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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1answer
34 views

Integrating Associated Legendre Polynomials

As part of a derivation for the question I asked here in Physics stackexchange, I am trying to calculate the following integral, but I am not sure how to proceed: ...
3
votes
0answers
27 views

Definite integral of arcsine over square-root of quadratic

For $a,b\in\mathbb{R}^{+}\land0<a<1$, define $\mathcal{I}{\left(a,b\right)}$ by the integral ...
8
votes
2answers
1k views

Proving and deriving a Gamma function

I'm having a hard time trying to prove this Gamma function and trying to derive the duplication formula: a.) Prove that $$\frac{\Gamma (p)\Gamma (p)}{\Gamma (2p)} = ...
3
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0answers
24 views

Identification of a function

I recently came across the following function $$\sum_{k=1}^\infty(\log(k))^n\frac{z^k}{k^n}$$ I found it while dealing with the polylogarithm function, $Li_n (z)$ (Notice that if instead of ...
1
vote
0answers
27 views

Integral involving Whittaker function

Consider the following integral: $$ \int_1^{\infty} \frac{e^{u/2}}{u}[-\mathrm{Ei}(-u)]\,W_{1,\imath p}(u)\,du, $$ where $\imath=\sqrt{-1}$ and $p>0$ selected so that $W_{1,\imath p}(1)=0$; here ...
7
votes
1answer
99 views

Can derivative of Hurwitz Zeta be expressed in Hurwitz Zeta?

Can the derivative of Hurwitz Zeta function by the first argument be expressed in terms of Hurwitz Zeta and elementary fuctions? There is a formula which expresses Hurwitz Zeta through its ...
4
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0answers
84 views
+50

Solving integral with spherical bessel functions

I would like to find if possible a solution (closed form) for the following integral: $$\frac{1}{2 \pi}\cdot\int\limits_0^{2\pi}\exp\bigg[-ia(\cos x+\sin x)\bigg]\,j_{0}(b\cos x)\,j_{0}(b\sin ...
3
votes
0answers
393 views

An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function

I think, here, I found $$ P_\color{red}x(\color{blue}s)=\sum_{p<\color{red}x} \frac{1}{p^{\color{blue}s}} =\sum_{\color{green}n=1}^{\infty}\frac{ \mu (\color{green}n)}{\color{green}n} ...
0
votes
0answers
15 views

Finding zeros of a function involving Gamma function.

I am looking for the zeros of following function ($a$ and $b$ are real): $$ F(a,b) = 4^{a+ib} \Gamma(a+ib) \Gamma(-a) \Gamma(-ib) + \Gamma(-a-ib) \Gamma(ib)\Gamma(a) $$ and I have no idea on ...
4
votes
0answers
68 views

integrate $\int \frac{1}{e^{x}+e^{ax}+e^{a^{2}x}} \, dx$

I've been trying to integrate $$ \int \frac{1}{e^{x}+e^{\omega x}+e^{\omega^{2}x}} \, dx $$ where $\omega=e^{2i\pi/3}$ but to no avail. I've tried substituting in $u=e^{(1+\omega)x}$ but ended up ...
7
votes
2answers
148 views

Solving $\ln{x}=\tan{x}$ with infinitely many solutions

Lets take $f(x)=\ln{x}$ and $g(x)=\tan{x}$ When $f(x)=g(x)$ that is $\ln{x}=\tan{x}$, we see that the graph is like: Hence we see that there are infinitely many solutions to $x$ but the two ...
11
votes
1answer
128 views

Evaluate the double sum $\sum_{m=1}^{\infty}\sum_{n=1}^{m-1}\frac{ 1}{m n\left(m^2-n^2\right)^2}$

As a follow up of this nice question I am interested in $$ S_1=\sum_{m=1}^{\infty}\sum_{n=1}^{m-1}\frac{ 1}{m n\left(m^2-n^2\right)^2} $$ Furthermore, I would be also very grateful for a solution ...
15
votes
1answer
325 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 ...
7
votes
6answers
160 views

Solve the following equation: $\sqrt {x + \sqrt {4x + \sqrt {16x + \sqrt {64x + 5}}}} - \sqrt x= 1$

A past examination paper had the following question that I found interesting. I tried having a go at it but haven't come around with any solutions. How would one go about tackling it? $$\sqrt {x + ...
2
votes
2answers
37 views

Finding all values of $\theta$ which describes a straight line

I am having quite a bit of trouble understanding the below question; my assumption is that I should bring the right-hand side in terms of $\sin \theta$ or $\cos \theta$ however am not able to proceed ...
3
votes
2answers
68 views

uniform bound for sine integral function

Prove that for any $0<a<b$, $$ \left|\int_a^b\frac{\sin x}{x}\,dx\right|\le4 $$ Here is my approach. I used integration by parts to prove that LHS is bounded by $3$ when $a\ge 1$. I will be done ...
2
votes
0answers
38 views

Conjecture of the general form of a power series

Relcently I met a power series(Source Link-Eq(4.1)) of the type $$ f(x)=1-x+\frac{1}{2}x^2+\frac{1}{4}x^3-\frac{1}{8}x^4-\frac{35}{128}x^5-\frac{157}{1024}x^6+\cdots $$ where $x$ is supposed to be a ...
1
vote
1answer
42 views

the roots & the limit of $2^{x^{\cos(x)}}\sqrt{\cos(x)}=2^{x}$

If $$2^{(2\pi)^{\cos(2\pi)}}\sqrt{\cos(2\pi)}=2^{2\pi}$$ Can you obtain or is it plausible to find the roots and the limit of $$2^{x^{\cos(x)}}\sqrt{\cos(x)}=2^{x}$$ if $0 < \cos(x)$ and $0 < ...
10
votes
0answers
105 views

The Laplace transform of $\frac{\ln(1+at)}{1+t}$

By expressing the square of the exponential integral as a double integral and then making a change of variables, one can show $$ \int_{0}^{\infty} e^{-2zt} \ \frac{\ln(1+2t)}{1+t} \, dt = \frac{e^{2z} ...
3
votes
0answers
46 views

What is $\int \frac{e^{a x}}{1+x^2} dx $?

In my answer to another question (here: Upper and lower bound on different of ${\rm erf}(\frac{x+c}{b})-{\rm erf}(\frac{x-c}{b})$), I came up with this integral: $\int \frac{e^{a x}}{1+x^2} dx $. I ...
2
votes
1answer
20 views

Simplification of Hankel functions

I have this Hankel function, $H_{1}(R_{1}+R_{2})e^{i\cos(a)}$. Would it be possible to simplify this function in terms of $H_{1}(R_{1})$ and $H_{1}(R_{2})$?
5
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0answers
130 views

Can these integrals be represented in closed form?

This paper in the formula F.3.6 (page 271) gives the following formula for the derivative of Hurwitz Zeta function: $$\frac ...
1
vote
1answer
52 views

transforming ordinary generating function into exponential generating function

I have seen a post here that says that you can convert an exponential generating function into an ordinary one with the aid of the Laplace transform. Is it possible to do the reverse transformation? ...
4
votes
2answers
947 views

Definition of the gamma function

I know that the Gamma function with argument $(-\frac{1}{ 2})$ -- in other words $\Gamma(-\frac{1}{2})$ is equal to $-2\pi^{1/2}$. However, the definition of $\Gamma(k)=\int_0^\infty t^{k-1}e^{-t}dt$ ...
0
votes
0answers
19 views

Solve $x$ in the equation: $a\cdot \textrm{arctanh} [b + a \cdot x] - c \cdot \textrm{arctanh} [d + c \cdot x] = e$

How to solve $x$ in the equation: $a\cdot \textrm{arctanh} [b + a \cdot x] - c \cdot \textrm{arctanh} [d + c \cdot x] = e$, where $\textrm{arctanh}(x) = \frac{1}{2} \log \left(\frac{1+x}{1-x} ...
25
votes
1answer
596 views

Weber-type integral

In connection with this answer, I came across the following integral: $$\int_{0}^{\infty} \frac{du}{u} \: \,e^{-t u^2} \frac{J_0(u) Y_0(r u)-J_0(r u) Y_0(u)}{J_0^2(u)+Y_0^2(u)}$$ where $r \gt 1$. I ...
1
vote
1answer
48 views

Gamma and Beta function proof.

I'm trying to proof the equality $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ when $x,y>0,$ without using calculus in many variables. I've investigated about the topic but all references make ...
1
vote
0answers
20 views

What does subcopula mean?

In copula concept, what does "subcopula" exactly mean? Does it mean a subset of copula? Would you please explain a little bit in details? Thanks in advance!
0
votes
1answer
17 views

Infinite Integral of a Bessel Function

I need to calculate the following integral $$ \int_0^{\infty}xdxJ_n(kx) $$ Integrating it by parts and using the normalization of Bessel functions, I find it (somewhat heuristically) to equal the ...
2
votes
1answer
67 views

Concerning Hurwitz Zeta function, how to prove the following identity?

It is claimed that $$\zeta'(0,s)=\ln\left(\frac{\Gamma(s)}{\sqrt{2\pi}}\right)$$ where the derivative is meant by the first argument (as usual with Hurwitz Zeta). How to prove this? Wolfram Alpha ...
0
votes
2answers
75 views

Prove that $f^{-1} (F)$ is closed

A set $F \subset \mathbb R$ is closed if for any convergent sequence $\{x_n\}$ in F converges, we have $\lim_{n \to \infty} x_n=x \in F $. How to Prove that if $f :\mathbb R \to \mathbb R$ is ...
1
vote
0answers
21 views

Asymptotic for Bessel Function

We have that, $$J_p(x) = \sqrt{\frac{2}{\pi x}} \sin \left( x - \frac{p\pi}{2} + \frac{\pi}{4}\right) + \frac{r_p(x)}{x\sqrt{x}}$$ We also know that there exists $M>0$ such that $|r_p(x)| \leq M$. ...
7
votes
1answer
862 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.
1
vote
1answer
32 views

Help Obtaining Numerical Approximation of Lambert W Solution

I am studying a particular generating function $$\frac{2e^x}{e^{2x}+1+2x}$$ and I thought I would try to solve the equation $$e^{2x}+1+2x=0$$ to determine for what value of $x$ if any the function ...
3
votes
1answer
37 views

On Lamda function

The Lambda function is defined as: $$\lambda(s)=\sum_{n=0}^{\infty} \frac{1}{(2n+1)^s},\; \mathfrak{Re}(s)>1$$ How to prove that $\lambda(s)=(1-2^{-s})\zeta(s)$? Basically, I was dealing with ...
0
votes
0answers
14 views

Orthogonality of Hankel functions, what are the relations?

What are the relations for orthogonality of Hankel's relations defined via: $H^{(1)}_{m}(z) \equiv J_{n}(z) + i Y_{n}(z)$ $H^{(2)}_{m}(z) \equiv J_{n}(z) - i Y_{n}(z)$ I have looked at some books ...
2
votes
1answer
309 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 ...
1
vote
0answers
28 views

Integrating a product of to error functions and an exponential

I have the following integral that I need to solve. $\int_{-\infty}^\infty \exp(-\frac{x^2}{2})*\text{erf}(x-\delta)*\text{erf}(x-\gamma)dx$ I was hoping I could use this: Integral of product of ...
1
vote
1answer
19 views

$x^2y''+(2x^2+x)y'+(2x^2+x)y=0$ A Bessel equation

$$x^2y''+(2x^2+x)y'+(2x^2+x)y=0$$ The solution is $$e^{-x}J_o(x)+e^{-x}Y_o(x)$$ How does one approach a problem like this?
21
votes
0answers
193 views

Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc… & Representation Theory of Special Functions

Many special functions of mathematical physics can be understood from the point of view of the representation theory of lie groups. An example of the power of this viewpoint is given in my question ...
6
votes
3answers
163 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
384 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 $$ ...
5
votes
1answer
166 views

Integral involving the confluent and the Gauss Hypergeometric functions with exponential functions

I am trying to compute the following integral: $$ \int _0^\infty x\exp[-\pi b x^2]{_1F_1}[n,1,\frac{-\pi b x^2}{2}]{_2F_1}[1,\frac{2}{aq},1+\frac{2}{aq},\frac{-sx^{-a}}{2}]dx$$ Is there any general ...
2
votes
1answer
43 views

elliptic curve isogeny class 14.a $L$-function Dirichlet coefficients

Are the Dirichlet coefficients $a(n)$ of the $L$-function associated with isogeny class 14.a the irrationals that the inverse symbolic calculator suggests they are? The Lcalcfile suggests that they ...
19
votes
1answer
378 views

Non-trivial values of error function $\operatorname{erf}(x)$?

The so called error function $\operatorname{erf}(x)$ is defined as $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt,$$ and it is well known that $\operatorname{erf}(\infty)=1$. Are ...
1
vote
1answer
39 views

prove an identity involving beta function and gamma function

We know that $B(p,q)=\Gamma(p)\Gamma(q)/\Gamma(p+q)$ where $p, q>0$, and $B(p,q)$ is related to binomial coefficients if one of $p,q$ is an integer. I want to prove the following identity. ...
3
votes
1answer
351 views

Gamma integrals

Is anything known about these integrals? Textbook suggestions are welcome \begin{equation*} f(n,p)=\int_{x=-0.5}^p \frac{n!}{x!(n-x)!} dx, \end{equation*} $n>0, p\le n+0.5$. For instance, as $n$ ...
9
votes
3answers
2k views

How to come up with the gamma function?

It always puzzles me, how the Gamma functions's inventor came up with it's definition $$\Gamma(x+1)=\int_0^1(-\ln t)^x\;\mathrm dt=\int_0^\infty t^xe^{-t}\;\mathrm dt$$ Is there a nice derivation of ...
0
votes
1answer
21 views

Integral over product of two bessel functions and power

I have searched the literature on integrals over bessel functions, but I couldn't find anything. The integral to be evaluated is, $\int_0^a J_{n}(bx)J_{\mu}(cx)xdx =: \mathcal{M}_r(a;n,\mu;b,c)$ ...
1
vote
2answers
244 views

Express complex Bessel function in terms of functions taking real arguements

I want to use the Bessel function in C++. Since this one is not implemented there for complex arguments, I am looking for a way to express the bessel function(first and second kind) as: ...