All Questions
Tagged with calculus-and-analysis special-functions
434 questions
4
votes
3
answers
562
views
Calculating $\lim_{x\to 1} \, \int -\frac{i \text{Li}_2\left(x-x^2\right)}{\sqrt{3} \left(x-\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)} \, dx$
How can we force Mathematica to compute this limit?
$$\lim_{x\to 1} \, \int -\frac{i \text{Li}_2\left(x-x^2\right)}{\sqrt{3} \left(x-\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)} \, dx$$
It seems it ...
- 62.1k
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:
$$
J_{m}\left( x \right) = \frac{ 1 }{ 2 \ \pi \ i^{m} } \int_{0}^{2 \pi} \ d\phi \ e^{i \left( x \cos{\phi} \ - \ m \ \phi \...
- 243
0
votes
1
answer
49
views
EllipticPi argument is complex and can not be plotted. How to handle this problem?
inttau[r_]=-(1/Sqrt[0.0345106153943703 - ((37.3042 - r) (-25.578 + r) (62.8822 +
r))/(3000 r)])
This is my function of r, now I integrated it w r t r
...
- 245k
0
votes
1
answer
33
views
Search for terms contatining error functions
question might have an answer in this post (Efficient Search for specific Terms in symbolic Expression) but i don't understand how to convert it to my specific case
Through some definite integrals I ...
- 163k
1
vote
1
answer
137
views
HurwitzLerchPhi
I am not sure why this is returned unevaluated:
HurwitzLerchPhi[1, 1, ∞]
Everything is returned unevaluated
...
- 67.7k
0
votes
1
answer
170
views
Integrate real function returns complex function [closed]
I want to compute the integral
$$
\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}...
- 56.9k
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 ...
- 163k
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(\...
- 2,914
12
votes
3
answers
639
views
Assumptions allowing to calculate an elliptic integral
When I feed Mathematica the following integral:
Integrate[Sqrt[(A - x) (B - x)/x], {x, 0, B}]
it spits it back out without evaluating it. However, it can ...
- 3,482
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 ...
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
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?)
...
- 10.4k
4
votes
1
answer
552
views
Mathematica Can Find the Primitive Function But It Cannot Find the Closed Form for Corresponding Definite Integral
Consider the following function
f = 1/2 (-3 + 4 u) Sqrt[-u (1 + u) (-1 - u + u^2)]
where it is assumed that
...
- 62.1k
10
votes
1
answer
923
views
Integrating over Bessel Function erroneous? (Hankel Transform)
Bug introduced in 8.0.4 or earlier and persists through 11.0.1 or later
The Hankel Transform is given by
Integrate[f[x] x BesselJ[0, x t], {x, 0, Infinity}]
It ...
1
vote
1
answer
200
views
How can I analytically integrate a BesselK function
I want to integrate the function below analytically, so that later on I can use the result for numerical calculations. But it seems Mathematica can not handle it the way I express it. However, if I ...
3
votes
1
answer
225
views
Evaluating an integral symbolically
I am trying to integrate following integral symbolically via integrate command:
$$(0.09)\Bigg[1 + \Bigg\{ \int_{3t-4k-7}^{2k+7/3} (0.09) \text{exp}\Bigg(\int_{3s-...
2
votes
1
answer
110
views
Verifying Summation form of Derivative of Hypergeometric1F1
First. Please read my code:
...
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-...
- 186
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
5
votes
1
answer
283
views
Bug: Wrong limit in Mathematica 11.1.1 and current WolframAlpha
Bug introduced around 11.1.1 and fixed in 11.2
...
12
votes
1
answer
394
views
Getting wrong limit with Bessel
Bug introduced in 7.0 or earlier and fixed in 11.3
I computed a following limit (related to the asymptotic expansion of the sequence A000009 - number of partitions ...
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 ...
- 67.7k
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.
...
- 50k
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 ...
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.
...
- 1,056
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 ...
- 2,914
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}...
- 3,482
0
votes
0
answers
33
views
Can't plot derivative of Hankel function [duplicate]
I am trying to plot the first derivative of $H^{(1)}_3 (ix)$ with respect to its argument $ix$, where $H^{(1)}_3 (ix)$ is the Hankel function of the first kind with order $3$, $x \in \mathbb{R}$ is a ...
4
votes
3
answers
306
views
Constant curvature surfaces. Revolution of the graphs of solutions to a nonlinear differential equation
I have the following differential equation:
$$\pi\cdot\text{y}(x)^2=\sqrt{1+\text{y}'(x)^2}\tag1$$
With the initial condition $\text{y}(0)=1$.
Now, I want to plot the solution in order to obtain the ...
- 57.9k
2
votes
1
answer
138
views
Cannot understand the meaning of Derivative[1, 0][BesselK][-M, 2]?
When I do the following integration
Integrate[(Log[x]/x)*x^M*Exp[-x-1/x],{x,0,\[Infinity]},Assumptions->Element[M,PositiveIntegers]]
Mathematica return a very ...
- 50k
2
votes
4
answers
201
views
Integration of LegendreP
I am trying to integrate a product of 2 Legendre polynomials as follows:
Integrate[LegendreP[1, x] LegendreP[2l+1, x], {x, -1, 1}]
I get the result:
...
7
votes
1
answer
266
views
Hypergeometric Function and Elliptic Integral
In the wiki page, complete elliptic integrals of first and second kind are related to the Hypergeometric function via:
$$
K(k)=\frac{\pi}{2} \mathstrut_{2}F_{1}(\frac{1}{2},\frac{1}{2};1;k^{2})\\
E(k)=...
2
votes
2
answers
1k
views
Integral giving a Dirac delta
I have the following type of integral
Integrate[ r BesselJ[n, a r] BesselJ[n, b r], {r, 0, Infinity}
(where a and ...
- 16.6k
1
vote
1
answer
416
views
Integral over Laguerre Polynomials
I would like to solve Integrals of the type
$\int_0^{\infty}dx e^{-x}x^{m+2}L^m_k(x)L^m_{k+j}(x)$,
for $m,k,j$ integers (and $m,k\geq0$).
On this page I found an expression which should do the job: ...
- 3,482
3
votes
3
answers
523
views
Summation of Legendre polynomials related to the zeta function
In the wake of my solution to discussion of the problem of summation of Legendre polynomials (https://math.stackexchange.com/questions/2111566/summation-of-legendre-polynomials-sum-l-2-infty-...
- 16.6k
3
votes
2
answers
228
views
Why can't I evaluate this integral and obtain a closed-form solution?
I have the following spherical density distribution:
$\rho(x, z) = \frac{1}{\sqrt{x^2 + z^2}\left(1+\sqrt{x^2+z^2}\right)^2}$
which I have broken into a "line of sight" dimension $z$ and a &...
CommunityBot
- 1
0
votes
1
answer
240
views
Partial derivative of an integral
I have the following function (it is the incomplete elliptic integral of first kind)
$$ F(b,g) = \int_{0}^{b} \frac{dx}{\sqrt{(1-x^2)(1-gx^2)}} $$
I would like to compute
$$\frac{\partial F}{\partial ...
- 8,448
4
votes
1
answer
158
views
Integration of product of BesselJ and BesselY not giving correct results
I am trying to integrate a product of Bessel functions as shown below. Where z is real valued and positive.
The integration yields MeijerG functions. Taking a ratio of the derivative of the MeijerG ...
- 245k
1
vote
2
answers
266
views
Why can't Mathematica evaluate this integral?
I want to work with the rectangle function, which I define by
f[x_, m_] := Limit[1/((2*(x - m))^(2*k) + 1), k -> Infinity];
(I know that in theory I can use <...
- 17.4k
3
votes
1
answer
120
views
Why does Integrate get this wrong?
Why does Integrate get this wrong?
...
3
votes
1
answer
112
views
Error in Creating Orthogonal Polynomials
I'm trying to create my own set of polynomials orthogonal to weight $w(x)=x^{14}$ on $[-a,a]$. My code:
...
- 245k
0
votes
1
answer
76
views
How to Integrate this expression?
All the parameters ($r, b,$ and $q$) are real and positive. Is it possible to do the below integration?
...
- 17.4k
3
votes
1
answer
73
views
Non-Convergence In Creating Legendre Series
I'm trying to use Mathematica to create a Legendre-Fourier series using this Wikipedia article. Here is my code:
...
- 8,448
0
votes
3
answers
421
views
Solving equations involving integrals
I need to find the value of $z$ for a particular value of $D_c$ (eg. $500$), but $z$ is inside an integral, and I'm not able to use Solve since the integral is ...
- 57.9k
2
votes
1
answer
157
views
Evaluate numerically derivatives of hypergeometric functions
I would like to evaluate numerically the coefficients of a series expansion. This is usually straightforward to do, however in this case I encounter terms of the following type:
$$^{\phantom{0}}_2F_1^{...
- 50k
4
votes
1
answer
301
views
Evaluating a hard integral related to the two-fluid model
The following definite integral describing the density of the normal part of a superfluid equals to
$$
\int_0^\infty dx\, x^4\, \frac{e^{x^2+a}}{\left(e^{x^2+a}-1\right)^2} = \frac{3\sqrt{\pi}}{8}Li_{...
- 57.9k
2
votes
2
answers
141
views
Different results with or without Assumptions in Integrate for an elliptic integral
Here are 2 examples I have examined.
1. Assumptions in Integrate.
...
- 57.9k
3
votes
3
answers
169
views
Transform integral to elliptic or hypergeometric forms?
I have two integrals that I suspect can be expressed as elliptic integrals:
$$\int_0^{2\pi} d\phi' \frac{1}{( 1 + \alpha^2 + \beta^2 - 2 \beta \cos(\phi') )^{3/2}} $$
$$\int_0^{2\pi} d\phi' \frac{\...
- 29.1k
0
votes
1
answer
185
views
Using Assumptions in Expressions that Evaluate to be Elliptic Integrals
I have the following integral that I am trying to evaluate in Mathematica:
$\int \sqrt{\alpha + m g l \cos(q)} dq$. If $\alpha > m g l$, then the result is a complete elliptic integral of the ...
- 15.3k