Questions on the special mathematical functions implemented in Mathematica.

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0
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2answers
59 views

Complete definition of special functions in Mathematica [on hold]

How can one display the actual integral involved in special functions like this MarcumQ[2, a, b] in Mathematica?
1
vote
1answer
40 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
81 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: ...
2
votes
2answers
105 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
47 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
68 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
163 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
100 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
77 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 ...
3
votes
2answers
113 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
90 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
169 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{$\frac{1}{1+i}<|x-t|\le \frac{1}{i}$} \\ ...
1
vote
0answers
48 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) ...
2
votes
1answer
102 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 Abs[MathieuS[4x, 4, Pi]] = 0, for 2 < x ...
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 ...
4
votes
0answers
53 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
73 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
26 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
30 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
votes
0answers
38 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 ...
2
votes
2answers
288 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
165 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
158 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
186 views

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

Consider this code: ...
2
votes
0answers
138 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
52 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
162 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
99 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
90 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
88 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
146 views

How to choose a solution of a differential equation?

I am trying to solve the following differential equation: ...
0
votes
2answers
243 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: ...
0
votes
1answer
121 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
132 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
111 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?
0
votes
3answers
128 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
148 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
179 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
201 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 ...
3
votes
1answer
321 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
217 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". $\frac{\Gamma(\frac{1}{2}+n)}{n-1}<\frac{\Gamma(\frac{1}{2}+k)}{k-1}$ ...
4
votes
1answer
91 views

Why do certain fractional values in TriangleWave not evaluate?

While answering another question I discovered that TriangleWave does not automatically evaluate when given certain fractional values, specifically fractions with a ...
1
vote
1answer
184 views

Integral over squared Hermite polynomial

I would like to calculate the uncertainty of the nth Eigenstate of a 1-dim harmonic oscillator. To obtain the result I have to solve the integral $$\int_{-\infty}^{\infty} \psi^* x^2 \psi \:dx$$ ...
4
votes
1answer
199 views

Ordinary differential equation invloving the function composition

I want to solve the following function: DSolve[(A1*Exp[B1*f[x]] + A2*Exp[B2*f[x]])*f'[x] == A1*Exp[B1*x] + A2*Exp[B2*x], f[x], x] And this is what I get as an ...
4
votes
2answers
133 views

Gamma function computation efficiency?

I wonder what kind of algorithm is used to compute the values for the gamma function. Specifically, I am interested in how the computational load increases when the complexity of the input grows. So, ...
1
vote
0answers
143 views

Minimizing a functional takes forever

I need help with minimizing a functional in Mathematica. I have a function $V(\xi)=\sum_{i=1}^\infty C_{3_i}J_0(\xi/C_i)+C_{4_i}Y_0(\xi/C_i),~~~\Sigma\leq\xi\leq\xi_1$, and want to find such ...
1
vote
1answer
123 views

Numerical integration of Hankel functions

I would like to know how to perform numeric integration for the following type of integrals in Mathematica. For the following integrand, we can not get the symbolic result. ...
0
votes
1answer
293 views

NDSolve + FindRoot for Bessel Zeros

I am trying to use a solution given by Michael E2 in this topic: ...
2
votes
1answer
204 views

Why FullSimplify doesn't work here?

Since the emphasis of this question is on finding a workaround, I decided to post this question with an emphasis on the explanation of the behavior of Mathematica. The Bessel function satisfies the ...
6
votes
0answers
193 views

Using FunctionExpand to evaluate symbolic derivatives

Some symbolic derivatives of certain special function are not expanded automatically, but FunctionExpand often helps to get a derivative-free closed form ...