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

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5
votes
2answers
150 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.
8
votes
3answers
128 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 ...
0
votes
1answer
29 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 ...
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.
3
votes
2answers
57 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 ...
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 ...
3
votes
1answer
163 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 ...
1
vote
1answer
29 views

Simplify $\int_0^{\infty}\,dk\,k^{\frac{1}{2}}R^{\frac{3}{2}}J_{\frac{3}{2}}(kR)\exp{(-ak^2)}$

I would like to rewrite this integral $$\int_0^{\infty}\,dk\,k^{\frac{1}{2}}R^{\frac{3}{2}}J_{\frac{3}{2}}(kR)\exp{(-ak^2)}$$ (where $a>\mathbf{R^+}$ and $J_{\frac{3}{2}}$ is the bessel function ...
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 $$ ...
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?
5
votes
2answers
223 views

Closed form for $\int_0^x \{1/t \}\,\mathrm{d}t$, $x \in \mathbb{R}_+$ and related.

After some tests I think that Conjecture 1 Let $x \in \mathbb{R}_+$ then $$ \int_{0}^{x}\left\{\,1 \over t\,\right\}\,{\rm d}t = 1 - \gamma + H_{\left\{1/x\right\}} - x\left\lfloor\, ...
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 ...
22
votes
2answers
505 views

Evaluating $\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]- ...
6
votes
2answers
127 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 ...
2
votes
1answer
229 views

Solve equation with lower gamma function: $A \gamma(2;x/B)=x$ for $x$

I need to find an expression for $x$ given: $A \gamma(2;x/B)=x$ where $\gamma(a,x)=\int\limits_0^x t^{a-1} e^{-t} \mathrm{d}t$ is the lower incomplete gamma function. $A$ and $B$ are real, positive ...
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 ...
2
votes
1answer
239 views

Incomplete Fermi-Dirac integrals and polylogs

The complete Fermi-Dirac integrals $$ F_s(x) = \frac{1}{\Gamma(s+1)} \int\limits_{0}^{\infty} \frac{t^s}{e^{t-x}+1} \: dt $$ are related to the polylogarithms, see http://dlmf.nist.gov/25.12#iii $$ ...
2
votes
2answers
383 views

Numerical approximation of the modified Bessel function $I_0$ with radical argument for integration purposes

I have to numerically calculate the following definite integral $$\int_{\alpha}^{\beta}I_0(a\sqrt{1-x^2})dx$$ for different values of $\alpha$ and $\beta$, where $a$ has a value of, say, $30$. I'm ...
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 ...
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
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 ...
7
votes
1answer
594 views

median of the F-distribution

Is the median of the F-distribution with m and n degrees of freedom decreasing in n, for any m? From experiments it looks like it might be, but I have been unable to prove it.
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 ...
3
votes
2answers
413 views

Does sinc function have any special inverse function defined?

We know that $y=xe^x$ cannot be solved for $x$ using elementary functions. The Lagrange inversion theorem can be used for finding a "new" function that would be the inverse function of the above ...
7
votes
3answers
463 views

Weierstrass $\wp$ function doubly periodic

I'm working my way through Silverman and Tate's Undergraduate Introduction to Elliptic Curves. I haven't yet been able to study complex analysis, so it comes as no surprise that I'm having a tough ...
8
votes
0answers
574 views

How to compute this integral of Bessel functions?

I have $\alpha_\max$ a real number between $0$ and $\frac\pi2$. Furthermore $\zeta$ and $\xi$ are positive real numbers. Now I would like compute the integral $$\int_0^{\alpha_\max} \mathrm{e}^{i ...
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 ...
4
votes
0answers
81 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 = ...
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 ...
-1
votes
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
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 ...
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) ...
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 ...
5
votes
2answers
201 views

Is $\eta^{24}(\tau)\,j(\tau) = {E_4}^3(q)$?

Given the j-function $j(\tau)$, $j(\tau) = 1728J(\tau)$, where $J(\tau)$ is Klein’s absolute invariant, the Dedekind eta function $\eta(\tau)$, and the following Eisenstein series, $\begin{align} ...
18
votes
2answers
518 views

Interesting log sine integrals $\int_0^{\pi/3} \log^2 \left(2\sin \frac{x}{2} \right)dx= \frac{7\pi^3}{108}$

Show that $$\begin{aligned} \int_0^{\pi/3} \log^2 \left(2\sin \frac{x}{2} \right)dx &= \frac{7\pi^3}{108} \\ \int_0^{\pi/3}x\log^2 \left(2\sin\frac{x}{2} \right)dx &= ...
10
votes
4answers
327 views

Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

Applying the Copson's inequality, I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it know any ...
1
vote
1answer
287 views

Solution of Bessel equation

Prove that for a Bessel function in its normal form that is: $$u'' + \left(1 + \frac{1-(4*p^2)}{4x^2}\right)u=0$$ if $p > \frac12$ then every interval of length $\pi$ contains at most one zero of ...
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 ...
4
votes
1answer
248 views

Integral involving bessel function/gaussian/rational function

I'd like to solve: $$\int_0^{\infty}\quad J_1(ak)\,\frac{b+k^2}{(k-\alpha_1)(k-\alpha_2)(k-\alpha_3)(k-\alpha_4)}\,\exp(-ck^2)\,\,dk$$ Is there any specific rule for it? Thanks!
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} ...
3
votes
1answer
123 views

An integral that might be related to the modified Bessel function of second kind

It is known that the modified Bessel Function $K_z(a)$ ($a>0$)can be expressed as a Fourier transform $$K_z(a)=\frac{1}{2}\int_{-\infty}^{\infty}\exp(-a\cosh t)\cosh(zt){\rm d}t=K_{-z}(a)$$ Can ...
0
votes
0answers
31 views

Integrating products of Hankel and Riccati Bessel functions

I want to do the integral: $$ \int_0^\infty dr h_l^+(kr)\hat j_l(kr) $$ where $h_l^+$ is the type 1 Hankel function, $\hat j_l$ is the type 1 Riccati-Bessel function. I would like a algebraic ...
1
vote
1answer
45 views

Satisfaction of Bessel equation by any other function.

Is it possible that any function $y(x)$ other than Bessel group of functions, satisfy Bessel's equation? $$x^2 \dfrac{d^2 y}{d x^2} + x \dfrac{d y}{d x} + (1-n^2/x^2) y = 0.$$
2
votes
1answer
20 views

Prove convexity of log modified bessel function

I need to prove that the modified bessel function of the second kind is log convex in the square of the argument. Specifically I'm interested in showing, $\log \mathcal{K}_0(\sqrt{x})$ (zero order) is ...