Questions on the special mathematical functions implemented in Mathematica.

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0
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0answers
24 views

Why constrained minimization involving gamma function does not work with an expression as argument

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 ...
3
votes
0answers
42 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 $\operatorname{Ce}_r(z,q)$, ...
2
votes
1answer
59 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: ...
1
vote
1answer
66 views

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 ...
6
votes
1answer
82 views

Sum over Binomials and Gammas

Given the function, ...
-1
votes
2answers
74 views

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?
1
vote
1answer
41 views

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, ...
1
vote
1answer
86 views

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: ...
0
votes
2answers
94 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 ...
2
votes
2answers
107 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 ...
0
votes
1answer
50 views

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 ...
3
votes
1answer
71 views

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 ...
2
votes
1answer
272 views

Inverse gradient operator

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 ...
2
votes
2answers
103 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: ...
1
vote
1answer
78 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 ...
4
votes
2answers
116 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 ...
3
votes
2answers
119 views

Lower branch of Lambert W function in mathematica

I am interested in values of Lambert W function, which is defined as the solution to equation $ z = W(z) e^{W(z)} . $ The solution is not, however, single valued, but branches into two solution for $z ...
3
votes
1answer
287 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 \begin{equation} f_t(x)=% \begin{cases} i& \text{$\frac{1}{1+i}<|x-t|\le \frac{1}{i}$} \\ ...
0
votes
0answers
61 views

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 ...
1
vote
0answers
49 views

MacDonald formula for Modified Bessel Functions

Mathematica seems to not know 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_{i t}(u) ...
3
votes
1answer
139 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 ...
0
votes
0answers
42 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 ...
5
votes
0answers
60 views

FindSequenceFunction for sum of hypergeometric terms

Mathematica's 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 ...
3
votes
1answer
77 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 ...
0
votes
0answers
27 views

Expanding Function Based on Congruence Classes

I'm trying to get Mathematica to expand $\quad \quad \beta_n=\sum_{r=0}^{n}\frac{\alpha_r}{(q^4;q^4)_{n-r}(q^4;q^4)_{n+r}},$ where $\alpha_r$ is an integer valued sequence that is given by ...
1
vote
1answer
32 views

Expand Expression with Nested Special Functions [closed]

I am trying to study the following sequence for $a=1$, $n \ge 1$: $$\alpha_n=(1-aq^{2n})\sum_{j=0}^{n}\frac{(aq;q)_{n+j-1}(-1)^{n-j}q^{\binom{n-j}{2}}}{(q;q)_{n-j}(q;q)_j^3}$$ using Mathematica, where ...
2
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0answers
39 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
101 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 ...
2
votes
2answers
339 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
173 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
174 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 ...
4
votes
2answers
190 views

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

Consider this code: ...
2
votes
0answers
143 views

Power::infy: error using FindFit [closed]

I have a 2D data to fit (the Airy type pattern from diffraction of Gaussian beam on the circular aperture). Please see the attached picture. I was going to use a function like ...
3
votes
1answer
53 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 ...
8
votes
1answer
181 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
100 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
92 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: ...
0
votes
1answer
95 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: ...
1
vote
1answer
152 views

How to choose a solution of a differential equation?

I am trying to solve the following differential equation: ...
0
votes
2answers
275 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: ...
1
vote
1answer
139 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 ...
2
votes
1answer
142 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 to ...
0
votes
2answers
115 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?
1
vote
3answers
132 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 ...
1
vote
1answer
150 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
0answers
192 views

Spherical harmonic derivative

Consider the following substitution Derivative[2, 0][S][th, ph] /. S -> Function[{th, ph}, SphericalHarmonicY[3, 0, th, ph]] which gives correct answer. While ...
3
votes
2answers
207 views

How to find the maximum of this function on the positive real line?

I need to maximize this function on the positive real line: $$ \frac{1}{\Gamma(x)^{14}}\cdot\frac{1}{{\frac{323.6}{14x}}^{14x}}\cdot(1.22578*10^{19})^{x-1}e^{-14x} $$ the correct answer should be ...
1
vote
1answer
286 views

Error Function Integral (Erf)

Any idea how to solve analytically this integral Integrate[(a Erf[a Sqrt[b/(a^2 + b)] c])/(a^2 + b)^(3/2), a] I tried substitution u=a^2 + b, but it didn't work. ...
3
votes
1answer
334 views

A Bessel & Struve functions related integral

I try to numerically compute this integral and I don't figure out why on earth Mathematica is not able to do it. Is my input correct? Does it possibly have a closed form? ...
2
votes
2answers
223 views

Solving Inequalities with Gamma Function

I am wondering why Mathematica outputs that the following system "cannot be solved with the methods available to Reduce". $\frac{\Gamma(\frac{1}{2}+n)}{n-1}<\frac{\Gamma(\frac{1}{2}+k)}{k-1}$ ...