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
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
243 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 f(x_0)-|f'...
1
vote
1answer
174 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
97 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
146 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
464 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
1k views

error function (erf) with better precision

Currently I'm using this C++ routine to approximate the error function ...
4
votes
2answers
556 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 $J_{iu} (ia)$ behave?...
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
782 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
487 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
578 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 BesselI(\nu,x_{2}\cdot125))-(x_{2}\...
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? $$ \frac{2^{m-k}\Gamma(n+1)\Gamma\left(\left[\frac{m+1-k}{2}\right]\right)\Gamma(m+1-n)}{\Gamma(m+1)\...
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
430 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
81 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?
3
votes
1answer
699 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
379 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
166 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 even ...
2
votes
3answers
257 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 : $$e^{iz\cot(\theta)}=\sum_{n=-\infty}^{\infty}\Lambda_{n}(z)e^{in\theta}$$ $\Lambda_{n}(z)...
17
votes
1answer
773 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
803 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
343 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: $$R^...
20
votes
1answer
717 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
539 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 ...
3
votes
0answers
85 views

Solutions of legendre equation for $\vert x\vert \leq 1$

Why books say that is necessary in Legendre equation to have $l$ integer if you want regular solutions in $\vert x\vert \leq 1$. It seems not necessary. Thanks in advance.
6
votes
1answer
314 views

Did Euler have an alpha function

I've heard of Euler Gamma function: $\Gamma(x)$, and Euler's beta function: $\text{B}(x,y)$. Did Euler have an alpha function?
2
votes
1answer
282 views

Question on legendre equation - part 2

I would like to know if is possible to have regular solutions of Legendre equation when the constant $l$ in the Legendre equation $(1-x^2)u''-2xu''+l(l+1)u=0$ is a non integer number? I am interested ...
2
votes
1answer
78 views

Question on Legendre equation

I have a doubt. If Legendre equation has a polynomial solution, is the constant $l$ in $l(l+1)$ necessarily a integer number? Asked in another way, is possible $l(l+1)$ be a integer if $l$ is not an ...
0
votes
1answer
318 views

Solve in terms of the Gamma function

Show: \begin{align*} \int\limits_0^1\sqrt{\frac{1-x^2}{1+x^2}}\,\mathrm d x &=\frac{\sqrt \pi}{4}\left(\frac{\Gamma \left(\frac14\right)}{\Gamma\left(\frac34\right)}-4\frac{\Gamma\left(\frac34\...
4
votes
0answers
394 views

Solving inhomogenous bessel equation

I have the following differential equation to be solved $\dfrac{d^2\psi}{dr^2}+\dfrac{d\psi}{rdr}+4\left(\omega^2-k_0^2-\dfrac{n^2}{r^2}\right)\psi=AJ_n^2(kr)+\dfrac{k}{r}J_n(kr)J_{n+1}(kr)-\omega k^...
6
votes
1answer
264 views

Definite integral involving Fresnel integrals

I am seeking to evaluate $\int_0^{\infty} f(x)/x^2 \, dx$ with $f(x)=1-\sqrt{\pi/6} \left(\cos (x) C\left(\sqrt{\frac{6 x}{\pi }} \right)+S\left(\sqrt{\frac{6 x}{\pi }} \right) \sin (x)\right)/\...
3
votes
1answer
1k views

Relationship between Legendre polynomials and Legendre functions of the second kind

I'm taking an ODE course at the moment, and my instructor gave us the following problem: Derive the following formula for Legendre functions $Q_n(x)$ of the second kind: $$Q_n(x) = P_n(x) \...
9
votes
0answers
478 views

An infinite series expansion in terms of the polylogarithm function

We have the complex valued function: $$f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)$$ We wish to recover the coefficients $a_{n}$. The only thing I though would ...