Questions on the special mathematical functions implemented in Mathematica.

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1answer
209 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
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1answer
812 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
323 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
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1answer
109 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
231 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
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2answers
271 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 ...
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1answer
200 views

I want change gamma function form into beta function form

I want to integrate below form ...
1
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1answer
476 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
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1answer
461 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
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2answers
310 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
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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
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1answer
334 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
463 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
311 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
388 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
223 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
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0answers
173 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
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0answers
236 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
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1answer
170 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
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3answers
257 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
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1answer
496 views

NDSolve + FindRoot for Bessel Zeros

I am trying to use a solution given by Michael E2 in this topic: ...
2
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1answer
323 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
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1answer
350 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
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1answer
896 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
649 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
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1answer
504 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
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1answer
314 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
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1answer
642 views

Incorrect result from Integrate

Bug introduced in 8.0 and fixed in 10.0 I attempted to calculate the following integral: ...
6
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1answer
519 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
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1answer
747 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
286 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
487 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
386 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
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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 \\ &...
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1answer
1k views

How to change machineprecision digits

I am trying to compute t0: ...
0
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1answer
263 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
572 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
461 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
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1answer
552 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
278 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
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2answers
378 views

Integrating special functions

I would like to integrate the following function with Legendre polynomial and Gamma function. I am open to suggestions. ...
3
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1answer
149 views

Strange NSum behavior

If I do: NSum[(i + 1)/(i + 2) LegendreP[i, 0] LegendreP[i, 0], {i, 0, Infinity}] I get: 1.25216 If I do: ...
0
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1answer
321 views

Why does the evaluation of this series fail?

The following series expression holds: ...
3
votes
1answer
257 views

Symbolic Integration of Special Functions

Sorry in advance if this formatting comes out strange, this is my first question! I can't find a way to integrate, e.g., a function of the Hermite polynomials for general (still integer) order. For ...
4
votes
2answers
606 views

Replacing gamma at half integers by double factorial

It is well-known that for any positive integer $n$ the equality $\Gamma(n+\frac12)=\sqrt\pi\,(2n-1)!!/2^n$ holds, where $!!$ stands for the double factorial. I am using ...
1
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1answer
194 views

Expansion of functions in the real domain

When I try FullSimplify[FunctionExpand[x^(1/12)(1/12) LerchPhi[x,1,1/12]]] I have a nice answer with many complex numbers. Is there any way to have the ...
3
votes
1answer
385 views

plotting hypergeometric functions

Does anyone know why a plot of a hypergeometric function turns out differently in Mathematica than in Maple? The function I'm plotting between x=-30 and +30 is: ...
5
votes
3answers
328 views

Implementation of a complex recurrence relation for polynomials

I would like to implement the recurrence relation for the polynomials $U_n(x)$ appearing in the large order asymptotics of the Bessel functions. The recurrence in question is: $$U_{n+1}(x)=\frac{1}{...
4
votes
1answer
95 views

SiegelTheta fails to evaluate when given proper arguments

SiegelTheta often returns error messages when I give it arguments that should be of the correct form. For instance, I have a numerical matrix like ...