All Questions
Tagged with calculus-and-analysis special-functions
83 questions
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
5
votes
3
answers
619
views
Integrating a real function I get a complex value, while after variable transformation the result is real. Bug?
I have the following integral:
Integrate[1/Sqrt[0.7 + 0.3*(1 + z)^3], {z, 0, ∞}, Assumptions -> z ∈ Reals]
>> -3.36354 - 3.85013 I
The output is complex, ...
- 1,201
2
votes
3
answers
485
views
Optimization of ODE with respect to the initial condition
One has a (system) of ODEs with a one-parameter family of initial conditions. For example,
...
- 207
22
votes
2
answers
2k
views
Incorrect results for elementary integrals when using Integrate
Bug introduced in 8.0 or earlier and persisting through 13.2 or later
There is a rather simple integral ($K_0$ is the 0-th order MacDonald function)
$$\int_0^\infty e^{-x \cosh\xi}\, d\xi = K_0(x)$$
...
- 1,424
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
1
vote
3
answers
726
views
Problem solving Third order non-linear differential equation in Mathematica
I am trying to find an analytical solution of the following 3rd order non-linear differential equation in Mathematica: $a (f'(x))^2+f'''(x)=0$ with boundary conditions $f(0)=0$, $f'(0)=0$, $f(1)=1$, $...
17
votes
2
answers
1k
views
What kind of hypergeometric function is it?
I found a formula for an integral of a product of three Bessel functions at The Wolfram Functions Site:
I cannot understand what kind of hypergeometric function it is.
The Mathematica code given for ...
- 7,333
10
votes
1
answer
853
views
Mathematica implementation of Zeilberger's algorithm (previously done in Maple)
I have this Mathematica code:
...
- 2,385
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
9
votes
2
answers
877
views
Analytical approximation of indefinite integral on a given interval to a given precision
I'm looking for an analytical approximation of
$\int_a^b e^{-x^2}\mathrm{erf}(x+c) dx$
that would be accurate to precision $\varepsilon$ for $a,b,c$ within a certain range. How do I ask Mathematica ...
- 767
23
votes
2
answers
2k
views
Why does Mathematica give the wrong answer when integrating?
Bug introduced in 8.0 or earlier and fixed in 9.0.0
I integrate
Integrate[Exp[I Cos[b - c]] Cos[b], {b, 0, 2 Pi}]
Mathematica gives:
...
- 993
20
votes
1
answer
1k
views
Incorrect result from Integrate
Bug introduced in 8.0 and fixed in 10.0
I attempted to calculate the following integral:
...
- 11.9k
19
votes
1
answer
1k
views
How to represent a continuous monotonic phase of Airy functions?
Note: In this question I am concerned only with real-valued variables and functions.
DLMF, §9.8 Airy Functions, Modulus and Phase, formula $9.8.4$ defines the phase of Airy functions:
$$\theta(x)=\...
- 7,333
15
votes
2
answers
1k
views
SphericalHarmonicY does not seem to be an eigenfunction of the spherical harmonic equation
I applied the spherical harmonic equation on the SphericalHarmonicY functions like this:
...
- 743
13
votes
3
answers
544
views
Find asymptotics of $\sum\limits_{i=0}^{n/3} 2^i \binom{n-i-1}{\frac{2n}{3}-1}$
I have an expression
2^n / Sum[ 2^i Binomial[ n - i - 1, 2n/3 - 1], { i, 0, n/3}]
...
- 1,119
13
votes
1
answer
341
views
Wrong Limit with LaguerreL
Bug introduced in 7.0 and fixed in 10.2.0
Limit[(n! LaguerreL[n,-1]/(n^(n+1/4)/Sqrt[2]/E^(n-2 Sqrt[n]+1/2))-1) Sqrt[n], n->∞]
Mathematica (wrong) output
<...
- 3,369
11
votes
1
answer
1k
views
Expanding derivatives of hypergeometric functions
Sometimes Mathematica expresses results of integration or summation in terms of symbolic derivatives of Hypergeometric2F1 function, and cannot further simplify ...
- 7,333
9
votes
1
answer
1k
views
Hankel Transform integrals won't work in Mathematica
I'm trying to do this integral, which is shown on the Wikipedia page on the Hankel transformation:
$$\int_0^{2\pi}\mathrm d\varphi\;e^{\mathrm im\varphi}e^{\mathrm ikr\cos(\varphi)}$$
The answer is ...
user854688
7
votes
1
answer
252
views
Analytical form of 2d integrals relevant to graphene
This question is continuation of my previous post.
Alex Trounev was very helpful in fixing a crucial typo in the analytic solution known from the article "Density Dependent Exchange Contribution ...
- 19.8k
7
votes
1
answer
266
views
Hypergeometric Function and Elliptic Integral
In the wiki page, complete elliptic integrals of first and second kind are related to the Hypergeometric function via:
$$
K(k)=\frac{\pi}{2} \mathstrut_{2}F_{1}(\frac{1}{2},\frac{1}{2};1;k^{2})\\
E(k)=...
- 420
6
votes
1
answer
168
views
Numerical computation of Caputo dervative
When trying to evaluate
CaputoD[MittagLefflerE[1/2, -t^(1/2)]^2, {t, 1/2}]
it returns
...
- 399
5
votes
1
answer
366
views
How to plot the result of this singular integral?
Please I open a new post here after this one : https://mathematica.stackexchange.com/a/59203/10158
Now I want to plot the function $f(a,b)$ as a function of $b$ for different values of $a$ : $a=0.5$ ,...
- 425
5
votes
2
answers
322
views
Only perform a symbolic differentiation once
I want to define a function that involves a differentiation step that Mathematica can do easily, which might be of the form
...
- 10.3k
4
votes
2
answers
205
views
Smallest positive real solution with InverseWeierstrassP
How can I get the smallest positive real solution with
InverseWeierstrassP[p, {g2, g3}] for real $g_2, g_3$ (if it exists)?
The documentation just says, that the ...
- 221
4
votes
0
answers
151
views
Puzzled by Assumptions [duplicate]
I don't know if this has already been discussed.
Integrate[BesselJ[2 m + 1, x], {x, 0, ∞}, Assumptions -> m ϵ Integers]
...
- 4,894
4
votes
1
answer
122
views
Transform expression involving Erfc back and forth with Laplace transform and its inversion
It's a problem I encountered when answering this question and I think it's worth starting a new question for it. Consider the following expressions:
...
- 68.4k
4
votes
2
answers
1k
views
How do I evaluate a symbolic integral involving Hermite polynomials?
I want to test a difficult integral : Integral on all reals of some complicated function involving the Hermitian polynomials, exponentials, squares, factorials, and being general considering any ...
- 63
4
votes
1
answer
246
views
Getting a series expansion for implicitly defined function
I have a function $f(r)$ defined as $x$ satisfying $g(r,x)=0$.
How do I get a series expansion of $f(r)$ around $r=\infty$?
Function $f(r)$ below is unexpectedly linear, and I'm trying to get a closed ...
- 6,697
3
votes
0
answers
497
views
Integrating the Associated Legendre Polynomials
I know the following identity:
$\qquad \int_{-1}^1 P_l^m(t)^2dt=\frac{2(m+n)!}{(2n+1)(n-m)!}$
I would like to verify this result using Mathematica. This is what I entered:
...
- 131
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
3
votes
1
answer
2k
views
Integrate Squared Legendre Polynomial
With the same purpose as this question, I wish to evaluate an integral that contains the squared Legendre Polynomials.
$\int_{-1}^{1}\left[P_n(x)\right]^2dx=\frac{2}{2n+1}$
I tried evaluating with ...
- 155
2
votes
2
answers
139
views
Derivative of integrated noise Gaussian likelihood
In a Bayesian problem with Gaussian likelihood with mean $\mu$ and a uniform prior on the standard deviation $\sigma$, it is possible to derive the marginal posterior (where $\sigma$ has been ...
- 3,167
2
votes
1
answer
158
views
How to speed up the calculation of a multi-dimension matrix involving symbolic integral?
The following program succeeds in getting matrix CC, but it takes time badly, especially in the case varNumber becomes larger just as the following ...
- 693
1
vote
2
answers
276
views
How to define a function that is related to derivative of Jacobi theta function
I would like to make 3D plot of the following function.
...
- 335
1
vote
3
answers
1k
views
Calculation of an integral with a Bessel function
How can I calculate below integral? Is it possible to calculate it in Mathematica? If yes, how? Am I doing something wrong?
...
- 331
0
votes
2
answers
1k
views
Using Mathematica to find poles of Gamma functions
I am concerned about the expression on the RHS of equation A.5 (page 19) in this paper:
$$\int\frac{d^d q}{(q^2)^{\nu_1}[(\vec{k}-\vec{q})^2]^{\nu_2}}=\frac{\Gamma(d/2-\nu_1)\Gamma(d/2-\nu_2)\Gamma(\...
- 227
16
votes
3
answers
1k
views
How do I numerically evaluate and plot the Fabius function?
The Fabius function is a well-known example in analysis of a non-analytic function that is infinitely differentiable. I want to be able to numerically evaluate the function for any real argument, as ...
- 161
13
votes
2
answers
661
views
Asymptotics of $\frac{\sum _{i=0}^{\lfloor n/2 \rfloor} {2(n-2i) \choose n-2i} {n \choose 2i} {4i \choose 2i}}{2^{3n - 1}}$
I am fairly sure that asymptotically $$\frac{\sum _{i=0}^{\lfloor n/2 \rfloor} {2(n-2i) \choose n-2i} {n \choose 2i} {4i \choose 2i}}{2^{3n - 1}} \sim \frac{2}{\pi n}.$$
I tried
Limit[n*Sum[...
- 1,119
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
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 ...
- 121
11
votes
1
answer
418
views
Vastly incorrect answers obtained by increasing WorkingPrecision with modified Bessel functions
Bug introduced in 7.0 or earlier and fixed in 11.0
This is a follow-up to this question regarding numerical instabilities occurring with modified Bessel functions. In trying to explore J.M.'s answer ...
- 743
11
votes
3
answers
421
views
Integrating a BesselJ integrand to obtain the same result as Maple 16
I would like to check the following integration:
Integrate[y Integrate[1/x BesselJ[1, x Exp[I π/4]] BesselJ[1, x Exp[-I π/4]],
{x, 0, y}], {y, 0, r}]
...
- 145
10
votes
3
answers
971
views
Integral Too Hard For Mathematica
I have a monstrous integral that I desperately want to solve with Mathematica. It takes the form of:
...
- 581
8
votes
4
answers
622
views
Inverse Laplace Transform of Hypergeometric Function
Any tips how to massage the following to get computed by Mathematica for $p>1$? I suspect the result should be expressible in terms of exponential integral
...
- 6,697
8
votes
3
answers
2k
views
Use Meijer-G function to represent elementary functions
I want to represent these elementary functions: $x^{2}\sqrt{x}$, $\sin{4x}$, and $x\ln{x}$ as cases of MeijerG. What arguments should I give to ...
- 183
7
votes
2
answers
1k
views
Mathematica 10 cannot solve definite integral [duplicate]
Bug introduced in 10.0 and fixed in 10.0.2
Mathematica 10 fails to solve the following integral, saying that it does not converge.
...
- 73
6
votes
2
answers
342
views
Why do these identical limits give different results?
Bug introduced in 12.0.0.0. and persisting through 14.0.0.0.
I wanted to calculate this limit:
...
- 3,369
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
6
votes
1
answer
423
views
Bad performance of Integrate (and WolframAlpha) for an Integral of Bessel function of the first kind: Version 11 edit
Version 11 Edit
The issue still remains:
Integrate[BesselJ[0, x], {x, 0, ∞}] // Timing
(* {29.8125, 1} *)
$Version
(* "11.3.0 for Microsoft Windows (64-bit)...
- 4,894
6
votes
2
answers
412
views
Imaginary terms in the derivative of Jacobi theta function (2) on the real line
I am trying to calculate/plot the derivative of the second Jacobi theta function $d\theta_2(0, e^{-\pi t} )/dt$.
Calculating or plotting the function itself works fine:
...
- 61