0
votes
1answer
57 views

Integrate square root of 4th grad polynomials

During some calculations for a program I came upon this Integral which I am not able to solve. I already tried Matlab but it didn't help me. Here is the Integral: $$\int\left(\sqrt{\sum_{0}^{5} 9 ...
0
votes
1answer
19 views

Interpolate the number of arrangements in a set

I am working the integral $$\int_0^\infty e^{x(k-\alpha) - e^x} dx$$ where $k$ is a positive integer and $\alpha$ a positive real. WolframAlpha shows that for $\alpha=0$ and $k=1,\ldots,7$ the ...
11
votes
1answer
147 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
0answers
37 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} = ...
18
votes
1answer
309 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 ...
3
votes
1answer
29 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 ...
16
votes
1answer
492 views

Fourier Transform $J^3_0(x)$.

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) ...
8
votes
1answer
184 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} ...
14
votes
2answers
153 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
2answers
131 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
26 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 ...
28
votes
3answers
440 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 ...
6
votes
5answers
382 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 ...
0
votes
0answers
83 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
33 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 ...
11
votes
1answer
244 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 ...
1
vote
0answers
30 views

Integral with incomplete gamma function and Modified Bessel Function

Can somebody suggest a technique to integrate this? all parameters (m, beta, v, k, A, B and n) are positive real constants
5
votes
1answer
52 views

Closed form for derivative $\frac{d}{d\beta}\,{_2F_1}\left(\frac13,\,\beta;\,\frac43;\,\frac89\right)\Big|_{\beta=\frac56}$

As far as I know, there is no general way to evaluate derivatives of hypergeometric functions with respect to their parameters in a closed form, but for some particular cases it may be possible. I am ...
0
votes
0answers
37 views

Interesting integral with Modified Bessel Function, Gamma Function

Is there anyway to integrate this monster? m, beta, v, k, A, B, and n are real positive constants.
0
votes
3answers
97 views

Gaussian-Like integral

What is the integral of this? $$\int_0^\infty xe^{-(ax^2+bx)}\,\mathrm{d}x$$ $a$ and $b$ are positive integers.
1
vote
2answers
67 views

How to find a closed form for the derivatives of $F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,dt,$ $F(0)=\frac12$?

I have been given the function $$F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,{\rm d}t$$ for $x\ne 0,$ $F(0)=\frac12,$ and charged with finding a Taylor polynomial for $F(x)$ differing from $F$ by no ...
0
votes
2answers
64 views

Equality of a function and Taylor Series

Does the following function have a Taylor series of the form given below: $$\frac{1}{(1+(\eta z)^n)^p} = ...
0
votes
1answer
57 views

taylor series expansion for a rational function

What is the Taylor Series Expansion (function of z ) for where $\eta$, $n$ and $p$ are positive real constants Based on the answers in the comments, does this mean that the taylor series is given ...
2
votes
1answer
41 views

Integral with the incomplete upper gamma function

Can anyone help me integrate this? $$\int_0^1 \frac{1}{x^{1/p}} \left[\frac{1-x^{1/p}}{x^{1/p}} \right]^{m/n-1} \Gamma\;\left(A, \left[\frac{1-x^{1/p}}{x^{1/p}} \right]^{1/n}\right) ...
0
votes
0answers
29 views

Taylor series of some stange function

Can somebody help find the Taylor series for: where p and n are real positive (not necessary integers)? Does it converge fast?
2
votes
1answer
40 views

Does $\int^1_0\frac{(1-s)^\alpha}{s^\beta} \operatorname d \!s$ converge?

I want to ask about the finitess of the following integral \begin{equation} \int^1_0\frac{(1-s)^\alpha}{s^\beta}ds \end{equation} when $\alpha>\beta>1$. This integral is very similar to the ...
2
votes
1answer
96 views

The identity $\int_0^\infty x^\alpha/(\exp(x)-1)dx=\zeta(\alpha+1)\Gamma(\alpha+1)$

I would like to prove that for every real $\alpha>1$ we have $$\int_0^\infty \frac{x^\alpha}{\exp(x) -1} \, dx=\zeta(\alpha+1)\Gamma(\alpha+1).$$ Proof: Let $0<a<b$; then we have $$\int_a^b ...
1
vote
0answers
42 views

Closed form double integral $ \int_{a}^{c}dr \int_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{r_<^{\ell}}{r_>^{\ell+1}}$

Is there a closed form expression for $$ S_\ell = \int\limits_{a}^{c}dr \int\limits_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{[\min( r , r')]^{\ell}}{[\max(r,r')]^{\ell+1}} ...
5
votes
2answers
259 views

Closed form integral $\int_b^c \frac{x^2}{\sqrt{(x-a)(x-b)(c-x)(d-x)}} dx$

Is there a closed form expression for the definite integral $$I=\int_b^c \frac{x^2}{\sqrt{(x-a)(x-b)(c-x)(d-x)}} dx$$ for $a<b<c<d$? Mathematica 9.0 can do it for special cases using ...
1
vote
1answer
53 views

integral with gaussian function

I am trying to evaluate the following integral: $$ \int_0^\infty{z^{m-1}\over\left[1+\left(\eta z\right)^n\right]^p}e^{-(z-b)^2\over c}\,{\rm d}z, $$ where the integration is w.r.t. to $z$, and the ...
2
votes
0answers
96 views

integral involving upper incomplete gamma function

I trying to evaluate the following integral $$\int_0^\infty \dfrac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$$ where the integration is w.r.t. $x$, and the ...
1
vote
1answer
74 views

Is there any infinite series representation of the sine integral?

Is there any infinite series representation of the sine integral? It is defined as $$\displaystyle \int\ \frac{\sin(x)}{x}\ dx$$
0
votes
2answers
57 views

Could a computer theoretically compute all integrals in terms of some special functions or it is not possible theoretically?

Could a computer theoretically compute all integrals in terms of some special functions or there need to be exist infinite number of such special functions to represent all integrals? I know there ...
0
votes
1answer
53 views

Fourier transform of Legendre

I am trying to figure out the Fourier transform of Legendre polynomial $P_\ell [\cos(\theta-a t )]$: $Q(\omega)=\int_{-\infty}^\infty P_\ell [\sin\phi\cos(\theta-a t )] e^{i \omega t} dt,$ where ...
2
votes
0answers
43 views

$\zeta(4)=\sum_ {k=1}^{\infty}{\frac{1}{k^4}}$ [duplicate]

How to Find $$\zeta(4)=\sum_ {k=1}^{\infty}{\frac{1}{k^4}}$$ the most basic way possible? I know it's $\pi^4/90$ but to arrive at this figure? Curious, because I need it to solve the integral ...
0
votes
1answer
56 views

Representation of heaviside step functions

Can the heaviside step function, $u(t)$ be represented like so: $$u(t)=\frac{1}{2}\left(\frac{|x|}{x}+1\right)$$
3
votes
0answers
56 views

Two properties about Bessel function

Let $J_\nu(x)$ be the Bessel function of the first kind. $\int_0^\infty J_\nu(x)dx=1 , (Re(\nu)>-1)$. $\lim_{\nu\to+\infty}J_\nu(x)=0$ for any fixed $x$. I think the above two properties of ...
8
votes
3answers
238 views

How prove this limit $\lim_{\alpha\to n}\dfrac{J_{\alpha}(x)\cos{(\alpha \pi)}-J_{-\alpha}(x)}{\sin{\alpha\pi}}$

let $$J_{\alpha}(x)=\sum_{m=0}^{\infty}\dfrac{(-1)^m}{m!\Gamma{(m+\alpha+1)}}\left(\dfrac{x}{2}\right)^{2m+\alpha}$$ show that: \begin{align*}&\lim_{\alpha\to n}\dfrac{J_{\alpha}(x)\cos{(\alpha ...
39
votes
2answers
396 views

Conjecture $_2F_1\left(\frac14,\frac34;\,\frac23;\,\frac13\right)=\frac1{\sqrt{\sqrt{\frac4{\sqrt{2-\sqrt[3]4}}+\sqrt[3]{4}+4}-\sqrt{2-\sqrt[3]4}-2}}$

Using a numerical search on my computer I discovered the following inequality: $$\left|\,{_2F_1}\left(\frac14,\frac34;\,\frac23;\,\frac13\right)-\rho\,\right|<10^{-20000},\tag1$$ where $\rho$ is ...
25
votes
1answer
305 views

Are elementary and generalized hypergeometric functions sufficient to express all algebraic numbers?

Are (integers) plus (elementary functions) plus (generalized hypergeometric functions) sufficient to represent any algebraic number? For example, the real algebraic number $\alpha\in(-1,0)$ ...
0
votes
0answers
62 views

Definite Integral of Modified Bessel function representation

I am trying to express the following integral of the Modified Bessel function either in closed form or even using other special functions. Any ideas ? $$ \int_{0}^{b}x\exp\left(-\,{x^{2} + z^{2} ...
18
votes
1answer
288 views

Derivative of the Meijer G-function with respect to one of its parameters

Are there any approaches that allow to find a derivative of the Meijer G-function with respect to one of its parameters in a closed form (or at least numerically with a high precision and in ...
20
votes
3answers
351 views

Integral $\int_0^\infty{_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right)\frac{dx}{1+4\,x}$

I need to evaluate this integral to a high precision: $$\large I=\int_0^\infty{_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right)\frac{dx}{1+4\,x}$$ Symbolic integration in ...
18
votes
2answers
250 views

A closed form for $\int_0^1{_2F_1}\left(-\frac{1}{4},\frac{5}{4};\,1;\,\frac{x}{2}\right)^2dx$

Is it possible to evaluate in a closed form integrals containing a squared hypergeometric function, like in this example? ...
7
votes
1answer
164 views

Simplification of $G_{2,4}^{4,2}\left(\frac18,\frac12\middle|\begin{array}{c}\frac12,\frac12\\0,0,\frac12,\frac12\\\end{array}\right)$

In this post Cleo gives a misterious result containing the following generalized Meijer G-function: ...
8
votes
0answers
76 views

Derivatives of the Struve functions $H_\nu(x)$, $L_\nu(x)$ and other related functions w.r.t. their index $\nu$

There are some known formulae for derivatives of the Bessel functions $J_\nu(x),\,$$Y_\nu(x),\,$$K_\nu(x),\,$$I_\nu(x)\,$with respect to their index $\nu$ for certain values of $\nu$, e.g. ...
7
votes
0answers
162 views

How find this value of $A$?

Question: Let $z\in C$ Find this value $A$,such $$\lim_{k\to +\infty}\left(k-\dfrac{W_{k^2}(z)}{W_{k}(z)}\right)= A\cdot i$$ where $i^2=-1$,and $w_{k}(z)$ is Lambert $W$ function:see ...
1
vote
1answer
69 views

Does a function that has an exponential analog to $log(xy) = log(x) + log(y)$ exist?

Similar to how $log(xy) = log(x) + log(y)$, does a nontrivial function exist that has the property $f(x^y) = f(x)f(y)$? How would one attempt to derive such a function?
19
votes
2answers
374 views

An integral $\int_0^\infty P_s(x-1)\,e^{-x}\,dx$ involving Legendre functions

Let $P_s(x)$ denote the Legendre functions of the $1^{st}$ kind, i.e. the Legendre polynomial generalized to an arbitrary (not necessarily integer) order $s$. It can be expressed using the ...
0
votes
2answers
171 views

Dirchlet Riemann Integrable in certain interval

Considering the Dirichlet function f : f (x) = { 1 if x is rational 0 if x is irrational } I want to know if this function can be Riemann Integrable in ...