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
65 views

Compute polylog of order 3 at $\frac{1}{2}$

How to compute the following series: $$\sum_{n=1}^{\infty}\frac{1}{2^nn^3}$$ I am aware this equals polylog of order 3 at $\frac{1}{2}$, but how to prove it using integral or Euler sum only (without ...
3
votes
2answers
58 views

There is a closed form for $\sum _{n=1}^{\infty }{\frac {{{\it J}_{0}\left(\,\alpha\,n\right)} {{\it J}_{0}\left(\,\beta\,n\right)}}{{n}^{2}}}$?

Using the method showed here proposed by Olivier Oloa with simplifications proposed by Anastasiya-Romanova, it is possible to prove that $$\sum _{n=1}^{\infty }{\frac {{{\it ...
0
votes
1answer
30 views

Proving an integration with a modified Bessel function and an exponential

I am trying to prove the following identity: where $\mu, h, H$, and $\tilde{\gamma}$ are real constants. The only hint that I have is use the relation between the modified bessel function of the ...
3
votes
1answer
164 views

On definition of gamma function.

We all know that gamma function's definition is $$\Gamma \left( x \right) = \int\limits_0^\infty {s^{x - 1} e^{ - s} ds}$$ and it is divergent for $x<0$. Yesterday, I was studying about Bessel ...
5
votes
2answers
154 views

What is the closed form of $\sum _{n=1}^{\infty }{\frac { {{\it J}_{0}\left(n\right)} ^2}{{n}^4}}$?

Using Maple I am obtaining the numerical approximation $$0.5902373619$$ Please, let me know what is the closed form. Many thanks.
1
vote
2answers
58 views

An Elliptic Integral - What's the Simplest Answer?

I have $$ \int_{0}^{2\pi}d\theta\left(R^{2}-\epsilon^{2}\right)\sqrt{R^{2}-\epsilon^{2}\sin^{2}\left(\theta\right)} $$ which Mathematica thinks is $$ ...
1
vote
1answer
69 views

What is the closed form of $\sum _{n=1}^{\infty }{\frac { {{\rm J_{0}}\left(2\,n\right)} ^{2}}{{n}^{2}}}$?

Using Maple I am obtaining $$\sum _{n=1}^{\infty }{\frac { {{\rm J_0}\left(2\,n\right)} ^{2}}{{n}^{2}}} = 0.09845497463 $$ Please check it. Many thanks.
9
votes
4answers
518 views

What functions satisfy this functional equation?

$$f(x)-g(x)=f(g(x))$$ How could I find an f(x) and g(x) that satisfy this?
8
votes
3answers
138 views
+300

Simplification of an expression containing $\operatorname{Li}_3(x)$ terms

In my computations I ended up with this result: $$\mathcal{K}=78\operatorname{Li}_3\left(\frac13\right)+15\operatorname{Li}_3\left(\frac23\right)-64\operatorname{Li}_3\left(\frac15\right)-102 ...
6
votes
2answers
129 views

$\sum_{n\geq 1}\frac{(-1)^n \ln n}{n}$

How can we compute the series $\displaystyle \sum_{n\geq 1}\frac{(-1)^n \ln n}{n}$? I know it is $\eta '(1)$ , where $\eta$ is the $\eta$ Dirichlet Function , i know its value. But I don't know how ...
0
votes
0answers
19 views

Eigenvalue of Heun's function and its computation

It is known that the Heun's differential equation: \begin{equation} \frac{d^2 w}{dz^2} + (\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-a})\frac{dw}{dz}+\frac{\alpha \beta z -q}{z(z-1)(z-a)} ...
4
votes
2answers
58 views

Closed form for $\int z^n\ln{(z)}\ln{(1-z)}\,\mathrm{d}z$?

Problem. Find an anti-derivative for the following indefinite integral, where $n$ is a non-negative integer: $$\int z^n\ln{\left(z\right)}\ln{\left(1-z\right)}\,\mathrm{d}z=~???$$ My attempt: ...
2
votes
0answers
25 views

Why do so many identities for the Logarithmic Integral begin with the terms $\log \log n + \gamma +…$?

Several identities for the log integral lead with the terms $\log \log n + \gamma$, where $\gamma$ is the Euler–Mascheroni constant. So, for example, there's the well-known $$\text{li}(n) = \log ...
0
votes
1answer
25 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
34 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 ...
3
votes
0answers
44 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
votes
0answers
23 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
25 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
49 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
43 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
82 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
106 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
223 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
75 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
26 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 ...
5
votes
1answer
82 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
80 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: ...
1
vote
1answer
25 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
69 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
28 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
72 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
37 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
34 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
95 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 ...
1
vote
2answers
120 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
512 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
57 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 ...