Questions on the special mathematical functions implemented in Mathematica.

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4
votes
2answers
144 views

Inverse of LogIntegral

I wanted the inverse logarithmic intgral, so I typed InverseFunction[LogIntegral] and received the expected symbolic answer. But when I try to integrate it or ...
5
votes
2answers
358 views

Lower branch of Lambert W function in mathematica

I am interested in values of the Lambert W function, which is defined as the solution to the equation $ z = W(z) e^{W(z)} . $ The solution is not, however, single-valued, but branches into two ...
0
votes
0answers
96 views

Spherical Hankel and Bessel in Explicit form

This is probably an easy question, when I type the spherical Hankel (first kind) and the Bessel function into Wolfram Alpha, it gives back an explicit form, the one you would get if you were to do it ...
5
votes
0answers
110 views

MacDonald formula for Modified Bessel Functions

How can I make Mathematica understand these two integrals? $$\int_0^{\infty} e^{-x \cosh{\xi}} d\xi = K_0(x)$$ $$\int_0^{\infty} e^{-\frac{1}{2} \Big( \frac{x y}{u} + u \frac{x^2+y^2}{x y} \Big) } K_{...
0
votes
2answers
113 views

Is this answer true?

I was using Mathematica to get the series solution for Legendre equation. But when I get the recurrence relation and use RSolve: ...
7
votes
2answers
924 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. ...
4
votes
1answer
278 views

Root finding: zeroes of Mathieu function

I am finding the roots of the Mathieu sine function, and find Mathematica and Maple do not agree on the solutions. For example, consider the solutions of ...
1
vote
2answers
153 views

Numerical integration of modified bessel function

I need to compute the following integral: NIntegrate[ BesselI[-nu, k x]/x ,{x, r1, r}] in which nu=-(2m-1)/2 and I have to ...
5
votes
1answer
311 views

Inverse error function

I solved some equation in Mathematica and I obtained something like $$y(t)=\exp \left\lbrace \left[ \text{erf}^{-1} (\text{i}t) \right]^2\right\rbrace, (1)$$ where $\text{i}$ is imaginary unit and $\...
0
votes
0answers
45 views

Should this be equal to Gamma function or not?

I define the following function: Nat2[s_] := InverseFourierTransform[FourierTransform[1/t, t, w] (-I w)^(s - 1), w, x]/Cos[Pi (s - 1)] /. x -> 1 I expected ...
1
vote
1answer
322 views

Plotting Integral equation

I want to plot the following indefinite integral : $C_l^{CC}=\int k^2\mathrm{d}k\: [e^{-2k^{2}}P_{Cl}^2(k\eta)|\dot{h}(\eta)|^2]$ with k from 0 to some large value (considered to be $\infty$), where:...
7
votes
2answers
136 views

`FindSequenceFunction` for sum of hypergeometric terms?

The built-in function FindSequenceFunction is quite good at recognizing hypergeometric terms, i.e. terms $c_k$ for which $c_{k+1}/c_k$ is a rational function of $k$....
4
votes
1answer
99 views

FreeQ and arguments of Hypergeometric2F1

I was trying to use FreeQ to test for the presence of Hypergeometric2F1 functions in my expressions. I encountered the following ...
6
votes
0answers
175 views

Fine tuning compiled code that computes dilogarithm function

As an exercise of writing a good Compile function, I want to do the simple task of coding a routine that outputs the real part of the dilogarithm function ...
2
votes
0answers
201 views

Mathematica not evaluating q derivative of Jacobi theta function

Jacobi theta functions, $\theta_a(u,q)$ for $a=1,2,3,4$ are defined in the unit disk $|q|<1$. For some reason that I would like to understand, Mathematica does not evaluate numerically the $q$ ...
2
votes
1answer
253 views

Definite Integral over Bessel Function

Hello I am interested in evaluating the following integral. ...
4
votes
0answers
60 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 ...
4
votes
1answer
125 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
217 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 ...
2
votes
1answer
73 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
262 views

How to plot the result of this singular integral?

Please I open a new post here after this one : http://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
305 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
648 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
279 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 $$\eta \frac{ J_{n+1}(\eta a)}{J_n(\eta a)}+\chi \frac{ I_{n+1}(\chi a)}{I_n(\chi a)}=0$$ ...
9
votes
1answer
251 views

Mathieu function periodicity problem

According to the documentation, the Mathieu characteristic function generates parameter a: The characteristic value Subscript[a, r] gives the value of the parameter a in y′′+(a-2q cos(2z))y=0 ...
7
votes
2answers
241 views

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

Consider this code: ...
3
votes
3answers
378 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
218 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. ...
4
votes
1answer
453 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...
3
votes
1answer
106 views

Hypergeometric function with large parameters [duplicate]

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 ...
6
votes
1answer
172 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 ...
10
votes
1answer
354 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 ...
2
votes
1answer
128 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
121 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: ...
2
votes
2answers
212 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
249 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
170 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: ...
3
votes
1answer
692 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
263 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 ...
0
votes
1answer
493 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 ...
4
votes
1answer
144 views

What is the formula Mathematica uses for ZetaZero?

What is the formula/algorithm Mathematica uses for the ZetaZero command?
3
votes
2answers
481 views

Bessel derivatives

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

How to choose a solution of a differential equation?

I am trying to solve the following differential equation: ...
1
vote
2answers
582 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
242 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 ...
1
vote
1answer
328 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 ...
5
votes
2answers
242 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
143 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
176 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 ...
2
votes
1answer
152 views

Equation involving hypergeometric functions

I want to solve this equation but nor NSolve nor Solve are able to do this. ...