Questions on the special mathematical functions implemented in Mathematica.

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18
votes
1answer
461 views

Incorrect result from Integrate

I attempted to calculate the following integral: ...
5
votes
1answer
371 views

How to express an expression with only ArcTan and ArcTanh?

I have an expression which is simply (j/k) x^(j/k) LerchPhi[x,1,j/k)] where 0 < j < k. Manually I have been able ...
-1
votes
1answer
361 views

Gamma Incomplete Function representation in Mathematica and Matlab

I am confused about Gamma Incomplete function calculation in Mathematica and MatLab: For example, in Mathematica: Gamma[5,3] = 19.56 But in MatLab: ...
3
votes
1answer
201 views

When does the real part of Zeta vanish on the critical line?

This seems to be a quite a simple problem but I cannot make it work. I am trying to find all values within a given range for which the real part of the Zeta function vanishes on the line: ...
3
votes
2answers
275 views

Assigning an analytical approximation to the error function erf(x)

Working with some iterative integral equations I have Gaussian density functions involved therein. Integrating the gaussian function I obrain the error function. When I take the second integration, ...
4
votes
2answers
284 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 ...
2
votes
2answers
881 views

Q-Function representation in Mathematica

I observed that there is no Q-function representation in Mathematica. The definition of Q-Function is: \begin{align} Q(x) &= \frac{1}{\sqrt{2\pi}} \int_x^\infty e^{-\frac{u^2}{2}}du \\ ...
0
votes
1answer
199 views

Why does Mathematica find a form for this general sum, but not for some special cases?

Today I found a Sum which Mathematica will simplify for a general parameter value $\nu$, but which will not simplify fully for special cases $\nu = 1, 3, ...$ despite the fact that the general answer ...
15
votes
1answer
393 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: ...
7
votes
2answers
302 views

Find asymptotics of Sum[2^i*Binomial[n-i-1,2*n/3-1],{i,0,n/3}]

I have an expression 2^n / Sum[ 2^i Binomial[ n - i - 1, 2n/3 - 1], { i, 0, n/3}] ...
1
vote
0answers
158 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, ...
3
votes
2answers
297 views

Integrating special functions

I would like to integrate the following function with Legendre polynomial and Gamma function. I am open to suggestions. ...
3
votes
1answer
195 views

Symbolic Integration of Special Functions

Sorry in advance if this formatting comes out strange, this is my first question! I can't find a way to integrate, e.g., a function of the Hermite polynomials for general (still integer) order. For ...
3
votes
0answers
55 views

SiegelTheta fails to evaluate when given proper arguments

SiegelTheta often returns error messages when I give it arguments that should be of the correct form. For instance, I have a numerical matrix like ...
5
votes
2answers
300 views

Numerical Integration with InverseErfc

I am trying to numerically integrate an equation that involves InverseErfc (embedded in the copula defined). The equation looks like the following: $$ \int_0^T \int_0^\infty \int_0^\infty ...
2
votes
0answers
159 views

Real integral evaluating as indeterminate

Mathematica evaluates the following integral as: ...
5
votes
1answer
227 views

Accurately evaluating the hypergeometric function

As part of another problem, I am working to evaluate hypergeometric functions such as Hypergeometric2F1[1, 1, n, -1] for large $n$. I am hoping to obtain at ...
8
votes
1answer
455 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 ...
5
votes
1answer
156 views

How can I program the RiemannR function using the LogIntegral command?

I would like to program the RiemannR function using the LogIntegral command because I would like to later experiment with a ...
0
votes
1answer
382 views

Searching for roots of complex function

I'm searching for roots of complex function $$ 2\imath q \ln(-2\imath k)+\imath\pi-2\imath \Im(\ln(\Gamma(1+2\imath q)))+\ln(\frac{\Gamma(1+\imath q-\imath q x/k)}{\Gamma(1-\imath q-\imath q ...
13
votes
2answers
361 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 ...
7
votes
1answer
528 views

Visualizing vector-spherical waves

This is a follow-up question to this one on visualizing vector-spherical harmonics. This time, I would like to visualize the vector spherical waves (including the radial dependence). The functions ...
2
votes
3answers
205 views

Why do these two different zetas produce the same value?

Zeta[-13] == Zeta[-1] == -1/12 Why do these two different zetas produce the same value?
8
votes
2answers
238 views

How to simplify an expression with special functions to zero

The following is a well-known Bessel function identity: $$J_{-n}(z)=(-1)^n J_n(z),\qquad n\in\mathbb Z$$ To check this, I used the following code and the result is as what I expected. ...
16
votes
2answers
2k views

Visualizing vector spherical harmonics

I have painstakingly derived the vector-spherical harmonics $\mathbf{V}_{J,\,M}^\ell(\theta, \phi)$, which are the generalization of ordinary spherical harmonics $Y_\ell^m(\theta, \phi)$ to vector ...
9
votes
1answer
536 views

Why can't I change the value of MaxRecursion in NIntegrate when integrating BesselJ?

I am trying to evaluate this integral numerically $$ \int_0^{\infty } J_0(q R) \tanh(q) \, \mathrm{d}q $$ for large values of $R$. This makes the integrand oscillate more quickly and Mathematica ...
9
votes
1answer
314 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. ...
2
votes
0answers
102 views

RSolve and incomplete gamma function

I am interested in how Mathematica uses the incomplete gamma function to solve difference equations. For example, if we have this inhomogeneous, 2nd order equation and use ...
7
votes
2answers
316 views

Problem with NIntegrate when WorkingPrecision is specified

I am trying to evaluate this integral numerically: $$ \int_0^{\infty } m \exp (-m) J_1(m){}^2 \, dm $$ Everything is OK when only the integration method is specified: ...
3
votes
3answers
303 views

Mathematica not able to confirm its own solution to differential equation

I type the following into Mathematica: DSolve[q''[x] + 2 x/(x^2 - 1) q'[x] - 4*q[x]/(x^2 - 1) == 0, q[x], x] It gives me the result ...
18
votes
1answer
761 views

Peirce's quincuncial projection

The Peirce quincuncial projection is the cartographic projection of a sphere onto a square. In short, I would like to see it implemented in Mathematica. Here is my code: ...
0
votes
1answer
109 views

Reflection transform of function [duplicate]

I am trying to find the reflection function. Here is my function and its graph. ...
1
vote
3answers
1k views

How to take conjugate of a function?

Naïvely this is what happens and it obviously is not helpful! ...
3
votes
1answer
121 views

Find point at which equation stops having roots (if it exists)

I am interested in the roots of this function: f[M_, b_] := 1 - (2 M Gamma[2, 0, (1/M + b M)/Sqrt[b]])/(1/M + b M) for fixed values of b. In particular I want ...
5
votes
2answers
179 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 ...
5
votes
5answers
1k views

Series expansion in terms of Hermite polynomials

I am trying to expand a polynomial in terms of orthogonal polynomials (in my case, Hermite). Maple has a nice built-in function for this, ChangeBasis. Is there a ...
5
votes
2answers
316 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 ...
1
vote
0answers
121 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 ...
4
votes
1answer
248 views

Simplifying numerical expressions involving special functions

I've encountered the following problem. There is the identity that the special functions EllipticK[x] and EllipticE[x] satisfy: ...
3
votes
1answer
227 views

Strange behaviour of PolyLog Function

I discovered some strange behaviour of the PolyLog[] Function in Mathematica which seems to me like a bug in the function implementation. It looks like ...
1
vote
2answers
680 views

Irregular Confluent Hypergeometric Functions (Spherical Coulomb Wavefunctions)

I want to program in the regular and irregular spherical Coulomb wavefunctions $F_\ell(\gamma,kr)$ and $G_\ell(\gamma,kr)$, respectively, which are defined in terms of the regular and irregular ...
5
votes
0answers
103 views

Expansion of $E(i c \mid m)$ at $c\to\infty$?

Currently, I am using a Windows machine with Mathematica 8. I noticed a difference in a series expansion of the function EllipticE[] in comparison with a result ...
0
votes
2answers
525 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 ...
6
votes
2answers
283 views

Cannot Get Numerical Results to Match

I try this numerical summation (in two parts) ...
6
votes
0answers
326 views

Reproducing the Integral Definition of the Modified Bessel function

I need to simplify some integral expressions in terms of special functions, such as the modified Bessel function of the first kind. See for example Eq. (5) on this page. Notice that the real ...
2
votes
1answer
273 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 ...
1
vote
1answer
364 views

Plotting Fresnel function

I am trying to plot the partial sums and the cesaro means of the function $\sqrt{|x|}$ and for $a_{n}$, I obtained the following code which contains FresnelS. ...
9
votes
2answers
2k views

problem with coloring spherical harmonics

I want to color a spherical harmonics. So I write as follows. ...