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
24 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
votes
0answers
10 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
34 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}...
15
votes
1answer
174 views

Family of definite integrals involving Dedekind eta function of a complex argument, $\int_0^{\infty} \eta^k(ix)dx$

The Dedekind eta function is denoted by $\eta(\tau)$, and is defined on the upper half-plane ($\Im \tau >0$). Put $\tau = i x$ where $x$ is a positive real number. The function has the following ...
52
votes
6answers
5k views

Is there a function whose antiderivative can be found but whose derivative cannot?

Does a function, $f(x)$, exist such that $\int f(x) dx $ can be found but $f' (x)$ cannot be found in terms of elementary functions. For example, if $f(x)=e^{x^2}$, then the derivative is easily ...
5
votes
4answers
145 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 ...
0
votes
2answers
77 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: ...
3
votes
2answers
82 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 ...
4
votes
0answers
101 views

Integral of combination of power, exponential, and confluent hypergeometric function

I am trying to solve a couple integrals of the form: \begin{equation} \int_{0}^{\infty} x \, e^{-a(gx-b)^{2}}\,e^{-\beta_{1}x}\, {_{1}}F_{1}(-\alpha_{1};-\alpha_{3};\beta_{3} x) \ \mathrm{d}x \end{...
5
votes
4answers
216 views

Prove that $2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)} \, \mathrm{d}x =\ln(\pi)-\gamma $

Let $\gamma$ be the Euler-Mascheroni constant. I'm trying to prove that $$2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)} \, \mathrm{d}x =\ln(\pi)-\gamma $$ I tried introducing a parameter to the ...
5
votes
1answer
238 views

Fabius function and equivalent

The Fabius function $F$ can be defined on $[0,1]$ by $F(0)=0$ $F(1)=1$ on $[0,\frac{1}{2}]$ $F'(x)=2.F(2x)$ on $[\frac{1}{2},1]$ $F'(x)=2.F(2(1-x))$ It's a known example of a not analytic $C^\...
1
vote
0answers
88 views

Ellipsoidal harmonics - A Series expansion for Lame functions of the second kind

Intro to skip In the theory of ellipsoidal harmonics, Lame functions of the second kind $F_n$ arise as the second linearly independent solution (the first being Lame functions of the first kind $E_n$)...
1
vote
0answers
20 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}...
2
votes
1answer
418 views

A Curious Binomial Coefficient Sum: $\sum_{j = 0}^{k} \binom{k}{j} \binom{j + n -\ell + 1}{n}$

Let $k, \ell \leq n$ be non-negative integers. Does the following identity simplify? \begin{align} \sum_{j = 0}^{k} \binom{k}{j} \binom{j + n -\ell + 1}{n} = \binom{n - \ell + 1}{n} \phantom1_{2}\...
5
votes
1answer
789 views

Inverse of elliptic integral of second kind

The Wikipedia articles on elliptic integral and elliptic functions state that “elliptic functions were discovered as inverse functions of elliptic integrals.” Some elliptic functions have names and ...
1
vote
0answers
63 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
25 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 $\varphi$. ...
32
votes
2answers
748 views

Evaluating the series $\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} $

Wolfram MathWorld states that $$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = \frac{ \pi \sqrt{3}}{18} \Big[ \psi_{1} \left(\frac{1}{3} \right) - \psi_{1} \left(\frac{2}{3} \right) \Big]- \...
3
votes
1answer
743 views

2-increasing functions

I'm trying to learn about copulas. Two definitions I've come across are the H-volume of a rectangle, which is defined as $V_H=H(x_2,y_2)-H(x_2,y_1)-H(x_1,y_2)+H(x_1,y_1)$. The function H, whose ...
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 ...
1
vote
0answers
22 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
45 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 ...
0
votes
0answers
23 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
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
38 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}\...
17
votes
1answer
395 views

Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I ...
5
votes
1answer
1k views

Gamma function has no zeros

I want to show that $\Gamma(z)$ has no zeros. My idea is to use the formula $$\Gamma(z)\Gamma(1-z)=\dfrac{\pi}{\sin(\pi z)}$$ which holds for all $z\in\mathbb{C}$. If $\Gamma(z)=0$, then the left-...
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 $...
19
votes
2answers
427 views

A closed-form expression for the integral $\int_0^\infty\text{Ci}^{3}(x) \, \mathrm dx$

Is there a closed-form expression for this integral: $$\int_0^\infty\text{Ci}^{3}(x) \, \mathrm dx,$$ where $\text{Ci}(x)=-\int_x^\infty\frac{\cos z}{z}\mathrm dz$ is the cosine integral? $\text{Ci}...
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.
1
vote
1answer
755 views

Heuman Lambda Function in MATLAB

I'm trying to implement the Heuman Lambda function in MATLAB, but i'm having problems getting a correct answer. The Heuman Lambda Function is: $$ \Lambda_0(\beta,k) = \frac{2}{pi}[E(k)F(\beta,k') + ...
9
votes
1answer
1k views

Derivative of the elliptic integral of the first kind

The complete elliptic integral of the first kind is defined as $$K(k)=\int_0^{\pi/2} \frac{dx}{\sqrt{1-k^2\sin^2{x}}}$$ and the complete elliptic integral of the second kind is defined as $$E(k)=\...
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
97 views

Verify $y=x^{1/2}Z_{1/3}\left(2x^{3/2}\right)$ is a solution to $y^{\prime\prime}+9xy=0$

This question is a sequel to this previous question. As before, some background information is needed first as follows from my textbook: The standard form of Bessel's differential equation is $$x^...
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 \...
1
vote
1answer
142 views

Definite Integral of Modified Bessel function representation

I am trying to express the following integral of the Modified Bessel function either in closed form or even using other special functions. Any ideas ? $$ \int_{0}^{b}x\exp\left(-\,{x^{2} + z^{2} \...
2
votes
0answers
47 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
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 '...
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
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 $c$...
17
votes
3answers
232 views

Fibonorial of a fractional or complex argument

Let $F(n)$ denote the $n^{\text{th}}$ Fibonacci number$^{[1]}$$\!^{[2]}$$\!^{[3]}$. The Fibonacci numbers have a natural generalization to an analytic function of a complex argument: $$F(z)=\left(\phi^...
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}_{...
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 ...
1
vote
1answer
109 views

$\int x\tan x$ and the Clausen Function

I have been attempting to evaluate $\int x \tan x \;\mathrm{d} x$. My first instinct was integration by parts, which produces $-x \ln|\cos x|+\int \ln|\cos x| \;\mathrm{d} x$. I have read online ...