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
13 views

Closed form for this incomplete gamma series?

The series I'm working with is $$\sum_{k=0}^\infty \binom{z}{k}(-1)^k ( 1-\frac{\Gamma(k,-\log n)}{\Gamma(k)} )$$ with $z$ a complex variable and $\Gamma(k, -\log n)$ the upper incomplete gamma ...
1
vote
2answers
29 views

Series expansion of incomplete gamma function ratio

I am interested in the series expansion of: $$S(k)=\frac{\Gamma(k+1,a)}{k!},$$ around $k=\infty$ where $\Gamma(x,z)$ is the incomplete gamma function and $a$ is some positive constant. In ...
2
votes
0answers
41 views

An integral with a decaying exponential with rational exponent

I was working on some mathematical derivations while I faced this integral: $$\Large \int_0^\infty x^{\alpha-1}e^{-\beta x} e^{-\lambda \left[\frac{x^2}{2x+\eta}\right]}\ \mathrm{d}x \quad .$$ Does ...
0
votes
2answers
51 views

Terminology for $1/(e^x+1)$?

$ \frac{1}{e^x+1} $ and $ \frac{e^x}{e^x+1} $ Just wonder if either of the above function has a term/name associated with it? Or they are just functions that look beautiful without names? Maybe they ...
0
votes
1answer
16 views

function undefined at odd inputs

I am a high-school student in pre-calculus. My teacher told me today that it is impossible to define a function using only multiplication, division, exponents, addition, subtraction such that it ...
-1
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0answers
22 views

How to find $b$ in this equation involving a Bessel function and a modified Bessel function?

How to find $b$ in this equation, $$v\sqrt{1-b}\frac{j_{-1}(v\sqrt{1-b})}{j_0(v\sqrt{1-b})}=v\sqrt{b}\frac{k_{-1}(v\sqrt{b})}{k_0(v\sqrt{b})}$$ where $ j$ is Bessel function of first order and $k$ ...
0
votes
1answer
24 views

Why are non-polynomial Legendre functions and Legendre polynomials not orthogonal?

The eigenfunctions of distinct eigenvalues for a Hermitian operator are proved to be orthogonal. Why does the same not apply to Legendre polynomials and functions that have different eigenvalues ? ...
0
votes
1answer
48 views

Solution of $\frac{dx}{dt}=-\frac{(\sigma+1)x}{\sigma x+1}$ in terms of Lambert $w$ function

Solution of $\frac{dx}{dt}=-\frac{(\sigma+1)x}{\sigma x+1}$ in terms of Lambert $w$ function. Should I first take the solution of ODE and then apply Laplace transform. Please give step by step ...
1
vote
1answer
40 views

Value of $\psi\left(\frac{1}{2}\right)$

I apologise if this is a dumb question, but I have trouble deriving $\displaystyle\psi\left(\frac{1}{2}\right)=-\gamma-2\ln{2}$. I have tried the following. \begin{align} \psi\left(\frac{1}{2}\right) ...
3
votes
1answer
69 views

Is anything known about $2\pi$ integer multiple arguments of the cosine integral?

I'm interested in $\text{Ci}(2\pi n)$ for integers $n\geq 1$. As the graph below shows, as $n$ increases the cosine integral seems to (strictly?) monotonically decrease. I've looked online but can't ...
2
votes
0answers
32 views

Solution of $\Pi(y(x)+1)+\sin(x)=y(x)+y'(x)$

How do we solve $$\Pi(y(x)+1)+\sin(x)=y(x)+y'(x)$$ I suspect it will be a function of many cases. The solution of $$\Pi(x+1)+\sin(x)=y(x)+y'(x)$$ is hard only at the evaluation of the last integral ...
4
votes
0answers
77 views

Second derivative of Hypergeometric function

I'm looking for the following second derivative $$ \kappa_2 := \left . \frac{d^2}{d\lambda^2} \ln \left({_2F_1}\!\left(\tfrac{1}{2},\,- \lambda;\,1;\,\alpha\right)\right) \right \vert_{\lambda = ...
0
votes
0answers
21 views

spherical Bessel function of the first kind

I'm trying to find the first few terms in the spherical Bessel functions of the $1^{st}$ kind and am not getting the third term correct. ...
2
votes
1answer
37 views

Numerical evaluation of Hurwitz zeta function

Is there a way to evaluate numerically the Hurwitz zeta function $$\zeta(s,a) = \sum_{n=0}^\infty \frac{1}{(n+a)^{s}}$$ that is more efficient (i.e., quick and precise) than simply explicitly adding ...
6
votes
3answers
100 views

A closed-form of product the gamma functions containing $\pi$ and $\phi$

Playing with gamma functions by randomly inputting numbers to Wolfram Alpha, I got the following beautiful result \begin{equation} ...
10
votes
1answer
214 views

Contour integration with branch points inside the contour.

In my scientific research I ran into an unpleasant situation with specific type of contour integrals. Being more specific I have problems not with integrals themselves (I can use various numeric ...
1
vote
2answers
71 views

A infinite sum with harmonic serie

Proof or disproof the folowing statment $$\sum_{n=1}^{+\infty}\frac{2n+1}{(n^2+n)^2}H_n=\sum_{n=1}^{+\infty}\frac{1}{n^3}$$ Where $\displaystyle H_n=\sum_{k=1}^{n}\frac{1}{k}$
0
votes
1answer
25 views

Mathieu function rescale problem

The Mathieu functions are the solutions for the equation $$ y''+(a-2q\cos(2z))y=0 $$ If we require the solution has the form $$ y(z) = e^{i r z}f(z) $$ where $f(z)$ is a periodic function with ...
4
votes
0answers
50 views

Simpler closed form for $\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}$

I'm trying to find a closed form of this sum: $$S=\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}.\tag{1}$$ WolframAlpha gives a large expressions containing multiple ...
6
votes
2answers
76 views

Trigonometric functions expressed as definite integrals with Bessel functions

Prove that $$\frac{\sin(x)}{x}=\int_0^\frac{\pi}{2}J_0(x\cos(\theta))\cos(\theta)\,d\theta \tag{a}$$ $$\frac{1-\cos(x)}{x}=\int_0^\frac{\pi}{2}J_1(x\cos(\theta))\,d\theta \tag{b}$$ Hint: ...
0
votes
0answers
22 views

Laurent series of $\psi(-2z)$ at positive integers and positive half integers.

The laurent series of $\psi(-z)$ at $z=n$ is $$\psi(-z)=\frac{1}{z-n}+H_n-\gamma-(H_n^{(2)}+\zeta(2))(z-n)+\cdots$$ However, may I ask how one can determine the laurent series expansion for ...
1
vote
1answer
24 views

Where is the mistake with my proof that $\sum\limits_{m=0}^{a-1}\operatorname{Li}_s(ze^{\frac{2\pi im }{a}})=a^{1-s}\operatorname{Li}_s(z^a)$

I tried to prove that $$\sum_{m=0}^{a-1}\operatorname{Li}_s(ze^{\frac{2\pi im }{a}})=a^{1-s}\operatorname{Li}_s(z^a)$$ where $\operatorname{Li}_s(x)$ is Polylogarithm function. ...
4
votes
2answers
68 views

Integral of the Square of the Elliptic Integral

Someone must know a good technique for $$ \int E^{2}(x)dx $$ Where $E$ is the complete elliptic integral of the second kind: $$ ...
2
votes
0answers
43 views

Simple Integral Involving the Square of the Elliptic Integral

I have, $$ \int uE^{2}\left(u\right)du $$ where $E$ is the complete elliptic integral of the second kind: $$ E\left(k\right)=\int_{0}^{\frac{\pi}{2}}d\theta\sqrt{1-k^{2}\sin^{2}\left(\theta\right)} ...
1
vote
1answer
46 views

Integrating a Ratio of Elliptic Integrals

Can anyone help evaluate $$\int dx\frac{\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-k^{2}\sin^{2}\left(\theta\right)}}}{x\int_{0}^{\frac{\pi}{2}}d\theta\sqrt{1-k^{2}\sin^{2}\left(\theta\right)}}$$ ...
0
votes
0answers
24 views

Mathieu characteristic function for non integer value in Maple

In Maple the Mathieu characteristic function can only evaluate integer values. But in Mathematica it can take non-integer values. And I have test that the integer values from both system seems ...
0
votes
1answer
45 views
0
votes
1answer
33 views

Limit of Mathieu function near the discontinuous point

Consider the Mathieu characteristic function, which is a piecewise function. The discontinuity happens at integer number. ...
2
votes
1answer
70 views

List of functions $f(cx) = C\cdot f(x)$

I was looking for some complex functions f(x), which satisfies the condition: $$\exists (c, C) \in \Bbb C^2 \backslash\{(1,1)\}, \forall x \in \Bbb C, f(cx) = C\cdot f(x)$$ Till now I have got ...
3
votes
1answer
36 views

Lambert function. Calculate $W(b)$ from $W(a)$.

The Lambert W function is defined as follows: $$z = W(z)e^{W(z)}$$ for any complex number z. Many equations involving exponentials can be solved using the W function. For example: $$ Y = X e ^ X ...
2
votes
0answers
33 views

Tractable indefinite integral of the exponentiation of some function

Consider the function $z(s)\in\mathbb{C}$ defined as $z(s)=\int_0^s \exp\left[i(q u+\lambda(u))\right]du$ for some $q\in\mathbb{Q}-\mathbb{Z}$ and $\lambda(s)$ a $2\pi$-periodic real differentiable ...
3
votes
2answers
93 views

Prove: $\int_{0}^{1}\frac{\ln{x}\,\mathrm{d}x}{\sqrt[3]{x(1-x^2)^2}}\stackrel{?}{=}-\frac18\left[\Gamma{\left(\frac13\right)}\right]^3$

I'd like to evaluate the following definite integral: $$\int_{0}^{1}\frac{\ln{x}\,\mathrm{d}x}{\sqrt[3]{x(1-x^2)^2}}\stackrel{?}{=}-\frac18\left[\Gamma{\left(\frac13\right)}\right]^3.$$ ...
2
votes
0answers
49 views

The integral $\int_{0}^\pi\frac{d\omega}{(2\sin(\omega/2))^{2\alpha}+2\lambda(2\sin(\omega/2))^\alpha\sin(\alpha\omega/2)+\lambda^2} $?

Can the integral \begin{equation} \int_{0}^\pi\frac{d\omega}{(2\sin(\omega/2))^{2\alpha}+2\lambda(2\sin(\omega/2))^\alpha\sin(\alpha\omega/2)+\lambda^2},\quad 0<\alpha<2,\quad \lambda>0 ...
0
votes
2answers
116 views

Is there any proof for this formula $\lim_{n \to ∞} \prod_{k=1}^n \left (1+\dfrac {kx}{n^2} \right) =e^{x⁄2}$

Some times ago, In a mathematical problem book I sow that this formula. I don't no whether it is true or not. But now I'm try to prove it. I have no idea how to begin it. Any hint or reference would ...
0
votes
0answers
36 views

Evaluation of $\int_{1}^{\infty} x^{-\frac{5}{3}} \cos \left( \left( x-1 \right) h \right)dx$ with Maple [duplicate]

I have calculated the Integral with the aid of some professors here and I get a problem: $$\int_{1}^{\infty} x^{-\frac{5}{3}} \cos \left( \left( x-1 \right) h \right)dx$$ I have done the Integral ...
10
votes
2answers
497 views

Inequality involving Pochhammer symbols

Let $m,S$ be integers satisfying $2\leq m\leq S$. I would like to show that $$h_1\left(x\right) h_3\left(x\right) \leq h_2^2\left(x\right)$$ for all $x\geq 0$ where $$h_k\left(x\right) \equiv ...
2
votes
3answers
67 views

Skewed Trigonometric Function

What would be an expression for a periodic function (period $2\pi$) that essentially behaves just like a negative sine function, but it has the following quirk: It's $0$s lie on the usual places ...
2
votes
1answer
55 views

Application of Bessel Function

I have read number of books and online literature on Bessel function. Theoretically, I have known about Bessel function. What is practical significance of Bessel function? How can Bessel function ...
4
votes
1answer
56 views

Spherical integral

Let $y \in \mathbb{R}^n$ be fixed. Is there a nice expression for the following integral taken over the unit sphere in $\mathbb{R}^n$? $$ \int_{\|x\|=1} e^{2\pi i (x \cdot y)}~dx $$
0
votes
0answers
22 views

What is the asymptotic behaviour of $n^3 \log(\Gamma(1 + 1/n))$ as $n\to\infty$?

What is the asymptotic behaviour of $n^3\log(\Gamma(1 + 1/n))$ as $n\to\infty$ ? I have deduced that $\log\Gamma(1+1/n)\sim\log(1-\gamma/n)$ but multiplying by $n^3$ then gives an error.
0
votes
0answers
29 views

Proving concavity for complicated function

I have a rather complicated function, $f$, that I am trying to demonstrate is log-log-concave, i.e., $$\frac{d^2\log f}{(d\log x)^2}\leq 0.$$ The reason I think it is concave is purely heuristic. ...
-1
votes
0answers
18 views

The spectral density of this signal

I want to calculate the spectral density of $m(t) = (a(k)-a(k-1))\times j(k) \times p(t-kT)$, where, $T$ is the bit period, $k$ is one point in time, $a(k) \in {\pm 1}$ and are equiprobable and $j$ is ...
13
votes
1answer
204 views

A Gamma limit $\lim_{n\rightarrow+\infty}\sum_{k=1}^n \displaystyle \left( \Gamma\bigl(\frac{k}{n}\bigr)\right)^{-k}=\frac{e^\gamma}{e^\gamma-1}$

Show that $$\lim_{n\rightarrow+\infty}\sum_{k=1}^n \displaystyle \left( \Gamma\bigl(\frac{k}{n}\bigr)\right)^{-k}=\frac{e^\gamma}{e^\gamma-1}$$ where $\gamma$ is the Euler-Mascheroni Constant. ...
0
votes
1answer
22 views

multiderivative of incomplete gamma function

I spend 4 hours trying to solve it. I believe that I am either stuck or I am approaching the problem at the wrong angle. Here is my challenge: $$\frac{\partial^m }{\partial x^m}\left [ x^{-a}\gamma ...
4
votes
1answer
80 views

Compute the following series $\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}$

Does the following series have a 'closed' form : $$\sum_{n=1}^{+\infty}\frac{1}{(n+a)(n+b)}.$$ Where $n\in \Bbb{N}$ and $a,b \in (0,+\infty)$ For $a,b$ integer we can use Partial fraction ...
9
votes
2answers
90 views

Trigamma identity $4\,\psi_1\!\left(\frac15\right)+\psi_1\!\left(\frac25\right)-\psi_1\!\left(\frac1{10}\right)=\frac{4\pi^2}{\phi\,\sqrt5}.$

I heuristically discovered the following identity for the trigamma function, that I could not find in any tables or papers or infer from existing formulae (e.g. [1], [2], [3], [4], [5], [6]): ...
3
votes
1answer
45 views

Find zero of sum of 4 modified Bessel functions

I am trying to find the (positive) root of the function $f(x) = I_{-3/4}(x) + I_{3/4}(x) - I_{-1/4}(x) - I_{1/4}(x)$ where $I_\alpha(x)$ denotes the modified Bessel function of the first kind. ...
0
votes
0answers
107 views

Why is it hold for types of operators?

We ‎let ‎the ‎state ‎space ‎be‎ ‏‎‎‎$ \mathcal{H} =‎ ‎‎H_{E}^{2}(0 , 1) \times L^2(0 , 1) $‎ equipped with the norm ‎ \begin{align} \| (f , g) \| = \int_{0}^{1} [ |f''(x)|^2 + |g(x)|^2] ‎\mathrm{d}x‎ ...
7
votes
1answer
75 views

Closed form for integral of integer powers of Sinc function

(Edit: Thank you Vladimir for providing the references for the closed form value of the integrals. My revised question is then to how to derive this closed form.) For all $n\in\mathbb{N}^+$, ...
2
votes
1answer
30 views

Uniqueness proof for a certain functional equation

Consider the following functional equation: \begin{equation}f(x)=kf(mx)\end{equation} where $x \in [0,1]$; $k>0$; $0<m<1$; and $f(1)=1$. It has at least one solution: $f(x)=x^l$, where ...