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

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0
votes
0answers
17 views

modern analysis: step functions with upper and lower sums [duplicate]

A function $f$ defined on $[a,b]$ is a step function if there is a partition $P$ such that $f$ is constant on each subinterval of $P$ a. Show that upper and lower sums are integrals of step ...
6
votes
1answer
154 views

analytic solution to definte integral

I am looking for Analytic solution to a definite integral. Or an approriate transformation to apply. the conditions on $\alpha$ , $\beta$ being positive real numbers while $n$ is positive integer.the ...
1
vote
2answers
44 views

non-sequential sequence function

if i remember correctly (i had one workshop on numerics years ago, sorry for my lack of knowledge) there is a way to create some sort of hash function that gives you a non sequential sequence. This ...
0
votes
1answer
43 views

Taylor series of a rational function

I am facing some complicated integral, which part of it is $$\frac{z^{M-1}}{(1+(\eta z)^n)^p}$$ I think if I find the taylor series of this part the integral might be solved. So, can someone help me ...
4
votes
3answers
152 views

Conjectured closed form of $G^{2~2}_{3~3}\left(1\middle|\begin{array}c1,1;b+1\\b,b;0\end{array}\right)$

In my answer to this question, I come across the following case of the Meijer G-function: $$F(b)=G^{2~2}_{3~3}\left(1\middle|\begin{array}c1,1;b+1\\b,b;0\end{array}\right), b>0$$ and based on my ...
3
votes
2answers
130 views

Real roots plot of the modified bessel function

Could anyone point me a program so i can calculate the roots of $$ K_{ia}(2 \pi)=0 $$ here $ K_{ia}(x) $ is the modified Bessel function of second kind with (pure complex)index 'k' :D My conjecture ...
12
votes
1answer
184 views

Closed form for $\int_{-\infty}^0\operatorname{Ei}^3x\,dx$

Let $\operatorname{Ei}x$ denote the exponential integral: $$\operatorname{Ei}x=-\int_{-x}^\infty\frac{e^{-t}}tdt.\tag1$$ It's not difficult to find that ...
0
votes
1answer
160 views

Integral of incomplete gamma function

I am trying to integrate this: $$\int_0^\infty z^{-|M|-1}\,\Gamma(A,z)\;dz$$ where $A$ is a real positive, and note that the power of $z$ is $-|M|-1$, i.e., is forced to be negative real.
0
votes
0answers
41 views

Help with taylor series as part of an integral involving gamma function

I am facing some strange problem regarding the Taylor series for this function: $$\frac{1}{(1+(\eta z)^n)^p} = ...
1
vote
1answer
77 views

I need an even function that grows faster than cosh(x)

Does anyone know of any even special functions that grow very fast, faster than $\cosh(x)$? (Not the exponential) (Further info): ...
18
votes
1answer
328 views

Fourier transform of $\operatorname{erfc}^3\left|x\right|$

(this is a follow-up on my another question) Could you please help me to find the Fourier transform of $$f(x)=\operatorname{erfc}^3\left|x\right|,$$ where $\operatorname{erfc}z$ denotes the the ...
0
votes
0answers
28 views

eigenfunctions in a Sturm-Liouville problem

I've found that the eigenfunctions in a certain Sturm-Liouville problem satisfy a differential equation whose general solution is $\phi(x)= x^{a}[C_1M(a,2a+2,x)+C_2U(a,2a+2,x)]$, $x\ge0$, where $M$ is ...
2
votes
1answer
32 views

asymptotics of tricomi function

What's the asymptotic behavior of the Tricomi confluent hypergeometric function $U(a,b,z)$ when $|z|\to0$ and $b$ is complex but with $Re(b)=1$. The Abramowitz and Stegun handbook does not seem to ...
1
vote
1answer
62 views

product of different order Bessel function integral

$\displaystyle w = \int_0^\infty r\; J_\mu(ar)\;J_\theta(br)\; \text{d}r $ I'd like to solve this integral ,where a and b are real and positive constant. any information regarding this integral help ...
3
votes
1answer
37 views

Limit of a hyperpower function

i have a question regarding this class of equations: Let $\gamma(x)=x^x$ Let $\Psi_n(x)=\underbrace{\gamma(x)\circ\gamma(x)\circ\gamma(x)}_n$, such that $\Psi_1(x)=\gamma(x)$ and ...
0
votes
0answers
67 views

Learning about the gamma function.

I have just started learning about the gamma function but the books I have are not sufficient to give me a complete picture of it. Can you guys suggest some online resources/free books where I can ...
0
votes
1answer
24 views

Connection between Expected power and Expected Energy over Frequency - Dirac Delta Squared?

I know math people don't like the Dirac delta, so feel free to answer with your measure theory - I'll try my best to understand. Suppose $x$ is a WSS stochastic process $\{x[n] : n \in ...
18
votes
1answer
626 views

Integral $\int_{-\infty}^\infty J^3_0(x) e^{i\omega x}dx $

Hi I am trying to evaluate the integral $$ \mathcal{I}(\omega)=\int_{-\infty}^\infty J^3_0(x) e^{i\omega x}dx $$ analytically. We can also write $$ \mathcal{I}(\omega)=\mathcal{FT}\big(J^3_0(x)\big) ...
0
votes
0answers
39 views

integral involving hypergeometric function

I've obtained that the eigenfunctions of a certain Sturm-Liouville problem are: $$ \phi(x,\lambda) = C\cdot(1/x)^{-1/2\pm i\lambda}\Psi(-1/2\pm i\lambda, 1\pm2i\lambda,1/x), $$ where $C$ is a ...
4
votes
2answers
152 views

Zeta function for negative integers

I already proved that $\zeta(z)=\frac{1}{\Gamma(z)}\int_0^\infty\frac{t^{z-1}}{e^t-1}dt=\frac{\Gamma(z-1)}{2\pi i}\int_{-\infty}^0\frac{t^{z-1}}{e^{-t}-1}dt$ Now the Benoulli numbers are defined by ...
8
votes
1answer
206 views

Strange closed forms for hypergeometric functions

So in the process of trying to find a derivation for this answer, the following interesting equalities arose (one can check with Wolfram Alpha/Mathematica): $$\frac{8\sqrt{2}G^4}{5\pi^2} ...
0
votes
0answers
29 views

integral equation with beta kernel

Is there any way to solve the integral equation $$ z(a,b;x) = 1+\dfrac{(1+x)^{b}}{B(a,b)}\int_0^c\dfrac{y^{a-1}}{(1+x+y)^{a+b}}z(a,b;y)\,dy,\;\;x\ge0, $$ where $a,b,c>0$ are parameters, and ...
0
votes
1answer
20 views

Beta function identity for $B(z,z)$

I would like to derive the identity $B(z,z)=2^{1-2z}B(z,\frac{1}{2})$ somehow. The Beta function is defined as $B(p,q)=\int_0^1 t^{p-1}(1-t)^{q-1}dt$ where $Re(p), Re(q)>0$ I used the ...
1
vote
0answers
18 views

An additive function with capability to retrieve individual components

Assume a set of points in a path $A,B,C,D$ and $E$. Starting from point $A$ assuming single direction there are 4 possible paths as $AB, ABC, ABCD, ABCDE$ with identifies 1 to 4 representing each ...
18
votes
4answers
505 views

Fractional Calculus: Motivation and Foundations.

If this is too broad, I apologise; let's keep it focused on the basics if necessary. What's the motivation and the rigorous foundations behind fractional calculus? It seems very weird & ...
15
votes
2answers
276 views

Integral $\int_0^\infty F(z)\,F\left(z\,\sqrt2\right)\frac{e^{-z^2}}{z^2}dz$ involving Dawson's integrals

I need you help with evaluating this integral: $$I=\int_0^\infty F(z)\,F\left(z\,\sqrt2\right)\frac{e^{-z^2}}{z^2}dz,\tag1$$ where $F(x)$ represents Dawson's integral: $$F(x)=e^{-x^2}\int_0^x ...
4
votes
1answer
103 views

Conjecture about integral $\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx$

I'm interested in the following integral: $$\mathcal J(n)=\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx,\tag1$$ where $K(z)$ is the complete elliptic integral of the 1ˢᵗ ...
3
votes
1answer
96 views

Solution of definite integrals involving incomplete Gamma function

The solution of the integral $$\int_0^{\infty}e^{-\beta x}\gamma(\nu,\alpha \sqrt x)dx $$ is given as ...
4
votes
2answers
162 views

Integral $ \int_0^1 \frac{\ln \ln (1/x)}{1+x^{2p}} dx$…Definite Integral

Calculate $$ I_1:=\int_0^1 \frac{\ln \ln (1/x)}{1+x^{2p}} dx, \ p \geq 1. $$ I am trying to solve this integral $I_1$. I know how to solve a related integral $I_2$ $$ I_2:=\int_0^1 \frac{\ln \ln ...
4
votes
2answers
144 views

Fourier transform of $\operatorname{erfc}^2\left|x\right|$

Could you please help me to find the Fourier transform of $$f(x)=\operatorname{erfc}^2\left|x\right|,$$ where $\operatorname{erfc}z$ denotes the the complementary error function.
0
votes
0answers
29 views

How to evaluate the derivate of a hypergeometric function w.r.t. one of its parameters?

I have to numerically evaluate the derivative of the hypergeometric function w.r.t. its first and second parameters $\large\frac{\partial}{\partial a}{_2F_1}\left(a , b ,c;z\right)$ and ...
0
votes
2answers
17 views

Simple function with a couple of properties

Please supply any simple function $f(x | p)$ which the following properties: $f(0 | p) = 0$ and $f(1 | p)=0$ $f'(0 | p) = 0$ and $f'(1 | p)=0$ $f(x | p)>0$ for $0<x<1$ For $0<x<1$ ...
0
votes
0answers
20 views

How to solve this Fourier-Bessel integral

I want to solve this integral: $\{\int_0^{a\cos{\phi}}\rho e^{-bk\rho}J_q(k\rho)d\rho\}$ where $b=\frac{jn\kappa \cos \phi}{k}$ and $q$ is integer. Since it has a form like Fourier-Bessel transform, ...
4
votes
0answers
165 views

Integral $=\int_0^\infty x^{\alpha -1}Li_n (-\sigma x) Li_m(-\omega x^r)dx$.

I am trying to calculate an integral that can be expressed in terms of infinite hypergeometric series by using transforms and Residue method, the integral is $$ ...
1
vote
0answers
46 views

Solution of some Bessel integrals

The solution of the integration $\int_0^\infty e^{-\alpha x}J_v(\beta x)x^{\mu-1}dx$ is given in a standard form. Can I use the same result when the upper limit of the integration is finite? The ...
3
votes
1answer
92 views

Proving Legendres Relation for elliptic curves

The legendre's relation can be stated as follows $$ K(k) E(k^*)+ E(k) K(k^*) - K(k) K(k^*) = \frac{\pi}{2} $$ where $k^* = \sqrt{1 - k^2}$ is the complimentary modulus, and $E$ and $K$ are ...
0
votes
1answer
85 views

Mean and variance of truncated generalized Beta distribution

The generalized Beta probability density function is given by: $$f(x) = \frac{(x-A)^{\alpha - 1} (B-x)^{\beta - 1}}{(B-A)^{\alpha + \beta - 1} \mathrm{B}(\alpha ,\beta)}$$ for $A<x<B$, and ...
0
votes
1answer
37 views

Differentiation of Bessel function

How can I represent the following differentiation in terms of $J_n$? The equation is: $\frac{\partial}{\partial x}[xJ_n'(x)]$
1
vote
1answer
32 views

What will be the solution of this equation?

What will be the solution of the equation. $(x^2+m^2)\frac{\partial^2y}{\partial(x^2+m^2)}+(x+m)\frac{\partial y}{\partial (x+m)}+(x^2+m^2-n^2)=0$ where $m$ may be a constant
1
vote
0answers
44 views

Closed-form expression for a hypergeometric series

What is the closed-form expression for $${}_2 F_1 \left(1+2\lceil n/2\rceil,-n;1/2;-z/4\right)$$ According to the book Concrete Mathematics (R.Graham, D.Knuth, O.Patashnik 2nd), the authors say the ...
30
votes
3answers
509 views

How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ dx$?

I successfully evaluated these integrals: $$\int_0^\infty\operatorname{erfc}x\ dx=\frac1{\sqrt\pi},\tag1$$ $$\int_0^\infty\operatorname{erfc}^2x\ dx=\frac{2-\sqrt2}{\sqrt\pi}\tag2,$$ but have problems ...
0
votes
1answer
65 views

Taylor series of the upper incomplete gamma function

What is the taylor series of the upper incomplete gamma function? I need it to approximate a difficult integration.
1
vote
1answer
41 views

Meaning of function $f(x) = [x]$

What does the function $f (x) = [x]$ mean? How is it different from least and greatest integer function ?
6
votes
5answers
414 views

Why has $\int \sin (\sin x) dx$ not been solved yet?

I have Calculus 2 background, so please try to keep your answers around that level. I inly want a brief explanation. What is it about $\sin (\sin x)$ that makes it difficult to integrate? Also, what ...
10
votes
1answer
162 views

How to prove $\int_1^\infty\frac{K(x)^2}x dx=\frac{i\,\pi^3}8$?

How can I prove the following identity? $$\int_1^\infty\frac{K(x)^2}x dx\stackrel{\color{#B0B0B0}?}=\frac{i\,\pi^3}8,\tag1$$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ kind: ...
4
votes
2answers
95 views

Does anyone know how to calculate the following integral?

Consider the function (coming from a joint probability density): $$ f(x,y) = \frac{1}{y}e^{-y-\frac{x}{y}}. $$ I want to evaluate the definite integral (marginal): $$ F(x) = \int_0^\infty f(x,y)\,dy. ...
0
votes
0answers
115 views

Inverse of the Modified Bessel function

Is there any chance of having a formula or approximation to inverse the Modified Bessel function of the first kind? I mean to solve $I_M(x)$ with respect to x: $I^{-1}_M(x)$? Thanks in advance
0
votes
0answers
38 views

help with complicated modified bessel function integral

I am trying to address the following complicated integral $$\int_0^{\infty} x^{m-1} e^{-(ax^2+bx+c)}I_v(kx)\text{d}x,$$ Where $I_v(x)$ is a modified Bessel function of the first kind. I did try to ...
12
votes
1answer
286 views

How to prove $\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right)$?

How can I prove the following conjectured identity? $$\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi\stackrel?=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right),\tag1$$ where $K(x)$ is the ...
0
votes
1answer
48 views

Asymptotic Expansion of Bessel $\frac{1}{\pi}\int_0^{\pi}e^{x\cos t}dt$

My question is how to find the asmyptotic expansion of $I(x)=\frac{1}{\pi}\int_0^{\pi}e^{x\cos t}dt$ as $x\rightarrow\infty$. I already got the expansion of $\int_0^{\pi/2}e^{-x\sin^2t} dt$ by using ...