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

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0
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0answers
10 views

How to simplify this special case of Meijer G-Function using 'duplication formula'?

I have the following expression, involving the Meijer G-Function: $$\frac{1}{\sqrt{a} (2 \pi)^{(a-1)/2}} G_{1,a+1}^{a+1,1}\left( \frac{c^a}{a^a} \left|\begin{matrix}0\\ 0,0,\frac{1}{a},\frac{2}{a},...
0
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0answers
22 views

Deducing Laguerre Polynomials

Studying for a final and came accross this problem in the textbook. Considering I have no idea how to even start im a bit scared :). Any explanation would be greatly appreciated. problem: If f(x)is a ...
1
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0answers
22 views

Getting from the ring of periods to the field of periods (which constant become periods in this case)?

The Ring of periods is a fascinating concept in number theory. However, it's rather restrictive, since many popular constants (such as $e$, $\gamma$) are not periods. Periods are integrals of ...
0
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0answers
18 views

Using Rodriguez to prove Legendre

Trying to prove that: $$\int_{-1}^1 P_n(x)P_k(x)\,dx=\frac{-1}{2^nn!} \int_{-1}^1 P_k'(x) \frac{d^{n-1}}{dx^{n+1}}(x^2-1)^n \,dx$$ I know that by the Rodriguez formula, $$P_n(x)=\frac{1}{2^nn!}\frac{...
1
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0answers
46 views

Prove that a certain hypergeometric function assumes either the value $\frac{1}{2}$ or 1

Numerics appear to indicate that the function \begin{equation} f(\alpha)= \end{equation} \begin{equation} \frac{\sqrt{\pi } 3^{-3 \alpha -1} \Gamma \left(2 \alpha +\frac{3}{2}\right) \, _5F_4\left(...
1
vote
1answer
33 views

Proving a Legendre function using generating function

I must prove that $\int_{-1}^1 (1-2xt+t^2)^{-1/2}P_n(x)dx=\frac{2t^n}{2n+1}$. I know that the generating function is $(1-2xt+t^2)^{-1/2}=\sum_{n=0}^\infty P_n(x)t^n$. I also know that the ...
3
votes
1answer
20 views

Variation on Hermite Generating Function

I am having trouble using the Hermite generating function to determine $e^{t^2}\cos(2xt)$. I know the generating function is $e^{2tx-t^2}=\sum_{n=0}^\infty (-1)^n \frac{t^n}{n!}H_n(x)$ but can't seem ...
0
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0answers
21 views

When is the incomplete Beta funtion finite?

I am interested in the incomplete Beta function as defined on Wolfram Mathworld, i.e. $$\text{Beta}(z;a,b)=\int_0^z u^{a-1}(1-u)^{b-1}.$$ I can't seem to find any results on the convergence of this ...
1
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2answers
46 views

Function is continous

If functionis continous at x=0 the we have to find the value of k I got a solution , but I am not able to understand what they have done in second step . Can anyone explain me
-3
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1answer
26 views

Stirling's formula problem [on hold]

Use stirling's formula to find: $$\displaystyle \lim_{n\to \infty}\dfrac {\ln(n!)}{n\ln(n)}$$ .
0
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0answers
12 views

Nonlinear odd real sinusoidal functions

I need a class of odd nonlinear sinusoidal functions whose graphs are given here: I got some example functions: 1) $x = \cfrac{x_{\max}}{2}\times\sin(\cfrac{\pi y}{y_{\max}})$ where $x_{max}$ and $...
0
votes
2answers
35 views

Beta function problem [on hold]

Write the following integral in the form of Beta function $$\int_{0}^{\pi/4} \tan(2x)\, \mathrm{d}x$$ I know that I can use this $$B(p,q)=2 \int_{0}^{\pi/2} \sin^{2p-1}(x) \cos^{2q-1}(x)\, \mathrm{d}...
0
votes
2answers
83 views

Exact value of a series

Is it true that $$\sum_{k=1}^\infty \left(\frac 3 2\sqrt{k}-\sqrt{k+1/2}-\frac{1}{2}\sqrt{k-1}\right)=\frac{(\sqrt2-4) \zeta(3/2)+4 \pi\sqrt 2}{8 \pi}?$$ (this is in regards to the question: ...
6
votes
4answers
148 views

A series with logarithms

Can we express in terms of known constants the sum: $$\mathcal{S}=\sum_{n=1}^{\infty} \frac{\log (n+1)-\log n}{n}$$ First of all it converges , but not matter what I try or whatever technic I am ...
3
votes
2answers
99 views

Solution of $f(x)^2\dfrac{d^2}{dx^2}f(x)=x$

I am stuck in finding the solution of this apparently simple differential equation: $$f(x)^2\dfrac{d^2}{dx^2}f(x)=x$$ with$f(0)=a$ and $f(0)'=b$ Using Maple the solution seems to be a combination of ...
1
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0answers
23 views

Find a hypergeometric formula embracing three specific cases

For a parameter value $a=\frac{1}{4}$, I have the result \begin{equation} Q(k,\frac{1}{4})=\frac{2^{-2 k-\frac{19}{4}} \Gamma \left(2 k+\frac{13}{4}\right) \, _3F_2\left(1,k+\frac{13}{8},k+\frac{17}...
1
vote
2answers
54 views

Proof that $\sum\limits_{k=1}^\infty\frac{2k}{(k^2+c^2)^2}\gt\frac{2}{2c^2+1}$

I tried to prove the following inequality which gives a lower bound to the Mathieu sum: $$S=\sum_{k=1}^\infty\dfrac{2k}{(k^2+c^2)^2}$$ where $c\neq0$. The Mathieu inequality states: $S\lt\dfrac{1}{c^2}...
1
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0answers
64 views

A series involving digamma function

I am trying to solve the series $$\sum_{k=1}^\infty\frac{1}{k(k^2+n^2)}$$ The best I got is $$\frac{\Re\left\{\psi(1+in) \right\}+\gamma)}{n^2}$$ I am not able to simplify it more. Maybe there ...
0
votes
1answer
26 views

Inverse of incomplete elliptic function of first kind

How to find the inverse of elliptic function of first kind in term of angle of integration $\varphi$? This link say that Jacobi amplitude $\varphi = \text{am}(u)$ gives the value of angle $\varphi$. ...
1
vote
0answers
29 views

Quotient of Confluent Hypergeometric Functions of the 1st Kind

I want to solve the following problem for x: \begin{equation} \frac{\mathrm{d}}{\mathrm{d}x}\ e^{-\beta_{1}x}\,{_{1}}F_{1}[-\alpha_{1};-\alpha_{3};\beta_{3}x]=0 \end{equation} where, $\alpha_{1},\...
1
vote
0answers
34 views

Line in Modified Bessel Function of the First Kind

Plotting the modified Bessel function of the first kind $I_\nu(x)$ as a function of two real variables, it looks like to one side of $x=\frac{2}{3}\nu$ the function falls rapidly to zero and on the ...
0
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0answers
47 views

Help with an Incomplete Gamma function-like integral

I was working on some mathematical derivations where I was struck with integrals of the form given below: The integral seems very close to the incomplete gamma function integral. Note here that m ...
1
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0answers
23 views

Construct a master (possibly hypergeometric) formula from a family of formulas indexed by the half-integers and integers

I have a set of individual formulas ($a=1/2, 1, 3/2,\ldots,6$), each itself a function of an integer variable $k$, of increasing complexity. I would like to find a "master" formula (conjecturally of a ...
0
votes
0answers
24 views

Integral of Product of modified Bessel function, exponential functions and power function

I am trying to evaluate the definite integral 1 to obtain a closed-form solution, with z being the integration variable, and the other parameters are real positive constants. The integral can be ...
0
votes
0answers
31 views

Meijer's G-function differentiation

I am trying to calculate the derivative of the Meijer's G function, Based on wolfram function identities I have found in (07.34.20.0003.01) that the derivative is expressed asl: $\frac{d}{dx}G^{m,n}_{...
2
votes
1answer
43 views

Integrating a Bessel function

I'm looking for help to show that $$\int_{0}^{\infty} e^{-at}J_{\nu}(bt)t^{\mu -1} dt $$ can be expressed in terms of the hypergeometric function, where $J_{\nu}$ is the Bessel function of $\nu$ ...
1
vote
1answer
70 views

Is it possible to identify this sequence?

Interested by this question, $j$ being a positive integer, I tried to work the asymptotics of $$S^{(j)}_n=\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+j)}=\frac{\, _2F_1\left(j,-n;j+1;-\frac{1}{n}\...
1
vote
1answer
36 views

aggregate two quadratic functions

I have a quadratic function$$W(x_1,x_2,\ldots,x_n)=A\sum_{i} x_i^2+ \sum_{i\neq j} x_ix_j.$$ Denote the input vector as $\textbf{x}$, in quadratic form, $W(\textbf{x})=\textbf{x}^TM\textbf{x}$, where $...
0
votes
0answers
15 views

Express Jacobi polynomial $P_n^{\alpha-1,\beta-1}$ into the sum of $P_m^{\alpha,\beta}$

I want to express Jacobi polynomial $P_n^{\alpha-1,\beta-1}$ into the sum of $P_m^{\alpha,\beta}$, where $m$ is $n,n-1,n+1...$. Can this be done? Equation (20,21) in this link might be useful.
2
votes
1answer
54 views

Bessel Function of the first kind

Could you please help me understand how to prove $$J_{(1/2)} (x) = \sqrt{\frac2{\pi x}}\cdot \sin⁡ x$$ using, $$J_p (x) = \sum_{(n=0)}^\infty \frac{(-1)^n}{(n! \Gamma(n+p+1) )} \left( \frac x 2 \...
2
votes
0answers
48 views

How to integrate the following definite integral?

$$\int_0^\infty\frac{(B+W)^{-k}}{\sqrt{W+\varphi}} \, dW$$ Is there any general result available . I referred to some table of integrals. There I didn't find a direct result. But found that there are ...
2
votes
1answer
55 views

Prove $\frac{\left(\Gamma(1 + 1/p)\right)^n}{\Gamma(1 + n/p)}\to 1$ for $p =\frac{\ln n}{\ln\frac n {n-2}}$

As the title says, I wish to prove the limit (as $n\to \infty$) $$\frac{\left(\Gamma(1 + 1/p)\right)^n}{\Gamma(1 + n/p)}\to 1\qquad \text{ for } p =\frac{\ln n}{\ln\frac n {n-2}}$$ Any hints? The ...
1
vote
1answer
50 views

A concern about an integral containing cosine integral function

How to prove that $$\int^\infty_0 \frac{\mathrm{ci}(px)}{q^2-x^2}\,dx = \frac{\pi}{2q}\mathrm{si}(pq)$$ The integral was taken from Table of integrals , series and products by Daniel Zwillinger. ...
0
votes
0answers
37 views

Asymptotics of Inverse Laplace transform of a function with a branch point and singularities

consider the inverse Laplace transform $f(x)=L^{-1}[\tilde{f}]$ of a function $\tilde{f}(s)$. I would really like to find the large-$x$ asymptotics of $f(x)$ for the following case: $$\tilde{f}(s)=\...
0
votes
0answers
16 views

Simplify $B(ix,2+iy,0)$ whre $B$ is the incomplete Beta function

Is there any way to re-write or simplify this function for $x,y\in\mathbb{R}$, in the limit $x\rightarrow\pm\infty$? Or any laws regarding symmetry with respect to $x\rightarrow -x$ or $y\rightarrow -...
2
votes
0answers
103 views

Help with a difficult trigonometric integral

let $s$ be a complex parameter. We have the integral : $$\int_{0}^{\pi/2}\tan^{-1}\left[\frac{\tan(sx)}{\tanh(s\log[\sec(x)])} \right ]\frac{\sec^{2}(x)}{e^{2\pi \tan(x)}-1}dx$$ This is a '...
2
votes
1answer
46 views

Function with limits only at irrational points

This is the starred example 5.1.8 in Krantz' Real Analysis and Foundations. Give an example of a function $f: \mathbb{R} \rightarrow \mathbb{R}$ so that $\lim_{x\rightarrow c} f(x)$ exists when $c$...
0
votes
1answer
41 views

Decomposing $\ln(x)$ into sum of even and odd function.

Can somebody help me break $\ln(x)$ into sum of even and odd function. As far as I know every function can be broken in such manner. Not being able to do this as $\ln(-x)$ and $\ln(x)$ cannot exist ...
8
votes
3answers
331 views

Limit involving the Sine integral function

$$ \mbox{Prove that}\qquad \lim_{x \to \infty}\left[\vphantom{\large A}% x\,\mathrm{si}\left(x\right)+ \cos\left(x\right)\right] = 0 $$ where we define $$\mathrm{si}\left(x\right) = - \int^{\infty}_{...
1
vote
0answers
27 views

Definite Integration of Hypergeometric function combined with two algebraic function?

Please help me to solve the integral: $$\int_0^{-w}\dfrac{{_2F_1(1,k,3/2;\phi/B)}}{\sqrt{\phi}\sqrt{-\phi-\omega}}d\phi$$ I have solved this in Mathematica.but I am not able to way a general result ...
1
vote
0answers
30 views

Combinatorial formula for Legendre Polynomials

Using the recursion formula for the solution of the Legendre equation: $$(1-x^2)y''(x)-2xy'(x)+n(n+1)y(x) = 0$$ With solution $P_n(x)$ such that $P_n(1) = 1$, show that $$P_n(x) = \sum_{k = 0}^{n}\...
1
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0answers
22 views

Expressing spherical harmonics as a combination of other spherical harmonics

Spherical harmonics are a useful tool in physics, particularly in classic electrostatics and electrodynamics. Given an integer $l$, the spherical harmonic $Y_{l,m}$, where $-l\leq m\leq l$, solves the ...
2
votes
1answer
86 views

Theta series and Jacobi theta functions

I have some difficulties with expressing the following series $\sum\limits_{b=-\infty}^{+\infty}\sum\limits_{a=-\infty}^{+\infty}q^{1 + 3 a + 3 a^2 - 3 b - 3 a b + 3 b^2}$ using standart theta ...
2
votes
0answers
31 views

Spherical harmonics: how's Laplace's equation related to spheres?

Many spherical harmonics derivations start from finding a solution to Laplace's equation and the results are in fact what are called spherical harmonics. However, how's Laplace's equation really ...
1
vote
1answer
19 views

Derivative of Incomplete Gamma Function

For the following incomplete Gamma function: \begin{equation} Γ(1+d,A-c \ln x)=\int_{A-c\ln x}^{\infty}t^{(1+d)-1}e^{-t}dt \end{equation} I am trying to calculate the derivative of $Γ$ with respect ...
0
votes
0answers
14 views

Solving the definite integral $\int_{-\psi/2}^{\psi/2}\,\exp{(A\sin(\theta+\phi))}d\theta$

I need to solve this definite integral: $$\int_{-\psi/2}^{\psi/2}\,\exp{(A\sin(\theta+\phi))}d\theta$$ where $A$ is a real positive constant and $\psi\in[0,2\pi]$. I know that for $\psi=2\pi$ the ...
5
votes
1answer
102 views

finding the series $\sum_{n=1}^\infty \frac{x^n}{n!} \frac{1}{n}$

My goal is to solve this series $$S(x) = \sum_{n=1}^\infty \frac{x^n}{n!} \frac{1}{n}$$ I did took the derivative first w.r.t $x$ $$S'(x) = \sum_{n=1}^\infty \frac{x^{n-1}}{n!}$$ which I ...
1
vote
1answer
34 views

Another Laplace transform of a function with square roots.

This question is very much related to this (one). Let $F\colon \mathbb{C}\to \mathbb{C}$ be defined as $$F(s) = \frac{1}{4+3s+\sqrt{s(4+s)}}.$$ My question is what is the inverse Laplace transform ...
2
votes
2answers
43 views

Problem on series expansion and Bessel functions

One way to define Bessel functions is $$ e^{\frac{x}{2}(t-\frac{1}{t})}=\sum_{n=-\infty}^{+\infty}J_{n}(x)t^n. $$ How do I prove that? I can't see a way of writing the L.H.S. as a geometrical (or ...
6
votes
3answers
232 views

Seeking closed form for infinite sum $\sum \limits_{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$

$\displaystyle \sum _{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$ is approximately $.5229461921333351$ but I've been assured that there is a closed form for this ...