All Questions
Tagged with calculus-and-analysis special-functions
434 questions
3
votes
1
answer
140
views
How reliable is AsymptoticIntegrate?
In new mathematica 12 there is a new function AsymptoticIntegrate. However, it seems that it gives me incorrect results in some cases. To be completely honest, it ...
- 139
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 ...
2
votes
1
answer
238
views
Integral for Bhattacharyya distance between two Cauchy distributions
I need to perform the following integral to calculate the Bhattacharyya distance between two Cauchy distributions:
$$
I = \frac{\sqrt{b_+ b_-}}{\pi}\int_{-\infty}^{\infty}dx\,\frac{1}{\sqrt{\left[(x-1)...
- 335
0
votes
0
answers
33
views
Can't plot derivative of Hankel function [duplicate]
I am trying to plot the first derivative of $H^{(1)}_3 (ix)$ with respect to its argument $ix$, where $H^{(1)}_3 (ix)$ is the Hankel function of the first kind with order $3$, $x \in \mathbb{R}$ is a ...
- 315
2
votes
2
answers
1k
views
Integral giving a Dirac delta
I have the following type of integral
Integrate[ r BesselJ[n, a r] BesselJ[n, b r], {r, 0, Infinity}
(where a and ...
- 21
0
votes
1
answer
240
views
Partial derivative of an integral
I have the following function (it is the incomplete elliptic integral of first kind)
$$ F(b,g) = \int_{0}^{b} \frac{dx}{\sqrt{(1-x^2)(1-gx^2)}} $$
I would like to compute
$$\frac{\partial F}{\partial ...
- 133
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 <...
- 2,335
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 ...
- 43
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:
...
- 6,837
3
votes
1
answer
120
views
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 &...
- 233
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:
$$
K(k)=\frac{\pi}{2} \mathstrut_{2}F_{1}(\frac{1}{2},\frac{1}{2};1;k^{2})\\
E(k)=...
- 420
3
votes
1
answer
112
views
Error in Creating Orthogonal Polynomials
I'm trying to create my own set of polynomials orthogonal to weight $w(x)=x^{14}$ on $[-a,a]$. My code:
...
- 231
0
votes
1
answer
76
views
How to Integrate this expression?
All the parameters ($r, b,$ and $q$) are real and positive. Is it possible to do the below integration?
...
- 703
3
votes
1
answer
73
views
Non-Convergence In Creating Legendre Series
I'm trying to use Mathematica to create a Legendre-Fourier series using this Wikipedia article. Here is my code:
...
- 231
2
votes
1
answer
157
views
Evaluate numerically derivatives of hypergeometric functions
I would like to evaluate numerically the coefficients of a series expansion. This is usually straightforward to do, however in this case I encounter terms of the following type:
$$^{\phantom{0}}_2F_1^{...
- 711
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
$$
\int_0^\infty dx\, x^4\, \frac{e^{x^2+a}}{\left(e^{x^2+a}-1\right)^2} = \frac{3\sqrt{\pi}}{8}Li_{...
- 143
2
votes
2
answers
141
views
Different results with or without Assumptions in Integrate for an elliptic integral
Here are 2 examples I have examined.
1. Assumptions in Integrate.
...
- 1,892
4
votes
3
answers
306
views
Constant curvature surfaces. Revolution of the graphs of solutions to a nonlinear differential equation
I have the following differential equation:
$$\pi\cdot\text{y}(x)^2=\sqrt{1+\text{y}'(x)^2}\tag1$$
With the initial condition $\text{y}(0)=1$.
Now, I want to plot the solution in order to obtain the ...
- 2,021
1
vote
1
answer
51
views
Comparing two equivalent definite integrals
Reading this question on Math.SE, I tried the following Mathematica instructions
...
- 1,094
0
votes
0
answers
69
views
1
vote
0
answers
42
views
Taking the limit of the derivatives of the Beta function [closed]
I tried using Limit[D[Beta[a,b],{a,1},{b,1}],{a -> 1},{b -> 1}] and Limit[D[Beta[a,b],{a,1},{b,1}],{a , 1},{b , 1}]
but it ...
- 11
4
votes
1
answer
353
views
Verification of a general solution to d'Alembert equation
I solved a nonlinear differential equation (d'Alembert one) by hand. Mathematica gives the same answer.
But I am not able to get Mathematica to verify the solution due to branch cuts.
Any one knows of ...
- 151k
1
vote
1
answer
123
views
Find analytic solution for integral only defined for even integers
I would like to calculate the integral $$\int_0^{2\pi} dx \sin^6\left(\frac{x}{2}\right) F\left(\frac{4-n}{2}, \frac{4+n}{2}, \frac{1}{2}, \cos^2 \frac{x}{2} \right)^2$$
where $F$ is the ...
- 289
0
votes
1
answer
78
views
3
votes
2
answers
907
views
Why doesn't Integrate evaluate an elliptic integral?
My code is
Integrate[ 1/Sqrt[(x - a) (x - b) (x - c) (x - d)], {x, a, ∞},
Assumptions -> 0 < d < c < b < a]
I know this can be ...
- 133
5
votes
2
answers
242
views
Evaluate integrals with parameters
I'm new to Mathematica so I apologize if the answer to my question is trivial.
I need to calculate the following integral
$$ \int_{-\infty}^{0}\frac{dx}{\sqrt{(x-a)(x-b)(x-c)(x-d)}} $$
with
$$ 0 < ...
- 133
1
vote
0
answers
57
views
Slow convergence of Fresnel integral evaluation
I am trying to solve a problem from optics, a so-called Fresnel integral. The argument of integration is defined with a function:
...
- 131
0
votes
0
answers
93
views
Numerical comparison of two integrals and a function :
Consider the following integral:
$$S(q)=\int_{x=2}^q\sin^2\left(\frac{π\Gamma(x)}{2x}\right)dx$$
And consider the functions :
$$R(q)=\frac{q}{\log(q)}$$
$$T(q)=\int_2^q\frac{1}{\log(x)}dx$$
I ...
- 223
3
votes
2
answers
195
views
Two identities involving contour integrals in the presence of a branch point where the integrand explodes, and the Kummer function
I need to understand how to establish two identities. The first is
$$ \int_{C} z^{-1-q}(1-z)^{-1-\lambda } dz=\frac{2 \pi \Gamma (q+\lambda +1)}{\Gamma
(\lambda +1) \Gamma (q+1)}, q\geq 0, \...
- 2,166
0
votes
3
answers
421
views
Solving equations involving integrals
I need to find the value of $z$ for a particular value of $D_c$ (eg. $500$), but $z$ is inside an integral, and I'm not able to use Solve since the integral is ...
- 19
0
votes
1
answer
56
views
Substitute gives me different result
I'm working with Legendre polynomials (& associated ones). When I do the following calculation:
...
- 343
3
votes
1
answer
162
views
Integrating Gamma and Pochhammer functions
Could anybody explain why Mathematica gives different results with:
...
- 71
6
votes
4
answers
993
views
Evaluate the defining Integral of the Bessel functions of the first kind
I am trying to evaluate the integrals
$$ \int\limits_{-\pi}^{\pi} \mathrm{e}^{\mathrm{i}(x\sin t - nt)} \mathrm{d}t $$
and
$$ \int\limits_{-\pi}^{\pi} \mathrm{e}^{\mathrm{i}x\sin t} \mathrm{d}t $$
...
- 343
6
votes
1
answer
339
views
DSolve: unable to solve the conditions
I have a second order differential equation with $2$ boundary conditions and want to use the following code to solve it:
...
- 838
0
votes
0
answers
87
views
Bessel Function Integration
I am trying (with no luck) to integrate this pdf that has a Bessel function within it...
...
- 437
9
votes
2
answers
285
views
Possible bug involving derivative of BesselI
Bug introduced in 12.0 or earlier. Fixed in 13.2 or earlier.
In Mathematica 12.0, I run the following code:
f[x_] = BesselI[0, x];
f'[x]
which returns ...
- 226
0
votes
0
answers
128
views
Teach Mathematica analytical continuation of the gamma function
If I ask Mathematica to compute the gamma function for me
Integrate[Exp[-s] s^(a - 1), {s, 0, Infinity}]
It dutifully returns to me
...
9
votes
2
answers
2k
views
The time-like geodesics (orbits) in the Schwarzschild spacetime
I am trying to plot Schwarzschild's orbit without invoking the geodesic equation.
As a reference I am using Chandrasekhar's Book (The Mathematical theory of Black Holes, Oxford University Press). In ...
- 196
4
votes
3
answers
325
views
How to simplify this formula?
$$
\int \frac{1}{\sqrt{1-2 x^3}} \, dx
$$
Integrate[1/Sqrt[1 - 2 x^3], x]// FullSimplify
The result is very complex.
Then I want to differentiate it with the ...
- 10.4k
4
votes
3
answers
585
views
Why does Mathematica refuse to evaluate my integral?
Here is the integral for which I want a symbolic result:
Integrate[x^(z - 1)PolyLog[2, x]/(1 + x), {x, 0, 1}]
But the output is the same as the input without any ...
2
votes
0
answers
204
views
Radial Mathieu functions, divergence problem
I am working on a project that requires the utilization of even Mathieu functions.
This is the definition of my functions:
Even Mathieu function: ...
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 ...
- 175
2
votes
2
answers
291
views
Need help with using Bessel functions [closed]
I'm new to Bessel functions, especially those of the first kind. I'm working with a problem that goes as such:
With that said, is my code for said problem correct?
...
- 105
12
votes
3
answers
778
views
Elliptic Integrals: Mathematica and Gradshteyn and Ryzhik
In Gradshteyn and Ryzhik, (specifically starting with the section 3.13) there are several results involving integrals of polynomials inside square root. These are given in terms of combinations of ...
- 420
1
vote
0
answers
57
views
How to solve an Integral analytically using a predefined definition for Besselfunctions (phi part of angular spectrum representation)
I'd like to use mathematica to calculate an Integral that is dependent on phi and theta (to obtain the intensity distribution of a tightly focused TEM20 mode using the angular spectrum representation)....
- 21
7
votes
2
answers
413
views
How can I inform Mathematica of an identity concerning Bessel functions?
I am doing some analytical work that includes the integral of $e^{i(n\, t - x \sin t)}$. I know the result of this integral is a Bessel function.
$$J_n(x)=\frac1{2\pi}\int_{-\pi}^\pi e^{i(x\sin\tau-n\...
- 71
5
votes
3
answers
511
views
How to evaluate theta function's derivative numerically?
I ran into this derivative that Mathematica won't evaluate:
...
- 1,755
0
votes
0
answers
126
views
After activating, inactive integral, output is coming same as input
I am trying to plot s w.r.t r (0,10). But because of inactive integral I am not able to. When I activate inactive integral, output is coming same as input. When I am trying to plot graph w.r.t r(0 to ...
- 55
1
vote
0
answers
269
views
How do I solve the integral over four spherical harmonics?
I want to solve this integral
...