Questions on the special mathematical functions implemented in Mathematica.

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1
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1answer
205 views

How to choose a solution of a differential equation?

I am trying to solve the following differential equation: ...
1
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2answers
597 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
247 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
336 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
256 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
144 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
156 views

Equation involving hypergeometric functions

I want to solve this equation but nor NSolve nor Solve are able to do this. ...
1
vote
1answer
212 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
834 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 ...
4
votes
0answers
328 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 ...
4
votes
1answer
113 views

Complex LogIntegral error

Going through Derbyshire's Prime Obsession & trying to take LogIntegral of 20^ZetaZero[1] & comes up with a value of <...
4
votes
1answer
234 views

Problems with series of generalized hypergeometric functions

I have been trying to do the following series: Series[HypergeometricPFQ[{1, 4, 4}, {4 - Sqrt[3], 4 + Sqrt[3]}, z],{z,1,0}] Mathematica says that the result is ...
3
votes
2answers
276 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
202 views

I want change gamma function form into beta function form

I want to integrate below form ...
1
vote
1answer
483 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
468 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
314 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". $\frac{\Gamma(\frac{1}{2}+n)}{n-1}<\frac{\Gamma(\frac{1}{2}+k)}{k-1}$ ...
6
votes
1answer
115 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
337 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 $$\int_{-\infty}^{\infty} \psi^* x^2 \psi \:dx$$ ...
9
votes
4answers
466 views

Numerical instability in cosh and sinh - integral functions [duplicate]

I'm trying to calculate the function: CoshIntegral[x] Sinh[x] - Cosh[x] SinhIntegral[x] Unfortunately Mathematica seems to hit a point (x~20) and things become ...
4
votes
1answer
320 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
1answer
397 views

Simplify doesn't simplify HypergeometricPFQ with exact arguments

Consider a series: $$\sum_{t=0}^\infty \frac{8^{-11-2t}(22+4t)!}{t!(11+t)!(11+2t)!(32+t)}$$ ...
6
votes
2answers
225 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
186 views

Converting a hypergeometric function to a Bessel function

Is there any way to get a Bessel function equivalent to a Hypergeometric function? Would it be possible in general? From Maple integration (as discussed in Integrating a BesselJ integrand to obtain ...
1
vote
0answers
240 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 $\{C_{...
1
vote
1answer
174 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. ...
11
votes
3answers
259 views

Integrating a BesselJ integrand to obtain the same result as Maple 16

I would like to check the following integration: Integrate[y Integrate[1/x BesselJ[1, x Exp[I π/4]] BesselJ[1, x Exp[-I π/4]], {x, 0, y}], {y, 0, r}] ...
0
votes
1answer
499 views

NDSolve + FindRoot for Bessel Zeros

I am trying to use a solution given by Michael E2 in this topic: ...
2
votes
1answer
328 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
1answer
355 views

Using FunctionExpand to evaluate symbolic derivatives

Some symbolic derivatives of certain special functions are not expanded automatically, but FunctionExpand often helps to get a derivative-free closed form ...
9
votes
1answer
905 views

how to simplify large expression with lots of special functions in it (BesselY, Hypergeometric, MeijerG etc…)

I saw this DE in Maple forum. When solving it using Mathematica 9.01, even though the result was correct (both solutions gave the same numerical answer for some random values), Mathematica's answer ...
3
votes
1answer
659 views

FindRoot giving false roots with Bessel Functions

I have read in some places about the errors associated with FindRoot, but the closest thing I can find on this website seems to be due to the imaginary unit. I am dealing with what should be a ...
5
votes
1answer
507 views

Strange result for the analytic integration leads to Hypergeometric2F1

The integration result for Integrate[1/(r^2 Sqrt[x/r^(4 - 2 \[Gamma]) + 1]), r] is: ...
4
votes
1answer
316 views

Numerical errors/inaccuracies in ProductLog

Context In cosmology, a fairly accurate model to describe the gravitational potential, $\psi(r)$ of dark matter halos is given by $\psi( r)=\log(1+r)/r$. ...
20
votes
1answer
650 views

Incorrect result from Integrate

Bug introduced in 8.0 and fixed in 10.0 I attempted to calculate the following integral: ...
6
votes
1answer
524 views

How to express an expression with only ArcTan and ArcTanh?

I have an expression which is simply (j/k) x^(j/k) LerchPhi[x,1,j/k)] where 0 < j < k. Manually I have been able (...
0
votes
1answer
774 views

Gamma Incomplete Function representation in Mathematica and Matlab

I am confused about incomplete gamma function calculation in Mathematica and MATLAB: For example, in Mathematica: Gamma[5,3] = 19.56 But in MATLAB: ...
5
votes
2answers
290 views

When does the real part of Zeta vanish on the critical line?

This seems to be a quite a simple problem but I cannot make it work. I am trying to find all values within a given range for which the real part of the Zeta ...
3
votes
2answers
492 views

Assigning an analytical approximation to the error function erf(x)

Working with some iterative integral equations, I have Gaussian density functions involved therein. Integrating the Gaussian function I obtain the error function. When I take the second integration, ...
4
votes
2answers
389 views

How do I evaluate a symbolic integral involving Hermite polynomials?

I want to test a difficult integral : Integral on all reals of some complicated function involving the Hermitian polynomials, exponentials, squares, factorials, and being general considering any ...
1
vote
2answers
2k views

Q-Function representation in Mathematica

I observed that there is no Q-function representation in Mathematica. The definition of Q-Function is: \begin{align} Q(x) &= \frac{1}{\sqrt{2\pi}} \int_x^\infty e^{-\frac{u^2}{2}}du \\ &...
0
votes
1answer
1k views

How to change machineprecision digits

I am trying to compute t0: ...
0
votes
1answer
266 views

Why does Mathematica find a form for this general sum, but not for some special cases?

Today I found a Sum which Mathematica will simplify for a general parameter value $\nu$, but which will not simplify fully for special cases $\nu = 1, 3, ...$ despite the fact that the general answer ...
16
votes
1answer
577 views

How to represent a continuous monotonic phase of Airy functions?

Note: In this question I am concerned only with real-valued variables and functions. DLMF, §9.8 Airy Functions, Modulus and Phase, formula $9.8.4$ defines the phase of Airy functions: $$\theta(x)=\...
14
votes
1answer
462 views

How to enlarge Mathematica's knowledge about certain functions?

I'm often troubled with the following task. I need to carry out symbolical computations involving certain special functions. Let me take as an example Barnes gamma-function. It is included in ...
11
votes
3answers
410 views

Find asymptotics of $\sum\limits_{i=0}^{n/3} 2^i \binom{n-i-1}{\frac{2n}{3}-1}$

I have an expression 2^n / Sum[ 2^i Binomial[ n - i - 1, 2n/3 - 1], { i, 0, n/3}] ...
1
vote
1answer
554 views

Sum[expr,{i,0,Infinity}] for power series of cumulative normal distribution gives exponential function?

The Taylor series (about 0) for the cumulative normal distribution has coefficients: ...
2
votes
0answers
282 views

How to prevent simplification of hypergeometric functions resulting from integrations?

Definite integrals from 0 to Infinity over a product of two hypergeometric (including exponential, trigonometric, hyperbolic, ...
3
votes
2answers
382 views

Integrating special functions

I would like to integrate the following function with Legendre polynomial and Gamma function. I am open to suggestions. ...