of the form : $$e^{iz\cot(\theta)}=\sum_{n=-\infty}^{\infty}\Lambda_{n}(z)e^{in\theta}$$ ...
17
votes
1answer
770 views

Prove that sum is finite

Let $j \in \mathbb{N}$. Set $$ a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!} $$ and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$. Please help me to prove that the following sum is ...
1
vote
1answer
784 views

Upper bound for a gamma function

Let $n \in N$. How to find a non-asymptotic upper bound for $\Gamma(n)$ and $\Gamma(\frac n2+1)$? Thank you
9
votes
2answers
3k views

Integrate $\sqrt{1+9x^4} \, dx$

I have puzzled over this for at least an hour, and have made little progress. I tried letting $x^2 = \frac{1}{3}\tan\theta$, and got into a horrible muddle... Then I tried letting $u = x^2$, but ...
0
votes
1answer
341 views

Zernike and Legendre polynomials

The even and odd Zernike polynomials are defined as follows: $$Z^{m}_n(\rho,\varphi) = R^m_n(\rho)\,\cos(m\,\varphi) \!$$ and: $$Z^{-m}_n(\rho,\varphi) = R^m_n(\rho)\,\sin(m\,\varphi), \!$$ with: ...
20
votes
1answer
707 views

Intuition why the volume and surface area of the unit sphere eventually decrease

The volume formula for a unit sphere, $$\frac{\pi^{n/2}}{\Gamma{(1 + n/2)}},$$ and the surface area formula, $$\frac{2\pi^{n/2}}{\Gamma{(n/2)}},$$ both attain maximum values for finite $n$. We can ...
5
votes
1answer
284 views

solution of Lagrange differential equation are square integrable

I was recently posing myself this question. Given the Lagrange DE $$[(1-x^2)u']'+\lambda u=0,$$ where $\lambda$ is a real parameter and $x\in[-1,1]$, it is well known that, if $\lambda=n(n+1)$ for ...
0
votes
1answer
89 views

Integral of Scaled Bessel Function With Linear Phase

I am trying to solve a problem part of which includes the following integral ($j=\sqrt{-1}$): $$\int_{k_1}^{k_2} k e^{-jk\sigma} J_n(\rho k) \, \mathrm{d}k$$ The $e^{-jk\sigma}$ term is making my ...
5
votes
1answer
531 views

Error Function limit

$$\prod_{n=1}^{\infty}{\frac{2}{\sqrt{\pi}}\int_0^n e^{-x^{2}} \mathrm{d}x} \approx 0.83874 $$ Is it a known constant? I couldn't find anything about it. Do you know ways to calculate the value ...