Questions on the special mathematical functions implemented in Mathematica.

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9
votes
2answers
279 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
239 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
242 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: ...
1
vote
2answers
73 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
290 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{...
0
votes
1answer
74 views

Can I tell DSolve to solve a first-order ODE by method of separation of variables?

I meet a first-order ODE $$\frac{dy}{dt}=\frac{a(\ln\frac{1-c}{1-y})^3}{\frac{b-y}{1-y}+\ln\frac{1-c}{1-y}},$$ where $a,b,c$ are constants. The ODE is subjected to the initial condition $y(t=0)=y_0$. ...
0
votes
1answer
78 views

Plotting complicated Bessel functions expression

I am trying to plot this function ...
3
votes
2answers
420 views

Solve stiff system by shooting method

I'm trying to solve a second order differential equation with the shooting method but, it appears to be a stiff system. This is worst for a larger parameter $\mu$. I've tried different methods: ...
4
votes
0answers
119 views

Puzzled by Assumptions [duplicate]

I don't know if this has already been discussed. Integrate[BesselJ[2 m + 1, x], {x, 0, ∞}, Assumptions -> m ϵ Integers] ...
4
votes
1answer
173 views

Bad performance of Integrate (and WolframAlpha) for an Integral of Bessel function of the first kind

The following returns unevaluated in WolframAlpha. Also in my machine Mathematica needs quite a lot of time to compute it. ...
2
votes
1answer
89 views

FullSimplify gives wrong answer when operating on HypergeometricPFQ

When I try to simplify this hypergeometric function FullSimplify[HypergeometricPFQ[{1/2 - n/2}, {-(1/2) - n/2}, -a^2]] Mathematica 10.0.1.0 returns ...
1
vote
1answer
134 views

Plot Meijer G function [closed]

How to plot using Mathematica the Meijer G function $$ G^{m l}_{p q} \left(\omega t \ \Bigg\vert \ {a_1,\cdots,a_p\atop b_1,\cdots,b_q} \right) $$ Thanks
3
votes
1answer
146 views

How can I plot the normalized distribution of the Riemann zeta zeros?

Given a list of eigenvalues or a list of Riemann zeta zeros, how can I plot this famous plot found here: On the page referred to, You need to click on "Programs", "The Riemann zeta function" and "...
1
vote
1answer
73 views

Failure of Series[] for hypergeometric functions

I am encountering peculiar errors when asking Mathematica for series expansions of certain hypergeometric functions. To give an example, consider the function $f(x) = {}_{5} F_{4}(3/2,3/2,3/2,2,2; 1,...
2
votes
2answers
87 views

Integration result with incomplete beta function

I'm doing some calculations (double integration) which results in the incomplete beta function occurring as an end result. My input is: ...
5
votes
2answers
85 views

Solution to Simultaneous Arithmetic/Geometric Mean Recursion Relations

I'm trying to solve the simultaneous convergent sequences of geometric/arithmetic means where $a_{n+1}=\frac{1}{2}(a_n+b_n)$ and $b_{n+1}=\sqrt{a_nb_n}$ and initial values are $a_0=1+x$ and $b_0=1-x$. ...
1
vote
1answer
94 views

Can anyone re-produce this result related to the spectrum of Riemann Zeta using error term generated from MangoldtLambda?

All: I tried to reproduce the results from this page: How to plot the Riemann-Zeta zero spectrum The following is the code that was posted on above page: ...
0
votes
1answer
45 views

Plotting precise intersections involving singular functions

I have pseudo-elliptic functions defined on a parallelogram within $\mathbb{C}$ and I would like to clearly highlight the path within this parallelogram which satisfies the real part of my pseudo-...
0
votes
0answers
40 views

Determine class of special function from algebraic constraints?

Consider vectors x_i in arbitrary dimension. Let's say I have an expression in six variables F[x_1,x_2,x_3][x_4,x_5,x_6], which ...
1
vote
0answers
59 views

Why can't I use OmegaPrime to find the Limit of Prime[n]? [duplicate]

I've looked at these links already; What are the limits of the Prime-functions? What is so special about Prime? which gave an answer for earlier versions of Mathematica. Yet when I try to input ...
0
votes
1answer
123 views

How to plot zeros of confluent hypergeometric function

I am fairly new to mathematica so I need a little bit of help. I need to plot the zeros of an equation containing confluent hypergeometric functions. The equation i need to solve is given by the ...
2
votes
3answers
436 views

Solve an equation that include Gamma

I want to solve the following equation Solve[Gamma[1 + x]/Gamma[x - 1/2] + 1 == 0, x] I have answer x=-0.25, but I can not obtain this answer with Mathematica. ...
2
votes
1answer
119 views

Plotting a partial sum

I am given the Legendre expansion of the first kind. $$f(x)=\sum_{n=0}^{\infty}A_{n}P_{n}(x).$$ I have worked the coefficient to be $$A_{n}=\frac{1}{\left \| P_{n}(x) \right \|^{2}} \int_{x=-1}^{x=1}...
5
votes
2answers
100 views

Expanding PolyGamma function error

Bug introduced in 6.0 or earlier and fixed in 10.3.0 For large half-integer arguments of the PolyGamma[] function, ...
1
vote
0answers
87 views

Legendre expansion of functions [closed]

I am given the expansion for the Legendre function of the first kind. It is just $$f(x)=\sum_{n=0}^{\infty}A_{n}P_{n}(x).$$ I have found the expression for the coefficient $A_{n}$. However I would ...
3
votes
1answer
108 views

Range of summation in simple Plot seems off

I was trying to reproduce a picture in a book by Havil of the sum, $$s = \sum_{r=1}^{\infty}\frac{\mu(r)}{r}\left(Li(x^{\rho_k/r})+Li(x^{\rho_k*/r})\right) $$ using ...
3
votes
0answers
82 views

Refining a density plot of the Eisenstein series argument [closed]

Thanks to amazing code from "Guess who it is" here: Eisenstein Series in Mathematica? I'm able to make some nice plots using Eisenstein Series. What I'd like is a color plot of the argument of certain ...
1
vote
0answers
75 views

Odd plotting/math issue (could be a precision problem) [closed]

I've got a pretty odd error on a project I'm working on and was hoping to enlist some advice to fix it. The goal of this notebook is to show that I can eliminate the non-normalizable (blowing up part) ...
4
votes
4answers
268 views

How to stop a summation when a variable is small enough?

I meet with a problem. I hope to get an infinite summation of $f1(x)/f2(x)$ which converges to zero. So my code is Sum[f1[x]/f2[x],{x,Infinity}] or ...
1
vote
1answer
50 views

problem about Root and Hypergeometric2F1

See these example, why does the output is different? ...
2
votes
0answers
97 views

Nasty integral advice

I have a pretty ferocious integral to solve, and since it doesn't seem I'll be able to do much analytically, I've taken to Mathematica to get some information. Mainly, I want to see if there are any ...
6
votes
2answers
145 views

Plot of The RiemannSiegelZ Function

I would appreciate your help to visualize the of the following function RiemannSiegelZ[x] with this range. { 18154980120849865 , 18154980120849885 } I tried this: ...
0
votes
1answer
57 views

Sum of hypergeometric functions, variable number of arguments

How can I write in Mathematica an expression like this? $$\sum_{k=1}^n {_{k}F_{k}} (1,1,\dots,1; \,2,2,\dots,2; \,z) ~,$$ where ${_{p}F_{q}}$ is the generalized hypergeometric function. My problem ...
0
votes
1answer
87 views

How to solve algebra equations containing integration and parameters?

I'm trying to solve two nonlinear algebra equations for two unknown parameters, U and Tf. Since some terms in these equations contain integration, and the integration also contains U and Tf. The main ...
1
vote
0answers
63 views
0
votes
0answers
44 views

Timing of associated Legendre polynomials

I encountered a strange issue with the associated Legendre polynomials implemented with LegendreP[l,m,z]. Quite simply, the time used for the numerical computation of those quantities depends on ...
6
votes
2answers
197 views

How does Mathematica calculate LaguerreL

About the function LaguerreL[n,a,x], the helping documents in Mathematica only say that this function satisfies equation $xy^{\prime\prime}+(a+1-x)y^\prime+ny=0$. ...
4
votes
1answer
95 views
6
votes
1answer
110 views

Spurious infinite limit of integration message in NIntegrate

Bug introduced in 7.0 and persisting through 10.2 NIntegrate returns an error complaining that ComplexInfinity is an invalid ...
12
votes
1answer
357 views

Eisenstein Series in Mathematica?

Mathematica doesn't seem to have built-in tools to deal with the Eisenstein series: $$\begin{align*} E_{2}(\tau)&= 1-24 \sum_{n=1}^{\infty} \frac{n e^{2 \pi i n \tau}}{1-e^{2 \pi i n \tau}}\\ E_{...
1
vote
0answers
74 views

Use Solve[] with Bessel, gamma, and hypergeometric functions?

I need to find values of {a,b,c} such that the 0th, 2nd, and 4th order moments of f[x]=Exp[-ax^4 - bx^2 - c] will equal respectively {1,2,10}. I didn't really expect this to work, and it didn't: <...
3
votes
2answers
232 views

Storing Variables in “Loops” and Point Plotting

Given the function $y=\sin x$ defined over the region $-\pi \leq x \leq \pi$, I need to implement a "do loop" such that I sweep over 100 or so points $-1 \leq y \leq 1$ and find precisely the two $x$ ...
-1
votes
1answer
208 views

Inverse of a Digamma/ Polygamma Function

How one can find the inverse of a digamma/ Polygamma function in Mathematica 10?
7
votes
2answers
123 views

Simplify expression to Integer

I have the following function: f[n_]:=(n-1) E Gamma[n,1]; I know that this expression always evaluates to an Integer, but Mathematica doesn't always output the ...
2
votes
1answer
134 views

Unexpected Weierstrass $\wp$-Function Behavior

In Mathematica, the Weierstrass $\wp$-function is expressed by WeierstrassP[u + I v, {g₂, g₃}] where g₂ and ...
0
votes
1answer
95 views

Why is the indefinite integral $e^\frac{x}{2}x^\frac{g}{2}dx$ the upper Gamma function in Mathematica? [closed]

I have an indefinite integral $e^\frac{x}{2}x^\frac{g}{2}dx$, I try to solve in Mathematica. Solve[Int[e^{x/2}x^{g/2-1}dx ,x]] (* Gamma[g/2,x/2] *) How does ...
3
votes
2answers
178 views

ParametricPlot with 2 variables

I want to try plotting this: As you can see the first axis is v and the second is just dependent of the angle of $\Phi$ . The function is ...
1
vote
0answers
90 views

Improve speed of evaluating a sum over four indices

I am trying to implement the Fox-H function with several variables as sum of residues. I have arrived at the following function; ...
3
votes
2answers
381 views

How I can integrate $\int_0^{a} x e^{-\frac{b^2 x^2}{2 c}} J_0(n x) dx$?

How can I get a solution to the integral given below ? $\quad \quad \int_0^{a} x e^{-\frac{b^2 x^2}{2 c}} J_0(n x) dx$ where $a,\,b,\,c$ and $n$ are constants and $J_0$ is a Bessel function of the ...