Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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4
votes
0answers
68 views

Example in Help File does not evaluate as claimed

In the help file, under BellB, I read at "Properties and Relations": Sum can give results involving ...
7
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2answers
999 views

Approximation to the prime counting function

Is there a function similar to PrimePi that gives approximate value for large numbers? In particular, I need a reasonably good approximation for $\pi(x)$, where $x\...
5
votes
1answer
179 views

Possible bug / numerical issues with HypergeometricU -- any suggestions for a fast workaround?

I've encountered some problematic behaviour with HypergeometricU. I have a probability distribution on the positive integers that takes the following form after ...
0
votes
1answer
588 views

Solution of differential equation in terms of incomplete gamma function

I need help in solving equation 15 and 16 either manually or in Mathematica to get the solution in terms of the incomplete gamma function. This is what Mathematica tells me. I can't understand ...
1
vote
1answer
96 views

Improving working precision of LegendreP[n,x]? [duplicate]

I was trying to evaluate N[LegendreP[5,0.1]] The cell gives me: N[LegendreP[5,0.1]]=0.178829 However I wanted more ...
5
votes
1answer
347 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 $a$ : $a=0.5$ ,...
8
votes
2answers
684 views

On the definition of the associated Legendre polynomials

Mathematica computes for n = 1,2,...: (-1)^n (LegendreP[n, -1, -3]/Sqrt[2]) -I, -3 I, -11 I, -45 I, -197 I, ... Maple ...
2
votes
2answers
2k views

Find roots of a function involving Bessel functions

I'm trying to find the roots of a function involving Bessel functions. Here is my code ...
2
votes
1answer
1k views

Find solutions of equation involving Bessel functions

I'm new in Mathematica and I'm trying to find the solutions of this equation involving Bessel functions $$\eta \frac{ J_{n+1}(\eta a)}{J_n(\eta a)}+\chi \frac{ I_{n+1}(\chi a)}{I_n(\chi a)}=0$$ ...
10
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1answer
497 views

Mathieu function periodicity problem

Bug introduced in 9.0 or earlier and fixed in 10.3.0 This is a documentation mistake in MathieuCharacteristicA, ...
1
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0answers
120 views

How to evaluate sums that have 0^0 = 1

In my code: ...
8
votes
2answers
378 views

Why does N[Re@f] give complex result?

Consider this code: ...
3
votes
4answers
905 views

Hypergeometric function with a matrix argument

I am looking for the evaluation of a Hypergeometric function with a matrix argument as for example in Koev and Edelman or as showcased in this Wikipedia article. From what I understand from ...
6
votes
1answer
381 views

Numerical error in Mathieu functions

Consider the MathieuCharacteristicA function, which is a piecewise function according to the documentation. The discontinuity happens at integer number. ...
7
votes
1answer
1k views

Fredholm integral equation of the second kind with kernel containing Bessel and Struve functions

I need to solve this Fredholm integral equation of the second kind: f[s]+integrate[f[t] K[s,t],{t,0,1}]=s where 0<=s<=1...
6
votes
2answers
418 views

Hypergeometric function with large parameters

I need an efficient and accurate method to evaluate hypergeometric ratios of the form: $$\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)}$$ for large positive values of ...
2
votes
0answers
159 views

Getting long complex-valued integrals when simpler real-valued expressions exist

I have a long list of real-valued functions I'd like to integrate symbolically. For many of them, Mathematica gives me results with long complex-valued expressions involving weird functions such as <...
0
votes
0answers
50 views

Use FindRoot solution as number in function [duplicate]

I would like to save a number that I gain with FindRoot so that I can use it in another function. I tried: ...
7
votes
1answer
285 views

Symbolic integration of elliptic functions

Is there a clever way to integrate products of elliptic functions like $\wp(z;g_2,g_3)$ or $\zeta(z;g_2,g_3)$ in Mathematica? Mathematica seems to be able to integrate functions like ...
9
votes
3answers
294 views

Substituting values in 3j symbols

Bug introduced in 8.0 or earlier and fixed in 10.2.0 I believe these three expressions should give the same answer of -1/Sqrt[3], but the first gives ...
4
votes
1answer
334 views

a mysterious error in multiple symbolical integration

I am trying to do the following integration: ...
2
votes
0answers
63 views

How to avoid joining MathieuCharacteristicA in a plot? [duplicate]

In the documentation As a function of the first argument, MathieuCharacteristicA is a piecewise continuous function So how to avoid joining different pieces of the function? For example, ...
11
votes
1answer
837 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 ...
3
votes
1answer
269 views

Finding a root of a parameterized integral

I have a function given as a parameterized definite integral: f[a_] := Integrate[BesselJ[0, x - a] BesselJ[0, x + a], {x, -∞, ∞}] I suspect it has a root near <...
2
votes
1answer
170 views

Numerical evaluation of ChebyshevT

When I evaluate the following Chebyshev series of the first kind in two different ways, I get two very different results: ...
3
votes
2answers
492 views

Define PolyLog so that positive reals evaluate on upper edge of branch cut

Generalizing the previous question: Define Log so that negative reals evaluate on lower edge of branch For real positive values $x>1$, Mathematica's polylogarithm function ...
0
votes
1answer
983 views

How to calculate the unknown quantity in an infinite series?

I'd like to calculate x value in this equation. Basically, I tried to 2 types of method which are FindRoot and NSolve. But, I have failed the calculation caused by these errors up to now. If there ...
0
votes
1answer
838 views

Cannot solve equation

I just started using mathematica and I'm facing a problem that I just can't solve. I want to solve the following equation: Code: ...
4
votes
1answer
2k views

HarmonicNumber problem

I am looking for all the roots of HarmonicNumber, in the domain [-30, 1] and [0, 6000] for the real and imaginary parts, respectively, and where the parameter ...
1
vote
2answers
603 views

A regularized hypergeometric function related question

I'm interested in finding a way (if possible) of expressing this specific value of the regularized hypergeometric function in terms of known constants. How might I use Mathematica to check this ...
5
votes
1answer
155 views

Strange NSum behavior for slightly larger number of terms

For some sums, NSum gives me the result nicely for some number of terms, but if I add one more term, it suddenly becomes much slower. I haven't found the minimum example for which this can be ...
0
votes
1answer
1k views

How to use `FindRoot` to solve an equation containing a parameter?

I'm trying to derive some of the results of the following paper: Electrodynamics of semiconductor-coated noble metal nanoshells, JT Manassah - Physical Review A In the paper there is matrix $\mathbf ...
1
vote
0answers
157 views

Finding Values of All Nine Constants in q-hypergeometric Equation

I suspect that an identity of the form $$\sum_{j=-n}^{j=n}(-1)^jq^{j(5j+1)/2}=\sum_{r=0}^{n}f_{ar+bn+c}(-1)^{n-r}q^{(n-r)(dr+en+f)/2}\frac{(q;q)_{gr+hn+j}}{(q;q)_{n-r}}$$ for $a, \dots , j$ integer ...
2
votes
0answers
560 views

Computing a sum

I'm trying to make Mathematica compute this sum: Sum[(-1)^k (n - k)^2 Binomial[2 n, k], {k, 0, n}] As is, I get an awful formula: ...
5
votes
1answer
210 views

What is the formula Mathematica uses for ZetaZero?

What is the formula/algorithm Mathematica uses for the ZetaZero command?
0
votes
1answer
243 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}] ...
3
votes
2answers
2k views

Bessel derivatives

I tried to expand BesselJ[k,x] function into a Taylor series with Series command. Here both k...
1
vote
1answer
411 views

How to choose a solution of a differential equation?

I am trying to solve the following differential equation: ...
19
votes
1answer
4k views

Creating a Mathieu stability diagram

I am attempting to re-create a Mathieu stability diagram like the one shown in a paper by Leary and Schmidt [1]: I expected that I could use MathieuC to generate ...
10
votes
3answers
2k views

Are solid spherical harmonics implemented in Mathematica?

In certain applications, solid spherical harmonics can be very useful. They are essentially the usual, 'surface' spherical harmonics, with the appropriate power of the radius inserted: $$S_l^m(x,y,z)=...
3
votes
4answers
589 views

Evaluating Erf[x] in arbitrary precision

Is it possible to evaluate Erf[200] with arbitrary precision? I only get 1. as result but I would like to know if a arbitrary ...
5
votes
1answer
1k views

Is there a function for falling factorial in Mathematica

If I had to construct a function for falling factorial in mathematica I'd do something like that (hope I'm not mistaken): fallfact[x_,k_]:=$\prod_{j=0}^{k-1}(x-j)$ But is there a built-in function ...
11
votes
2answers
447 views

Binomial[-1,-1]

According to various sources e.g. http://www.proofwiki.org/wiki/Definition:Binomial_Coefficient and Wolfram themselves http://functions.wolfram.com/GammaBetaErf/Binomial/02/ , the binomial ...
7
votes
2answers
416 views

Symbolically Expanding One-Variable $q$-hypergeometric Series

In general, how do I get Mathematica to symbolically expand various infinite series to an arbitrary degree of accuracy? I deal with $q$-hypergeometric series quite frequently, and specifically want ...
0
votes
2answers
188 views

How can I compute the real part of $\zeta^2$ numerically? [duplicate]

I want to compute and plot $\Re(\zeta(x+iy)^2)$ and $\Im(\zeta(x+iy)^2)$. How can I do that with Mathematica?
3
votes
3answers
225 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 ...
0
votes
2answers
900 views

What is wrong in my calculation with Clebsch-Gordan coefficients?

According to theory of angular momentum here, this must equal one. What's wrong? ...
2
votes
1answer
343 views

Equation involving hypergeometric functions

I want to solve this equation but nor NSolve nor Solve are able to do this. ...
1
vote
1answer
478 views

How to calculate the following integral of the multiplication of two Bessel functions?

This integration has an analytical solution and its behavior is described by 1/r^2 function, but Mathematica gives some weird oscillating answer. Can anybody explain this and help me overcome this ...
1
vote
1answer
1k views

Trying to solve a transcendental equation involving Bessel functions

I've never used Mathematica before and am trying to numerically solve equation (12) from this paper. Ideally I'd be able to find the smallest value of $x_{n\nu}$ for $\exp(-kr\pi)$ close to 1, and ...