Questions on the special mathematical functions implemented in Mathematica.
28
votes
2answers
912 views
What is the difference between a few simplification techniques?
I am trying to understand the difference between Refine, Simplify and FullSimplify, and when ...
25
votes
3answers
449 views
Is there a Mathematica API for the functions.wolfram site?
Is there a Mathematica API for the functions.wolfram site?
If there's not, has anyone implemented a web scraper for it?
For example it would be nice to be able to access
...
16
votes
1answer
269 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:
...
15
votes
0answers
353 views
Fast Spherical Harmonics radiative transfer
This is a rather specific question and I apologize for spamming you with some lengthy code. But it could be interesting for some reader and maybe you can help out, so please bear with me.
I am using ...
12
votes
2answers
348 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 ...
12
votes
1answer
237 views
Extra factors appear when evaluating Euler integrals
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 give
$$B(z+1,z+1)^2 = ...
11
votes
1answer
436 views
Incorrect results for elementary integrals when using Integrate
There is a rather simple integral ($K_0$ is the 0-th order MacDonald function)
$$\int_0^\infty e^{-x \cosh\xi}\, d\xi = K_0(x)$$
which mathematica cannot solve. This even though the documentation ...
10
votes
6answers
2k views
About multi-root search in Mathematica for transcendental equations
I have some questions for multiroot search for transcendental equations. Is there any clever solution to find all the roots for a transcendental equation in a specific range?
Perhaps ...
10
votes
4answers
690 views
Numerical underflow for a scaled error function
I calculate scaled error function defined as
f[x_] := Erfc[x]*Exp[x^2]
but it can not calculate f[30000.]. ...
10
votes
2answers
164 views
What kind of hypergeometric function is it?
I found a formula for an integral of a product of three Bessel functions at The Wolfram Functions Site:
I cannot understand what kind of hypergeometric function it is.
The Mathematica code given ...
10
votes
1answer
279 views
Is my expression too complicated for FullSimplify or am I doing something wrong?
I have a messily defined function $v(h, w)$ with $h, w \in \mathbb{R}$ and with a removable singularity at $h=1/2$, and I am trying to prove some of its properties using Mathematica. In particular I ...
10
votes
1answer
219 views
Why can't I change the value of MaxRecursion in NIntegrate when integrating BesselJ?
I am trying to evaluate this integral numerically
$$
\int_0^{\infty } J_0(q R) \tanh(q) \, \mathrm{d}q
$$
for large values of $R$. This makes the integrand oscillate more quickly and Mathematica ...
9
votes
3answers
291 views
Multi-Factorial and Series with Triple-factorial terms
Let $n!^{(k)}$ denote a multi-factorial which is defined by
$$
n!^{(k)} = \begin{cases}
1 & n \leqslant 0, \\
n, & 0 < n < ...
8
votes
2answers
134 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:
...
8
votes
1answer
460 views
solve an integral equation numerically
I am trying to find a numerical solution for an equation of the form:
$$
f(t)=\int_{t_{min}}^{t} {\exp[2 (t^\prime - t)] E(t^\prime) f(t^\prime)}
+ \int_{t}^{0}{ \exp(t^\prime - t) E(t^\prime) ...
7
votes
4answers
435 views
Finding the roots of Hypergeometric1F1[]
I am trying to find the roots, λ, for this equation:
Hypergeometric1F1[1/4 (2 - λ /β), n + 1, β]
for certain ...
7
votes
2answers
521 views
problem with coloring spherical harmonics
I want to color a spherical harmonics. So I write as follows.
...
7
votes
3answers
446 views
Checking if the roots of a function are real
I'm trying to determine if the roots of a function are real. How would you do that?
(In particular I'm interested in verifying that the roots of LegendreP[6, x] ...
7
votes
2answers
101 views
How to simplify an expression with special functions to zero
The following is a well-known Bessel function identity:
$$J_{-n}(z)=(-1)^n J_n(z),\qquad n\in\mathbb Z$$
To check this, I used the following code and the result is as what I expected.
...
7
votes
1answer
98 views
Visualizing vector-spherical waves
This is a follow-up question to this one on visualizing vector-spherical harmonics. This time, I would like to visualize the vector spherical waves (including the radial dependence). The functions ...
7
votes
1answer
341 views
Implementation of Incomplete Fermi-Dirac Integral in Mathematica
I'm working on a special algorithm to implement a more accurate effective mass calculation for hole carriers in silicon in Mathematica. This rather involved algorithm uses incomplete Fermi-Dirac ...
6
votes
2answers
399 views
6
votes
1answer
83 views
How to calculate this integral? Integrate[BesselJ[0, x - BesselJZero[0, 1]]/x, {x, -Infinity, Infinity}]
I tried to calculate the following integral, but it returned unevaluated.
...
6
votes
1answer
414 views
Hankel Transform integrals won't work in Mathematica
I'm trying to do this integral, which is shown on the Wikipedia page on the Hankel transformation:
$$\int_0^{2\pi}\mathrm d\varphi\;e^{\mathrm im\varphi}e^{\mathrm ikr\cos(\varphi)}$$
The answer is ...
6
votes
2answers
138 views
6
votes
0answers
252 views
Inverse Laplace transform not obtained
I can't seem to be able to compute the inverse Laplace transform of a Laplace transform:
...
5
votes
5answers
301 views
Series expansion in terms of Hermite polynomials
I am trying to expand a polynomial in terms of orthogonal polynomials (in my case, Hermite). Maple has a nice built-in function for this, ChangeBasis.
Is there a ...
5
votes
2answers
193 views
Solve for $a$ as a function of $\beta$?
I am trying to solve this equation:
$$\beta^{-a} \Gamma(a) \sin(a \pi) +
e^\beta \beta^{2 a - 1} \Gamma(1 - a) \sin(a \pi) = 0$$
I tried the following:
...
5
votes
2answers
109 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
...
5
votes
2answers
135 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
0answers
173 views
5
votes
0answers
81 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 ...
5
votes
0answers
233 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 ...
4
votes
1answer
98 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:
...
4
votes
2answers
140 views
Problem with EllipticE documentation
The complete elliptic integral of the second kind, EllipticE, is defined as,
Integrate[Sqrt[1-m Sin[t]^2],{t,0,z}]
According ...
3
votes
2answers
109 views
LevinRule and SphericalBessels
I'm currently looking at a simplified problem that approximates another problem I'm looking into. In this simplified problem I at least have an analytic integrand and can easily provide all info on ...
3
votes
2answers
216 views
3
votes
3answers
145 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
...
3
votes
1answer
99 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 ...
3
votes
1answer
75 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 ...
3
votes
0answers
110 views
Why doesn't Log[Gamma[]] simplify to LogGamma[] where it could?
I have been playing with various equations involving amount of permutations in relatively large sets. Easiest way to look at these is something like Log[10, bignumber!] . Often expressions, even ...
2
votes
3answers
130 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?
2
votes
0answers
42 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 ...
2
votes
1answer
149 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
2answers
163 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 ...
1
vote
1answer
168 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.
...
1
vote
2answers
77 views
How to take conjugate of a function?
Naïvely this is what happens and it obviously is not helpful!
...
1
vote
0answers
69 views
Getting poles of a Gamma functions
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 ...
1
vote
0answers
105 views
A curious double zeta evaluation [closed]
While investigating the evaluation of a double Euler sum (a.k.a. a double zeta function),
$$\zeta(r,s)=\sum_{j=1}^\infty \sum_{k=1}^{j-1}\frac1{j^s k^r}$$
in Mathematica, I chanced upon most ...
0
votes
1answer
195 views
How can I define a Step-Wise function in Mathematica (Not using Heaviside Step Function)? [closed]
I need to define a function, which has very different behaviour in different regions. There are about 13 different regions. A sample of my function is the foloowing table:
I want to define it as a ...
