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

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10
votes
2answers
794 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
497 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
583 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
153 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
433 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
704 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
386 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
170 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
258 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
821 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
349 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
722 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
553 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
318 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
79 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
395 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
268 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
488 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 ...
1
vote
0answers
129 views

Question on arguments of the Gamma Function

I came accross with this site about Gamma function. I just want to verify, clarify, whatever you may want to call it. It says you can compute for the gamma value for a negative argument using $$\Gamma(...
0
votes
1answer
203 views

Expressing solution to an inequality with Lambert W function

I'm new to Lambert functions, any ideas on how to solve this are welcome: $$ \theta \rho^{\theta}+r \theta>v $$ where $\theta \in \mathbb{R}^{+}, -1<r,v<1, \ 0<\rho<1$. I've tried ...
3
votes
0answers
142 views

Modified Bessel function

I use the standard notations. When $x$ is real then by definition $$ I_{\nu}(x)=e^{-\nu\pi i/2}J_{\nu}(ix). $$ I want to define $I_{\nu}$ for complex $z$. Watson (Treatise of the Theory of Bessel ...
2
votes
1answer
90 views

To solve $U''_{n}(x)-\frac{2n}{x}U'_{n}(x)+(\frac{2n}{x^2}-1)U_{n}(x)=0 $

$$e^{x\sqrt{1+t}}=\sum \limits_{k=0}^\infty \frac{U_k(x)t^k}{k!}$$ $$\frac{\partial}{\partial t }(e^{x\sqrt{1+t}})=\frac{\partial}{\partial t }(\sum \limits_{k=0}^\infty \frac{U_k(x)t^k}{k!})$$ $$\...
6
votes
2answers
239 views

Property of sum $\sum_{k=1}^{+\infty}\frac{(2k+1)^{4n+1}}{1+\exp{((2k+1)\pi)}}$

Is it true that for all $n\in\mathbb{N}$, \begin{align}f(n)=\sum_{k=1}^{+\infty}\frac{(2k+1)^{4n+1}}{1+\exp{((2k+1)\pi)}}\end{align} is always rational. I have calculated via Mathematica, which says \...
43
votes
9answers
18k views

Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$ ?

It seems as if no one has asked this here before, unless I don't know how to search. The Gamma function is $$ \Gamma(\alpha)=\int_0^\infty x^{\alpha-1} e^{-x}\,dx. $$ Why is $$ \Gamma\left(\frac{1}{2}...
1
vote
2answers
966 views

How do I show the Wronskian of $(J_{a}(x),Y_{a}(x)) = \dfrac {2} {\pi x}$

Based of using my undergrad class notes. I know that the wronskian of $(J_{a}(x),Y_{a}(x))$ is $ W(J_{a}(x),Y_{a}(x)) = \left| \begin{matrix} J_{a}(x) & Y_{a}(x) \\ J_{a}'(x) & Y_{a}'(...
6
votes
2answers
416 views

Bounding the Gamma Function

I'm trying to verify a bound for the gamma function $$ \Gamma(z) = \int_0^\infty e^{-t}t^{z - 1}\;dt. $$ In particular, for real $m \geq 1$, I'd like to show that $$ \Gamma(m + 1) \leq 2\left(\frac{...
4
votes
3answers
437 views

Limiting behavior of gamma function

I am trying to determine whether $\Gamma(x+iy)\rightarrow 0$ as $y\rightarrow\infty$. How should I go about doing it? I was trying to see if I could get anything from $\Gamma(z)\Gamma(1-z)=\frac{\pi}{...
1
vote
0answers
78 views

for what $\nu$ does Riemann-Liouville differintegral of digamma function $\psi(z)$ exist?

For what values of $\nu$ does the Riemann-Liouville differintegral $_{-\infty}D_{z}^\nu$ of the digamma function $\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}$ exist, with $c=-\infty$? All I've got so far is ...
3
votes
0answers
307 views

How to prove Gegenbauer's addition theorem?

How can one prove the following identity: $$ V_k(r_1, r_2) = {2k+1\over 2 r_1 r_2}\int_{|r_1 - r_2|}^{r_1+r_2} e^{-{r\over D}} P_k\left(r_1^2 - r^2 + r_2^2 \over 2 r_1 ...
0
votes
1answer
137 views

How can I get $B(x,y+1)= \frac{y}{x+y} B(x,y)$ using integration by parts?

So far I have $B(x,y+1) = \int_{0}^{1} t^{x-1}(1-t)^{(y+1)-1} dt =\int_{0}^{1} t^{x-1}(1-t)^{y} dt $ for $x,y>0$. I tried doing integration by parts by letting $dv = (1-t)^y$ and $u=t^{x-1}$ but it ...
0
votes
1answer
56 views

How to find $\lim_{x \rightarrow -N} J_{a} = (-1)^N J_{N}$?

So far I have $$\lim_{a \rightarrow -N} J_{a} = \lim_{a \rightarrow -N} \mid \frac{x}{2} \mid^a \sum_{k=0}^{\infty} \frac{(-x^2/4)^k}{k! \; \Gamma(a+k+1)} = \mid \frac{x}{2} \mid^{-N} \sum_{k=0}^{\...
5
votes
0answers
150 views

Has the $\Gamma$-like function $f_p(n) = 1^{\ln(1)^p} \cdot 2^{\ln(2)^p} \cdot \ldots \cdot n^{\ln(n)^p} $ been discussed anywhere?

In an older fiddling with the gamma-function (expanding on the idea of sums of consecutive like-powers of logarithms, similarly as the bernoulli-polynomials for the sums of like powers of consecutive ...
1
vote
1answer
315 views

Limit of a summation wth Gamma function

Can anyone prove this (I'm very confident that it is correct) or have any idea how this can be handled: $$ \lim_{n \rightarrow \infty} \frac{1}{n-1}\sum_{i=1}^{n-1} \frac{1}{(\alpha-1)(n-i) -1} \frac{...