All Questions
Tagged with complex-analysis special-functions
344
questions
-3
votes
0
answers
41
views
How to get the following estimate of integral invoving Airy function [closed]
$$
\mbox{Define}\quad
G(x,y)=\frac{Ci\left(\gamma(x-y_c)\right)Ai\left(\gamma(y-y_c)\right)}{\epsilon\gamma},
$$
where $y>x,$ $y_c$ is a complex number such that $\Re(y_c)>0,$
$\epsilon$ is a ...
- 55
2
votes
2
answers
105
views
Asymptotics of modified Bessel function of second kind
Let denote $K_\nu$ the modified Bessel function of second kind of argument $\nu\in(0,\infty)$. It is kown that for $x\in\mathbb{R}$, we have:
$$K_\nu(x) \sim \sqrt{\frac{\pi}{2x}} e^{-x}$$
as $x\to+\...
- 506
0
votes
0
answers
14
views
Question regarding terms with negative coefficients in multivariate Fox H function
I was studying a paper where an integral expression in terms of Fox H function of multiple variable were used. The definition of multivariate Fox H (extracted from appendix A-1 of Mathai-Saxena) is as ...
- 3,263
4
votes
0
answers
182
views
Definite integral involving K Bessel function and a square root
I have recently been trying to evaluate some integrals involving the modified Bessel function $K_0(x)$. The specific integrals are
$$L(x,u) = \int_0^{1} K_0\left( 2x \sqrt{r(1-r)} \right) \exp(2ixur) ...
- 51
5
votes
0
answers
101
views
Zeta Lerch function. Proof of functional equation.
so I'm trying to prove the functional equation of Lerch Zeta, through the Hankel contour and Residue theorem, did the following.
In the article "Note sur la function" by Mr. Mathias Lerch, a ...
3
votes
0
answers
139
views
Is there a website that has all the special functions? [closed]
There are a lot of special functions, and I wonder if there is a website that collects all of them, similar to how the Encyclopedia of Triangle Centers collects information on triangle centers.
...
- 6,620
1
vote
0
answers
13
views
Continuity of confluent hypergeometric function in terms of its parameters
The confluent hyper geometric function of the first kind (or the Kummer's function) is defined as
$${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)\Gamma\left(b-a%
\right)}\int_{0}^{1}e^{...
- 3,263
12
votes
3
answers
542
views
Complex analysis or real analysis books that have these special functions.
I saw an integral question that involved the digamma function, which I know nothing about, and I want to learn more about it, its properties, and other functions like the polylogarithm function and ...
- 6,620
1
vote
0
answers
123
views
How to evaluate the integral $\int_0^\pi e^{i(a\sin x + b\cos x)} dx$
We know that (from e.g. here)
$$
\int_0^{2\pi}{\rm e}^{{\rm i}\left[a\sin\left(x\right) + b\cos\left(x\right)\right]}{\rm d}x =
2\pi\operatorname{J}_{0}\left(\sqrt{a^{2} + b^{2}}\right)
$$
where $\...
- 123
1
vote
1
answer
53
views
Why is the Jacobi amplitude real when $k>1$?
I understand the Jacobi amplitude is defined as the inverse function of the incomplete elliptic integral of the first kind
$$
\mathrm{am}(u,k) = F^{-1}(u,k),
$$
where
$$
F(u,k) = \int_0^u \frac{d\phi}{...
4
votes
2
answers
176
views
Integration of hypergeometric function on complex plane
I have come across an integral that involves a hypergeometric function, which can be expressed as follows:
$$I = \int_0^1 x^{1/2}(1-x)^{\epsilon-1} {_{2}F_1}(\frac{1}{2}+\epsilon,1+\epsilon;\frac{3}{...
- 109
0
votes
0
answers
48
views
$\log \Gamma(z + 1) = \log \Gamma (z) + \log z$ issue
I was reading proof of Binet's first expression for $\log \Gamma(z)$; that is for $\Re z > 0$,
$$
\log \Gamma(z) = \left(z - \frac 1 2\right)\log z - z + \frac 1 2 \log (2\pi) + \int_0^\infty \left(...
- 2,376
0
votes
2
answers
188
views
Proper Way to Calculate Value of Riemann Zeta function?
I understand that an Analytic Continuation of a function will extend its domain into areas that it previously wasn't defined in.
I've been looking at one of the Analytic Continuations of the Zeta ...
0
votes
0
answers
82
views
Does any closed form of this integral exists or is related to some special function integral?
$$\int_{-\infty}^\infty e^{i \omega x-i b \tanh( x)}dx$$
I have to solve this integral for my research project .I substituted $ \tanh{x} = y $ and $ \tanh^{-1}{x} = \frac{1}{2}\log{\frac{1+y}{1-y}} $, ...
1
vote
1
answer
59
views
Connection between the polylogarithm and the Bernoulli polynomials.
I have been studying the polylogarithm function and came across its relation with Bernoulli polynomials, as Wikipedia site asserts:
For positive integer polylogarithm orders $s$, the Hurwitz zeta ...
- 812