# Tagged Questions

Questions on the special mathematical functions implemented in Mathematica.

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### Is MathieuC for moderately large imaginary arguments broken?

I'm trying to plot MathieuC[-3,0.3,I x] for $x\in[0,10]$, and here's what I get even with arbitrary precision arithmetic (here I use ...
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### Using Assumptions in Expressions that Evaluate to be Elliptic Integrals

I have the following integral that I am trying to evaluate in Mathematica: $\int \sqrt{\alpha + m g l \cos(q)} dq$. If $\alpha > m g l$, then the result is a complete elliptic integral of the ...
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### Having trouble interpreting the results DSolve gives for the Laguerre equation

I am trying to solve a second order ODE using Mathematica. Before I get into solving my (more complicated) problem, I am trying to use DSolve on known ODEs to check that the answer that Mathematica ...
78 views

### Why minimization does not work with symbolic array as arguments

If I try to minimize with constraints a function of several variables, with Gamma regularized function involved in the constraints it seems to works, as shown below (this is just a dummy example ...
102 views

### How to define $\operatorname{Mc}$ and $\operatorname{Ms}$ Mathieu functions in terms of MathieuC and MathieuS?

Reading multiple books about Mathieu functions, I always come across notation like $\operatorname{ce}_r(z,q)$, $\operatorname{se}_r(z,q)$ for angular functions and $\operatorname{Ce}_r(z,q)$, ...
116 views

### Imaginary terms in the derivative of Jacobi theta function (2) on the real line

I am trying to calculate/plot the derivative of the second Jacobi theta function $d\theta_2(0, e^{-\pi t} )/dt$. Calculating or plotting the function itself works fine: ...
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### Why is the Spherical Bessel Function acting strangely at this point?

I'm doing some computation that requires the use of Spherical Bessel Functions of the 1st kind, at high orders and values. So, I managed to find this, while running it over a wide range of values. I ...
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### Computational Complexity of HypergeometricPFQ

What is the computational complexity of HypergeometricPFQ? I got a result of a product of a multinomial and HypergeometricPFQ and I was wondering if that would be considered a closed form solution ...
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### Sum over Binomials and Gammas

Given the function, ...
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### Expansion of a hypergeometric function takes ages with Mathematica 9 and 10 (regression?)

Mathematica 8 (Linux version) can evaluate AbsoluteTiming[Series[Hypergeometric2F1[1, 1 - eps/2, 3 - eps, 1/2], {eps, 0, 0}]] in no time. On one of the ...
257 views

### 3D Gamma function

I try to plot the following function: Plot3D[Gamma[1+0.5*(n+m)]/Sqrt[Gamma[1+n]*Gamma[1+m]],{n,0,1000},{m,0,1000}] I expect that for m=n and near to it the value ...
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### Complete definition of special functions in Mathematica [closed]

How can one display the actual integral involved in special functions like this MarcumQ[2, a, b] in Mathematica?
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### Perturbation theory with Mathematica: Definite integral of polynomial times exponential times hypergeometric function of imaginary argument

I would like to ask also Mathematica users about my question from the math forum. To expand, I'm adding the code which calculates the full double integral for $n=0$ and $\mu=0$ (the second in the ...
117 views

### What are the terms of the sequence generated by Zeta(3s)/Zeta(s)?

The LiouvilleLambda function has Dirichlet generating function of Zeta[2s]/Zeta[s]. I am curious about an analogous function with Dirichlet generating function of Zeta[3s]/Zeta[s]. Can Mathematica ...
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### Defining functions that depend upon other functions which depend upon other functions and so on

The problem is that I have to evaluate functions, that take output from previous functions, and combine them to form new functions, and do some operations and evaluate the value at new function. But, ...
117 views

### Is there an easy way to let mathematica print out every Erfc and InverseErfc as F and F^{-1}

Mathematica uses complementary error function and its inverse as functions for example when integral of a Gaussian is taken. Therefore, all output expressions of Mathematica involve Erfc and ...
127 views

### Solving equation containing Erf expressions

Given the equation below, how do I find the value of b so that the function is equal to 21. I tried solve but I keep getting an error message. ...
118 views

### Trouble using Solve and NSolve with functions involving Erf

I have the following functions: R[k_, x_, t_] := -.5*(k - x)*(1 + Erf[-(k - x)/t]) L[k_, c_, x_, t_] := .5*c*(k - x)*(1 + Erf[(k - x)/t]) I'm interested in ...
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### How to find and verify relationships between functions?

MMa gave me a complicated result involving Hypergeometric0F1's. It was much less complicated after I discovered this identity: ...
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### NSolve can not handle a PolyGamma equation

I would like to solve the following equations numerically: ...
128 views

### Computing Poincaré symbolic solution for an arbitrary integer order polynomial

In the 1880s, PoincarĂ© created functions which give the solution to the nth order polynomial equation in finite form. These functions turned out to be "natural" generalizations of the elliptic ...
58 views

### Plotting the Schwarz D minimal surface

With the aid of the following paper I'm trying to plot the Schwarz D minimal surface: paper. So far I have followed all the instructions to the best of my abilities and have come up with the following ...
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### Does Mathematica support Laguerre Polynomials of Matrix Argument?

I am wondering whether Mathematica can also compute the Laguerre polynomials of matrix argument as the appear in multivariate statistics? For example, they appear in this paper here: ...
129 views

### NIntegrate and Integrate of a Spherical Bessel function

I am trying to integrate over a spherical bessel function. I have used both the Integrate and NIntegrate functions in Mathematica but the values given by each do not match. Any reason why this ...
151 views

### Probability of multivariate normal being positive on each coordinate

How can I find the probability that each coordinate of a specified multivariate normal distribution is positive? I tried the following, which I believed should work ...
116 views

### Bug in associated Legendre Polynomials?

Mathematica's definition of the connection of associated Legendre polynomials with $m$ and $-m$ is: $P_l^{-m}=(-1)^m \frac{(l-m)!}{(l+m)!} P_l^m$. We also now that $|m|>l \Rightarrow P_l^m=0$. ...
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### EllipticF reduction

Is it possible to reduce the following Z to Legendre form Elliptic integral of the First kind? $$\int \dfrac {\sec u\; du } {\sqrt{ (1- {(\nu \tan u)}^2 }} ..(1*)$$ With ...
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### Why is PolyLog[] giving weird answers for ordinary values? [duplicate]

Possibly related to this question, but it seems slightly different: Strange behaviour of PolyLog Function Wikipedia says that for real s, z<1 should be real. So I was confused when MMa returned: ...
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### Strange timings of integrals involving Hermite's polynomials

I have used Mathematica to calculate tunneling for quantum harmonic oscillator. The code is simple: ...
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### How to make the result of Piecewise be a closed interval?

Description This question comes from two questions. Namely Q1 and Q2 The defintion of B-Spline basis function as shown below: Let $\vec{U}=\{u_0,u_1,\ldots,u_m\}$ a nondecreasing sequence of real ...
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### Integrating a compound expression

I have an integral of the form I[r]=âˆ«(arExp[-r]-brSin[k(r-d)]Exp[-r])BesselJ[0,kr]dr where Besse1J[0,kr] is the modified ...
530 views

I found a nice paper about inverse vector operators here. I have successfully implemented a Mathematica function for most of them, however I can't figure out how to do inverse gradient (page 7 in the ...
175 views

### Number of divisors visualized with the QPochhammer function, how improve performance of code?

I have this code that is originally Jeffrey Stopple's code for the Riemann zeta function in the complex plane. Because I discovered yesterday that the number of divisors can be generated with the ...
106 views

### Using Fold to show stages of Euler product formula

I would like to recursively replicate each stage of the Euler product proof. This does it: ...
95 views

### Is there a LogBeta function like the LogGamma?

In a computation, I need Log[Beta[alpha+j,beta-j+n]] of some kind. Is there a LogBeta function built-in to avoid any under/over ...
122 views

### Get rid of Error Function: How to get rid of sequential appearances of error function?

We have a function as e[t_] :=(E^(-t^2)) Cos[0.1 t] and we must evaluate below integration (However I used the variable x ...
126 views

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 ...
189 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 ...
583 views

### Define and plot a piecewise function with parameters [closed]

I have this function , but i do not know how to define it in mathematica f_t(x)=% \begin{cases} i& \text{$\frac{1}{1+i}<|x-t|\le \frac{1}{i}$} \\ ...
68 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 WolframAlpha it gives back an explicit form, the one you would get if you were to do it by ...
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### Real or Imaginary result of spherical bessel and hankel functions of imaginary arguments

I am trying to calculate a rather complicate expression involving Spherical Bessel and Hankel functions. My problem is that somehow for pure imaginary arguments the functions are not pure real or ...
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### 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 ...
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 $\infty$), ...
### 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 ...