Questions on the special mathematical functions implemented in Mathematica.

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64 views

Another question about inverse Laplace transform

Given $r = 0.06;\quad \theta = 105;\quad \kappa = 1;\quad x_0 = 100;\quad K = 100;\quad \sigma = 0.10;\quad T = 0.25;$ Define $ \nu = -\kappa/\sigma^2 - 0.5;\quad p = \kappa*\theta/\sigma;\quad q = -\...
1
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1answer
58 views

Derivative wrt to order of MacDonald function

I'm trying to get the following result confirmed in Mathematica: $$ \left.\frac{\partial\mathop{K_{\nu}}\nolimits\!\left(x\right)}{\partial\nu}% \right|_{\nu=\pm\frac{1}{2}}=\pm\sqrt{\frac{\pi}{2x}}\...
0
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0answers
38 views

Find Roots of expression with modified Bessel functions [duplicate]

I would like to solve an equation of this type $$A-B\,K_0(\kappa\,r)-C\,K_1(\kappa\,r)=0\ \ \quad \text{with}\ A,\,B,\,C,\,\kappa\in\mathbb{R},\ \text{known}$$ for $r$. I am not aware whether this ...
5
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0answers
77 views

SiegelTheta gives misleading message when the dimensions don't match

Bug introduced in 6.0 and persisting through 10.4 SiegelTheta is new in 6.0 In order to test the SiegelTheta function, I ...
2
votes
1answer
51 views

Getting rid of sub-exponential terms in an asymptotic expansion for a modified Bessel function

I am trying to get Mathematica to produce suitable asymptotic expansions for some modified Bessel functions at large argument (more specifically, the expansion in the DLMF's eq. (10.40.1)), and I'm ...
10
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1answer
147 views

Teaching Mathematica more about DiracDelta and KroneckerDelta

As the documentation and some experimentation indicates, Mathematica contains little information about representations of the DiracDelta and ...
1
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2answers
134 views

How to convolve the unit box function and the modified Bessel function of the second kind in 2D?

In 1D the convolution of the unit box function and the modified Bessel function of the second kind $K_0(x)$ works very well. ...
3
votes
2answers
92 views

DifferenceRoot question

I was doing the following sum: $$\sum_{i=2}^k \frac{(-1)^i}{i-1} \binom{2k-i-1}{k-1}x^i$$ First, Mathematica simplifies it to some DifferenceRoot function: ...
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1answer
52 views

Unexpected behavior in symbolic integration with GenerateConditions->False

Consider the following two symbolic integrations: ...
4
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0answers
55 views

Integral Form of Modified Bessel Function of the Second Kind

Why can't Mathematica integrate r = Integrate[Exp[-x Cosh[t]], {t, 0, Infinity}]; r = Assuming[Element[x, Reals], Simplify[r]]; Together[r] From Wikipedia, it ...
2
votes
2answers
135 views

Why is LegendreQ[1/2,x] complex-valued for x>1?

Something is strange with $\sf LegendreQ$. Let $x>1$. I wonder why $\sf LegendreQ[\frac12,x]$ is complex-valued, and the following two codes do not give the same results: ...
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0answers
100 views

Product of two Meijer's Function

I want to evaluate an integral $I_1$ defined in $Eq.(1)$ as \begin{align} I_1=\int_{0}^{\infty}\frac{x\exp(-\beta x)K_1(\alpha x)}{1+x}dx\tag{1} \end{align} Where $\alpha\geq0$, $\beta\geq0$, and $...
0
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0answers
61 views

Riemann Zeta function definition was expanded by Euler with an infinite product series [duplicate]

The Euler infinite product series definition for Riemann's zeta function requires that Mathematica use all prime numbers in the product series. Can anyone help me with the code that will give a ...
0
votes
1answer
60 views

Plotting a function based on complicated integral

I have this function : f[Lambda_] := K Integrate[x Exp[- x^2-Lambda x] HypergeometricU[-Lambda,1/2,(x+ Lambda/2+2)^2],{x,0,Infinity}]; where ...
3
votes
1answer
80 views

Interval arithmetic for DawsonF (or other special functions)

I am currently trying to estimate a complicated expression involving DawsonF using interval arithmetic. The interval arithmetic is partially supported by ...
4
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1answer
84 views

Simplify Class invariant $G(25)$

How to simplify $$\frac{\sqrt[3]{\vartheta _3\left(0,e^{-5 \pi }\right)}}{\sqrt[12]{2} \sqrt[6]{\vartheta _2\left(0,e^{-5 \pi }\right) \vartheta _4\left(0,e^{-5 \pi }\right)}}$$ This is a ...
0
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0answers
94 views

Solving Fredholm equation of the 2nd kind

While I was using the code from here to solve this integral equation: ...
0
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1answer
61 views

$\tt DiracDelta$ behaves incorrectly on multidimensional integral [duplicate]

Is there a reason why this seems to work: Integrate[DiracDelta[x] F[x], {x, -Infinity, Infinity}] F[0] But this does not: ...
7
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2answers
125 views

Portion of Curve Omitted by Plot

In the course of addressing question 104559, I encountered the following problem with Plot. Begin with ...
6
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2answers
199 views
3
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2answers
210 views

Use Meijer-G function to represent elementary functions

I want to represent these elementary functions: $x^{2}\sqrt{x}$, $\sin{4x}$, and $x\ln{x}$ as cases of MeijerG. What arguments should I give to ...
0
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0answers
79 views

How to evaluate this integral

I am trying to evaluate the following integral in Mathematica: ...
9
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3answers
294 views

Labeling solutions of an Eigenvalue equation involving Bessel functions

I'm solving the Schrödinger equation for a particle in an annular geometry with hard wall boundary conditions and I've reduced it to the following equation: $$J_m(k\,R_1)\,Y_m(k\,R_2) - J_m(k\,R_2)\,...
5
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1answer
103 views

Somewhat Irreproducible Integrate Results

Backslide introduced in v10 and persisting through v10.3.1. In the course of considering question 102922, I encountered erratic results from a particular integration. It is illustrated as follows. ...
3
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1answer
100 views

Is the real spherical harmonic (l = 1, m = 0) really 'bigger' than (l = 1, m = 1)?

Using SphericalPlot3D to plot the real spherical harmonic with l = 1 and m = 0: ...
5
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2answers
163 views

Symbolic integration of SphericalBesselJ

Backslide introduced in v10 and persisting through v10.3.1. Consider the following integral ...
7
votes
2answers
303 views

Can your Mathematica do the following integral?

Backslide introduced in v10 and fixed in v10.4.1. I was trying to do the following integral in Mathematica: Integrate[(z-2) PolyLog[2,z]Log[1-z]/z^3,z] What I ...
13
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2answers
231 views

$\tt SphericalHarmonicY$ does not seem to be an eigenfunction of the spherical harmonic equation

I applied the spherical harmonic equation on the SphericalHarmonicY functions like this: ...
4
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1answer
68 views

convert MeijerG to form Standard Functions in Mathematica

I'd like to convert MeijerG[{{}, {1., 1.}}, {{0., 0., 4.}, {}}, -1. T λ] to its Standard Functions (For example Bessel function or ...). Any suggestion?
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1answer
84 views

How to make this code involving Hypergeometric functions to run faster?

This question is followed up from this Question. I would like to thank Dr. Hintze and I_Mariusz for the comments and help. I am pretty new to mathematica ( I just learned it 4 days ago) so I would ...
6
votes
2answers
336 views

Inverse Laplace transform

Let $r=\mu = 0.15; \sigma = 0.05; T = 1; S_0 = 100; K = 95;$ Let $\nu:=\frac{2\mu}{\sigma^2}-1$ and $\eta \equiv\eta(\alpha):=-\frac{\nu}{2}+\frac{1}{2}\sqrt{\nu^2+\frac{8\alpha}{\sigma^2}}$. ...
1
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1answer
85 views

Integrating sinc function over discontinuity

I would like to analytically integrate the sinc function. First of all, if I just perform the integration the following way, everything is as expected: ...
3
votes
1answer
69 views

Convert an expression to use a specific analytic form

I have an expression that evaluates to an expression containing multiple ExpIntegralEi expressions. However, I would prefer that Mathematica use ...
2
votes
1answer
56 views
3
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2answers
128 views

Show Factorial instead of Gamma in the result of RSolve

Now I wanted to solve for a recurring function with RSolve. Here's how I tried: ...
1
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0answers
84 views

How to understand the symbolic integral in a result returned by DSolve?

I am trying to solve an ODE like $$\frac{dF}{dx}=\frac{a}{\ln^3 x+[c+d(ax+b)][2\ln x+c+d(ax+b)]\ln x},$$ where $a,b,c,d$ are constants. I guess it could contain a special function, say, the ...
0
votes
1answer
73 views

Implementing AiryAiPrimeZero function

There are some functions implemented in the Wolfram Language related to Airy functions. For example, AiryAi, AiryAiZero or ...
4
votes
1answer
87 views

Question about PrimeZetaP

The PrimeZetaP function appears to give results for complex s with real part > 0. Apparently, the analytic continuation is built ...
1
vote
3answers
160 views

Find zeros of function in 2 variables

I have two functions $f(r,\phi)$, and $g(r,\phi)$. What is the best way to find the curve in the plane $(x,y)$ or $(r,\phi)$, over which $f(r,\phi)=g(r,\phi)$? I know how to plot it, using ...
7
votes
2answers
200 views

Derivative of the Dedekind eta function fails to compute with errors I don't understand

When trying to understand better the question Eisenstein Series in Mathematica? I stumbled on the following: issuing Derivative[1][DedekindEta][.11 I] gives ...
5
votes
1answer
280 views

Find solution of nonlinear ODE in terms of JacobiCN

I am trying to find a specific solution for this differential equation: $-\frac{1}{2}\frac{d^2}{dx^2}\psi(x)-2k \; \psi(x)^3 + \frac{1}{2}k^2\; \psi(x)=0$ MMA gives me a solution in the form of a <...
1
vote
1answer
54 views

Integrating the product of two imaginary error functions

I am trying to evaluate the following integral: Integrate[Erfi[y] Erfi[z + y] , {y, -L, L}] which simply returns the input. How could I force ...
1
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0answers
43 views

Problem on special functions

What does the symbol PolyLog^{(0,1)}(0,1/e) mean? I know the meaning of the Polylogarithm, but what is that exponent? It happens the same with the Lerch zeta function!! Thanks in advance.
2
votes
2answers
69 views

How to solve or plot roots of the equation involves Bessel function of first and second kind?

Here is my equation x^2 + BesselJ[m,k*x^2]*x + k*BesselK[m,k]==0. I would like to solve this equation for different initial guesses of ...
2
votes
1answer
98 views

making 3d listplot smoother? [closed]

This is a continuation of my previous two questions: this one and this one. I would like to plot the following function $$ p(x,t) = \frac{e^{-1/A}}{A}\sum_{i=1}^{500}e^{c_i\, t}m_i(x)\frac{z_i}{w_i}, $...
9
votes
2answers
277 views

How can I get the solutions of $x^n-x-t=0$ in hypergeometric form?

$x^3-x-t=0$ has three roots that can be expressed in hypergeometric form, for example $$x_1=-1-\frac{t}{2}{_2F_1\left(\frac{1}{3},\frac{2}{3};\frac{3}{2};\frac{27t^2}{4} \right)}+\frac{3t^2}{8} {...
7
votes
2answers
236 views

Confusion regarding the incomplete elliptic integral of the first kind

I am trying to manipulate a conformal map from the half-plane to a square $z \rightarrow w(z)$ defined by: $$ w(z) = \int \limits^{z} dx \frac{1}{\sqrt{(1-x^2)x}} = \sqrt{2} \; F\left(\sqrt{z+1},\...
6
votes
1answer
226 views

Precompiling a Whittaker function

Is there a way to speed up the evaluation of special functions in Mathematica? I am particularly interested in the Whittaker W function. For instance, the following piece of code: ...
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vote
2answers
65 views

Calculating the numerical value of the regularized generalized hypergeometric function

I'm trying to calculate the numerical value of the regularized generalized hypergeometric functions: $\qquad \sf{HypergeometricPFQRegularized}^{(\{1\},\{0,0\},0)}(\{-1.5\},\{-1.,-0.5\},3600.)$ I ...
6
votes
2answers
287 views

How to implement a numerically efficient Airy Zeta Function

Define the Airy zeta function for $n=2,3,\dots$, by $$ Z(n) := \sum_r \frac{1}{r^n}. $$ where the sum is over the zeros $r$ of the Airy function $\operatorname{Ai}$. In Mathematica the $\operatorname{...