All Questions
Tagged with calculus-and-analysis special-functions
434 questions
2
votes
2
answers
374
views
Integration by parts for deriving gamma function
Well, there is an integral that has quite a lot to do with $n!$, and that is the following :
$f(n)$ = Integrate[x^(n - 1)/E^x, {x, 0, Infinity}] =$\int_0^{\infty } \...
- 669
2
votes
1
answer
84
views
Radial integral involving WhittakerM and Hypergeometric0F1
I am trying to find the definite integral of $$I=\int_{0}^{\infty}dx\left(\frac{\sqrt{n^{2l-1}}M_{n,l+\frac{1}{2}}(\frac{2x}{n})}{\Gamma(2l+2)}\right)x\left(\frac{(2r)^{l_f+1} {}_0F_1(2l_f+2;-2r)}{\...
- 23
4
votes
1
answer
196
views
How to calculate the sum of the series of Hermite polynomial?
I want to calculate the infinite sum of Hermite polynomials, which works fine with older versions of Mathematica, but with version 13 it doesn't.
The infinite sum is:
...
- 43
0
votes
3
answers
728
views
Solving elliptic integrals in Mathematica
I have an integral
$$\int_{a2}^{a1}\frac{dx}{\sqrt{(a1 -x)(a2 - x)(a3 - x)}}$$
And I'm trying to integrate it with
...
- 157
2
votes
1
answer
134
views
How to calculate an `InverseMellinTransform` up to its definition in Mathematica?
I have a question about how to calculate Inverse Mellin Transformation's in Mathematica, one way or the other.
Look at these findings.
The following integral results in Gamma[s].
...
- 9,687
4
votes
2
answers
468
views
How do I get other representations of the Gamma function?
In my studymaterial I see different definitions of the Gamma function
In Mathematica it is for example:
...
- 669
5
votes
1
answer
211
views
How to get the expression of a special function?
How to get the expression of a special function? Like gamma function:
Gamma[z]=Integrate[t^(z-1)/e^t, {t, 0, Infinity}]
- 2,425
3
votes
2
answers
294
views
Convergence of a complete elliptic integral
How to prove that the following integral
assuming $k^2 < 1$ $$\int_{0}^{1} \frac{d x}{\sqrt{1-x^2}\sqrt{1-k^2 x^2}}$$ is convergent?
...
- 2,425
0
votes
1
answer
446
views
Numerical integration of a function with Dirac delta
I have a little question about the numerical integration of a function that includes a Dirac delta.
I have the following function:
...
0
votes
1
answer
62
views
Two equivalent functions give two different indefinite integrals [closed]
I consider the following indefinite integral
$\int \frac{1}{\sqrt{P^2 s^{-2p} +1}} ds$
for $p,P>0$. By multiplying the integrand with $\frac{s^p}{s^p}$ this can be written as
$\int \frac{s^p}{\sqrt{...
- 101
0
votes
0
answers
108
views
Calculating complex integral representation of the Hankel function of the first kind
I was wondering if any one can help me calculate this integral using Mathematica.
This is a form of integral representation of Hankel function of first kind:
...
- 420
1
vote
2
answers
259
views
Error complex function ERFI(X): looking for alternative function representations?
I have some analytical results from a physics problem, where the Mathematica gives the results in terms of complex error function. I would like to explore another function representation using ...
- 545
1
vote
0
answers
67
views
Why does Mathematica only solve this indefinite integral for two values of n?
I'm trying to estimate this integrate:
where "a" and "b" are real constants and n> 0.
I use the code
...
- 11
1
vote
0
answers
77
views
How to use only real values in a notebook?
I am wondering how to make Mathematica only evaluate real numbers by creating an initialization line in the beginning of a notebook.
I would like to prevent the following:
complex trigonometric ...
- 1,765
4
votes
3
answers
483
views
Why can't Mathematica evaluate the limit of this hypergeometric function? Is there any way to find the answer?
I have the following function
...
- 1,067
1
vote
1
answer
196
views
Strange result for an integral containing Meijer G-function
The question is about Mathematica's result for the integral
$$ \int_0^\infty \frac{16}{\pi^2} \left( \frac{\sqrt{\pi}}{4} r^2
%
G_{1,3}^{2,1}\left(\frac{r^2}{4}\Bigg|
\begin{array}{l}
0 \\
0,0,-\...
- 155
0
votes
1
answer
144
views
Expectation Value of a Potential Using Radial Solution of Hydrogen Atom
I am trying to input $R_{nl}$, which is the radial solution of the hydrogen atom and I would like to obtain an expectation value of particular potential. This is my code:
...
3
votes
1
answer
218
views
Closed Form of Integration [duplicate]
I have tried this integral in Mathematica,
Assuming[m > 0,Integrate[Cos[m k] Exp[ Cos[k]], {k, 0, 2 Pi}]]
And Mathematica failed to perform this. Whereas if any ...
- 33
2
votes
1
answer
160
views
Integrate of HypergeometricPFQ gives the wrong result
Good morning,
I computed the following integral using Integrate in version 12.2 and it gives the wrong result. Can you help me understand what I am doing wrong? Here is the integral:
...
- 31
0
votes
1
answer
87
views
How do I code up nested derivatives?
I want expressions for the spherical Bessel functions in terms of sinusoids using Rayleigh's formulae:
$$
j_{n}(x)=(-x)^{n}\left(\frac{1}{x}\frac{d}{dx}\right)^{n}\frac{\sin(x)}{x}\\
y_{n}(x)=-(-x)^{n}...
- 1,043
0
votes
1
answer
191
views
Elliptic Integral simplification
Integrate[Sqrt[(roh^2 - r^2)/(r^2 - rb^2)], {r, rb, roh},
Assumptions -> r > rb \[And] roh > rb \[And] roh > 1]
Outputs to:
...
- 3,234
1
vote
0
answers
76
views
New symbollicaly integral operator [closed]
I have an u=u(x,t) that is an unknown function.
Suppose that the function u can be written as follows:
**
...(Eq. 1)
**
In here, $\Psi(x)$ is an Nx1 known vector, and C is an NxN matrix. (N is an ...
- 543
4
votes
1
answer
348
views
Proof associated Legendre polynomials are orthogonal: integral doesn't solve
Wondering why Mathematica can't solve this integral:
Integrate[
LegendreP[l1, 1, x] *
LegendreP[l2, 1, x], {x, -1, 1}]
Mathematica outputs: $\int_{-1}^1 P_{\...
- 181
4
votes
1
answer
106
views
MeijerG function: numerical evaluation of its derivative
Mathematica does not evaluate the derivative of the following MeijerG function.
...
- 103
2
votes
2
answers
171
views
Unpacking a Mathematica expression returned by DSolve
I was trying to solve a system of differential equations in Mathematica and had troubles understanding what the solution looked like. So I wanted help to unpack it.
I had a system of two coupled ...
3
votes
0
answers
123
views
Spherical harmonics Y (l,m,theta,phi) for general l, m
I am trying to solve integrals involving spherical harmonics Y(l,m, theta, phi) and their derivatives. I do not have any particular l,m, theta, phi values. I need to solve it for general l,m. When I ...
- 307
0
votes
2
answers
374
views
Integrating an exponential with upper incomplete gamma functions
I would greatly appreciate calculating an integral consisting of an upper incomplete gamma function and an exponential function.
...
- 21
6
votes
3
answers
392
views
Express MeijerG as integral
For definite integrals MMA gives identities in terms of Meijer G-functions, e.g.
$\begin{align}\sqrt{\pi}\int_0^\infty \textrm{e}^{-4x/t^2-t}\ \textrm{d}t &=
G_{0,\,3}^{3,\,0} \left( x\left.
\...
- 593
3
votes
1
answer
251
views
Is this a bug in mathematica for integrals of multiple error functions?
I'm scratching my head over the the following result in Mathematica (v11.3)
I'm considering the function
B = Erfc[x] Exp[-x^2/2] + Sqrt[2] Erfc[x/Sqrt[2]] Exp[-x^2]
...
- 155
1
vote
0
answers
65
views
Why is this integrand not integrating to a Bessel function? [duplicate]
I know from the identities of Bessel functions that the following is true:
$$
J_{m}\left( x \right) = \frac{ 1 }{ 2 \ \pi \ i^{m} } \int_{0}^{2 \pi} \ d\phi \ e^{i \left( x \cos{\phi} \ - \ m \ \phi \...
- 243
0
votes
1
answer
49
views
EllipticPi argument is complex and can not be plotted. How to handle this problem?
inttau[r_]=-(1/Sqrt[0.0345106153943703 - ((37.3042 - r) (-25.578 + r) (62.8822 +
r))/(3000 r)])
This is my function of r, now I integrated it w r t r
...
- 175
0
votes
1
answer
33
views
Search for terms contatining error functions
question might have an answer in this post (Efficient Search for specific Terms in symbolic Expression) but i don't understand how to convert it to my specific case
Through some definite integrals I ...
- 155
1
vote
1
answer
129
views
why integrating over error function gives nothing?
I try to integrate over an error function and evaluate the following integral
...
- 1,308
1
vote
1
answer
137
views
HurwitzLerchPhi
I am not sure why this is returned unevaluated:
HurwitzLerchPhi[1, 1, ∞]
Everything is returned unevaluated
...
- 1,708
0
votes
1
answer
170
views
Integrate real function returns complex function [closed]
I want to compute the integral
$$
\int_0^c \exp(-cx+x^2) \mathrm{d}x,
$$
where $c>0$ is an unknown constant. In Mathematica Version 12.2.0
...
- 239
1
vote
1
answer
127
views
Evaluating this generalised integral
I have the following integral
$$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\exp \left(a u^2+b v^2+c u v\right) \; dvdu,$$
which returns the following solution:
$$\frac{2 \pi }{\sqrt{4 a b-c^2}...
- 997
1
vote
2
answers
72
views
Error in Integration of special functions using mathematica 12.0
When I try to integrate the following,
Integrate[-GegenbauerC[22,-1/2,x]/(1+k*x),{x,-1,1}]
where -1<k<1 and k!=0, Mathematica gives different results if I ...
- 11
0
votes
0
answers
53
views
Evaluation of a double summation invovlving hypergeometric and exponential functions
I am trying to compute the following double summation over the indices, $m$ and $n$, which involves the hypergeometric function, ${}_2 F_1$, an exponential function and, factorials as a part of a ...
- 85
1
vote
1
answer
81
views
limit of an expression including BesselK function
i want to calculate the limit of the following expression when 'w' tend to zero. I have used the Limit function, it takes a lot of time for running without any result. could you please help me how to ...
1
vote
1
answer
158
views
How does Mathematica evaluate these sum and integral?
How does Mathematica internally evaluate the following (interrelated) sum and integral, and how does it do the subsequent simplification? (I mean, based on what mathematical facts?)
...
- 855
2
votes
1
answer
205
views
Program for efficient computation of given functional:
I need to plot the following functional with accuracy:
$$
I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy,s) − F(x −\mathrm iy,s)}{\mathrm e^{2πy}-1},
$$
Where $ F(z,s) = \dfrac{1}{z^s\Gamma(\...
- 223
4
votes
1
answer
157
views
How to expand Lie characters?
The following involves characters of affine Lie algebras, and I will be using as reference the book on CFT by Francesco et at (here are some screenshots if useful). But hopefully the post will be self-...
2
votes
1
answer
110
views
2
votes
0
answers
101
views
How to do a fast numerical computation of an oscillatory integral including HeunC function using Mathematica?
I am trying to numerically compute the following integral in Mathematica
...
- 119
0
votes
0
answers
89
views
How to simplifying the following integral that involves Bessel functions and Exponential integral function?
I have obtained the following as a solution of heat conduction equation of a semi-infinite model in cylindrical coordinates which is initially subject to non-homogenous initial condition and isolated ...
- 43
1
vote
1
answer
202
views
How to simplify the following integral to be in terms of Bessel functions?
I have evaluated the following integration using Mathematica. I obtained a solution in terms of Meijer G function. I wonder if it can be simplified to be in terms of Bessel functions.
...
- 43
0
votes
0
answers
58
views
How to evaluate the following integrals using Mathematica?
I have the following integrals obtained during solving heat diffusion equation for semi-infinite system that is subject to non-homogenous initial condition in Laplace domain. I want to simplify the ...
- 43
1
vote
2
answers
332
views
How to prove the following integration identity?
I have the following integration that I want to evaluate it using Mathematica.
...
- 43
7
votes
1
answer
256
views
Keeping Phase Factors in Sqrt
I am trying to plot certain holomorphic functions that contain square and higher roots. In the complex analysis sense, the function $f:z\mapsto z^\alpha$ for some $\alpha\in\mathbb C$ has a phase ...
- 287
3
votes
2
answers
1k
views
Calculating a double integral
I want to calculate the following integral:
$$\int^{10}_{0}\int^{\pi}_{0}\sqrt{(37-\frac{45\cdot37\cdot x^2}{74\cdot 150})^2\cdot \sin(t)^2-(40-\frac{27\cdot37\cdot x^2}{16\cdot 150})^2\cdot \cos(t)^2}...
- 39