All Questions
Tagged with calculus-and-analysis special-functions
434 questions
0
votes
1
answer
264
views
Symbolic integral including Hermite polynomial does not evaluate
I am trying to evaluate the following integral:
Integrate[HermiteH[n, Sqrt[a]* x] * Exp[- (c/2)* (x^2 + y^2) + b* x * y - (a * x^2)/2], {x, - Infinity, Infinity}]
...
- 103
3
votes
3
answers
236
views
Symbolic derivatives of special functions yield incorrect results
When I evaluate with Mathematica this expression
D[ Abs[ Zeta[x + I y]], {x, 2}] + D[ Abs[ Zeta[x + I y]], {y, 2}]
0
the ...
- 61
4
votes
0
answers
598
views
Spherical harmonic derivative
Consider the following substitution
Derivative[2, 0][S][th, ph] /. S -> Function[{th, ph}, SphericalHarmonicY[3, 0, th, ph]]
which gives correct answer. While ...
- 3,528
1
vote
1
answer
1k
views
Error Function Integral (Erf)
Any idea how to solve analytically this integral
Integrate[(a Erf[a Sqrt[b/(a^2 + b)] c])/(a^2 + b)^(3/2), a]
I tried substitution u=a^2 + b, but it didn't work. ...
- 367
3
votes
1
answer
760
views
A Bessel & Struve functions related integral
I try to numerically compute this integral and I don't figure out why on earth Mathematica is not able
to do it. Is my input correct? Does it possibly have a closed form?
...
- 1,415
3
votes
1
answer
1k
views
Integral over squared Hermite polynomial
I would like to calculate the uncertainty of the nth Eigenstate of a 1-dim harmonic oscillator. To obtain the result I have to compute the integral
$$\int_{-\infty}^{\infty} \psi^* x^2 \psi \:dx\,,$$ ...
- 393
8
votes
1
answer
623
views
Prove an identity in quantum harmonic oscillator
Problem:
In the context of quantum harmonic oscillator the eigenfunctions are given by:
$
u_n(x) = (N_n/\sqrt{b}) H_n(x/b) \exp\left[-x^2/(2b^2)\right]
$, where $N_n$ is the normalization factor:
$
...
- 3,074
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
5
votes
1
answer
653
views
Strange result for the analytic integration leads to Hypergeometric2F1
The integration result for
Integrate[1/(r^2 Sqrt[x/r^(4 - 2 γ) + 1]), r]
is:
...
- 1,067
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
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
3
votes
0
answers
100
views
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:
$$\theta(x)=\...
- 7,333
13
votes
3
answers
544
views
Find asymptotics of $\sum\limits_{i=0}^{n/3} 2^i \binom{n-i-1}{\frac{2n}{3}-1}$
I have an expression
2^n / Sum[ 2^i Binomial[ n - i - 1, 2n/3 - 1], { i, 0, n/3}]
...
- 1,119
3
votes
0
answers
528
views
How to prevent simplification of hypergeometric functions resulting from integrations?
Definite integrals from 0 to Infinity over a product of two hypergeometric (including exponential, trigonometric, hyperbolic, ...
4
votes
2
answers
625
views
Integrating special functions
I would like to integrate the following function with Legendre polynomial and Gamma function. I am open to suggestions.
...
- 435
3
votes
1
answer
276
views
9
votes
2
answers
2k
views
Numerical contour integrations in the complex plane - contour deformation gives different answer for analytic kernel
I am trying to do a contour integration in Mathematica numerically. In particular, I'm checking the identity:
$$ H_m^{(1)}(z) =\frac{i^{-m}}{\pi}\int_{-\pi/2 + i \infty}^{\pi/2 - i \infty} \exp[i m \...
- 390
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
9
votes
1
answer
518
views
How to calculate this integral? Integrate[BesselJ[0, x - BesselJZero[0, 1]]/x, {x, -Infinity, Infinity}]
I tried to calculate the following integral, but it returned unevaluated.
...
- 503
5
votes
2
answers
220
views
Why does Integrate return a solution that is not defined at a particular point when it actually is well defined at that point?
I am trying to compute
Integrate[Sqrt[x^4 + (y - y^2)^2], {x, 0, y}]
Mathematica 8 gives
...
- 1,119
7
votes
2
answers
714
views
Mathematica cannot calculate a limit
When I evaluate
Limit[E^(-n)*Sum[n^k/(k!),{k,0,n}], n -> ∞]
Mathematica gives me the result
...
- 161
1
vote
0
answers
154
views
Getting poles of a Gamma functions [duplicate]
Why do the following 2 sequences give different answers?
n = 1.5
Series[Gamma[0.5 - n - x], {x, 0, 2}]
Series[Gamma[-1 - x], {x, 0, 2}]
(..clearly the output from the second expression is correct ...
- 1,191
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
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:
$$\int\frac{d^d q}{(q^2)^{\nu_1}[(\vec{k}-\vec{q})^2]^{\nu_2}}=\frac{\Gamma(d/2-\nu_1)\Gamma(d/2-\nu_2)\Gamma(\...
- 227
2
votes
1
answer
496
views
Integrate[(1 + x/n)^n*Exp[-x], {x, 0, Infinity}]
I evaluated
Integrate[(1 + x/n)^n*Exp[-x], {x, 0, Infinity}] .
thinking the answer should be approximately $\sqrt{\pi n/2}$. Mathematica gave me
...
- 143
4
votes
1
answer
723
views
Directional derivative of SiegelTheta
I'm working on a problem where I have to integrate both the Mathematica function SiegelTheta and some of its second order directional derivatives. Using the function works well but something goes ...
- 43
10
votes
1
answer
923
views
Integrating over Bessel Function erroneous? (Hankel Transform)
Bug introduced in 8.0.4 or earlier and persists through 11.0.1 or later
The Hankel Transform is given by
Integrate[f[x] x BesselJ[0, x t], {x, 0, Infinity}]
It ...
- 345
19
votes
1
answer
851
views
Speeding up trigonometric integral
Context
On a possible non trivial toric topology for the Universe (nothing less!).
Problem
I would like to carry out the following integral for $\ell=2,4\cdots 20$.
$$\int _0^{\pi }\int _0^{2 \pi }...
- 23.1k
13
votes
1
answer
384
views
Extra factors appear when evaluating Euler integrals
Note: this is fixed in version 9.
When I perform the double integral in Mathematica,
Integrate[(x (1 - x))^z (y (1 - y))^z, {x, 0, 1}, {y, 0, 1}]
which should ...
- 161
3
votes
2
answers
466
views
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
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
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 ...
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