1
vote
1answer
59 views

Function Approximation

I need to solve the following equation $$-\frac{\partial S(x,y,t)}{\partial t}=ax^2+bx\frac{\partial S(x,y,t)}{\partial x}+c\Big[\frac{\partial S(x,y,t)}{\partial ...
0
votes
0answers
18 views

Numerical evaluation of an infinite 3D sum of cosine?

Consider the following function: $$f\left(x, y, z\right) = \sum_{\left(n, m, l\right)\in \mathbb{N}_*^3}e^{-\alpha\left(n^2+m^2+l^2\right)}\frac{\cos\left(\omega nx\right)\cos\left(\omega ...
3
votes
2answers
130 views

Real roots plot of the modified bessel function

Could anyone point me a program so i can calculate the roots of $$ K_{ia}(2 \pi)=0 $$ here $ K_{ia}(x) $ is the modified Bessel function of second kind with (pure complex)index 'k' :D My conjecture ...
1
vote
0answers
26 views

Kummer U function: U(a,a+1/2,z)

Is there any way to simplify $U(a,a+1/2,z)$ or relate it to any other common special functions such as the incomplete beta function or the incomplete gamma function? Here, $U(a,b,z)$ is Kummer's U ...
1
vote
0answers
58 views

Find bound on order of Bessel function

I need to evaluate and store the first significant ($n$) orders for a given abscissa for the Bessel functions of the first kind. That is, $J_v(x), \ v \in [0, n]$. Here significant means that I want ...
1
vote
0answers
75 views

Calculation of integral with Bessel function

I have a trouble with to calculating (or bounding from above) the following integral: $$ \int_{-\infty}^{\infty}\left(\frac{J_2(x)}{x^2}\right)^p\, dx, \quad p\geq 1, $$ where $J_2(x)$ is a Bessel ...
3
votes
1answer
217 views

How to calculate Bessel Function of the first kind fast?

I have wrote a C++ code to calculate the first kind of Bessel Functon by its infinite series definition. I took the sum of the first 20 series as the value of Bessel Function, which is same as MATLBA ...
6
votes
1answer
175 views

Numerical approximation for log of incomplete beta function

Is there any known numerical approach to directly compute the log of the incomplete beta function? I would like to be able to compute $$ \log\left( \int_0^u x^{a-1} (1-x)^{b-1} dx \right) $$ ...
4
votes
1answer
109 views

Computing the inverse Jacobi function $\mathrm{arccd}$ with elliptic integrals

According to page 42 of 1, $\operatorname{arccd}(x, k)=F\left(\arcsin\left(\sqrt{\frac{1 - x^2}{1 - k^2x^2}}\right), k\right)$, where $F(\phi, k)=\int_0^\phi \frac{dt}{\sqrt{1 - k^2\sin^2t^2}}$, and ...
0
votes
1answer
366 views

How to compute complete elliptic integral of the first kind in explicit form using elementary functions?

How to compute complete elliptic integral of the first kind in explicit form using elementary functions? If it is not possible to compute complete elliptic integral of the first kind in explicit way, ...
2
votes
1answer
102 views

Numerical methods for integrals involving product of Bessel functions of the first kind (1st order)

I am looking for the best (in terms of low computation times) numerical methods for calculating the following integrals: $$\int_0^{\infty}\,f(k)\,J_1(ak)\,J_1(bk)\,dk$$ with for instance ...
1
vote
1answer
258 views

Numerical values of the Jacobi elliptic function sn: Wolfram Alpha vs. Maple vs. C++?

I have a problem to check the validity of an algorithm I've implemented in C++ to compute the Jacobi elliptic function $\mathrm{sn}(u, k)$ (inspired and improved from Numerical Recipes 3rd edition). I ...
1
vote
1answer
154 views

How does one calculate the amount of time required for computation?

For example, to compute the zeroes of the Riemann zeta function using the Euler-Maclaurin summation method one has to do O(T) work. The Euler-Maclaurin summation method for zeta is given by $ ...
2
votes
1answer
60 views

Function space with difficult boundary conditions for Galerkin method

I am using Galerkin's method for solving a generalized eigenvalue problem numerically, which results from a linear stability analysis. Due to the physics of the problem the following boundaries of the ...
1
vote
0answers
210 views

Approximations of the incomplete elliptic integral of the second kind

For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$. I know that this is a ...
2
votes
2answers
347 views

Numerical approximation of the modified Bessel function $I_0$ with radical argument for integration purposes

I have to numerically calculate the following definite integral $$\int_{\alpha}^{\beta}I_0(a\sqrt{1-x^2})dx$$ for different values of $\alpha$ and $\beta$, where $a$ has a value of, say, $30$. I'm ...
1
vote
1answer
148 views

How to approximate $\text{li}(z)$ numerically?

I'm trying to implement a function to calculate $\pi(x)$ via Riemann's formula: $$ \pi(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan ...
1
vote
0answers
88 views

Bessel functions with complex coefficients

Are there any known algorithms and libraries to numerically evaluate Bessel functions of the first kind $J_\nu(z)$ with complex coefficients, i.e. $\nu \in\mathbb{C} $
3
votes
1answer
80 views

Calculating the divisor, known to be small, of two Stirling approximations of the logarithmic Gamma function without overflows

Earlier, I asked a question on MathOverflow regarding how one might analytically approximate a function of the form: $f(n) = \prod_{i=1}^{n-1} (1-ai)$ for $a \ge 0$, $(ai) < 1$, and $n > 10^5$ ...
1
vote
0answers
83 views

fastest way to evaluate $\arg\zeta\left(\frac{1}{2}+i\text{t}\right) $ [duplicate]

Possible Duplicate: evaluation of $ \operatorname{Arg}\zeta (1/2+is) $ ?? If we consider $$\arg\zeta\left(\frac{1}{2} + i\text{t}\right) = \text{Im ...
4
votes
3answers
915 views

Estimating the Gamma function to high precision efficiently?

I know there are several approximations of the Gamma function that provide decent approximations of this function. I was wondering, how can I efficiently estimate specific values of the Gamma ...
1
vote
0answers
404 views

solving Bessel function equation by hand

I have a Bessel function of the first kind given by the equation $$J_\alpha (\beta) = \sum_{m=0}^{\infty}\frac{(-1)^m}{m!\Gamma(m+\alpha +1)} \left(\frac{\beta}{2}\right)^{2m+\alpha}$$ I am trying to ...
2
votes
1answer
123 views

What representation should I choose for numerical computation of hypergeometric function ${}_2 F_1(1+i\eta, 2; 2+i\eta; x)$ where $|x|=1$

I have a task - to plot graphics of the function: $$ I(E) = \frac{16i \pi k \mu}{(\beta - ik)^{4}} \frac{1}{1 + i\eta} {}_2 F_1(1+i\eta, 2; 2 + i \eta; x) $$ where $$ x = \left( \frac{\beta + ...
4
votes
4answers
722 views

Calculation of Bessel Functions

I want to calculate the Bessel function, given by $$J_\alpha (\beta) = \sum_{m=0}^{\infty}\frac{(-1)^m}{m!\Gamma(m+\alpha +1)} \left(\frac{\beta}{2}\right)^{2m}$$ I know there are some tables that ...
6
votes
1answer
480 views

series including infinite sum

I am looking for the approximation of the following function: $$\rho(a,b)=1-e^{-(a+b)}\sum_{m=1}^{\infty}\left(\sqrt{\frac{b}{a}}\right)^m I_m(2\sqrt{ab})$$ where $I_m(x)$ is the modified Bessel ...
0
votes
2answers
1k views

Easy approximation of the incomplete beta function $\text{B}_x(a,b)$

I need to calculate $\text{B}_x(a,b)$ on the cheap, without too many coefficients and loops. For the complete $\text{B}(a,b)$, I can use $\Gamma(a)\Gamma(b)/\Gamma(a+b)$, and Stirling's approximation ...
4
votes
1answer
339 views

Approximating Lambert W for input below 0

As a small part of a much bigger project, I need to be able to approximate the numerical output of the Lambert W function. I have found decent approximations (good up to at least 4 decimal places), ...
2
votes
2answers
364 views

Representing affine transform of Legendre polynomials

I have a function defined as a set of weighted Legendre polynomials: $f(x)=\alpha_0 P_0(x) + \alpha_1 P_1(x) + \alpha_2 P_2(x) +\ldots$. I have another function similarly defined with Legendre basis ...
3
votes
3answers
1k views

How to solve $n$ for $n^n = 2^c$?

How to solve $n$ for $n^n = 2^c$? What's the numerical method? I don't get this. For $c=1000$, $n$ should be approximately $140$, right?
6
votes
5answers
4k views

Algorithm to compute Gamma function

The question is simple. I would like to implement the Gamma function in my calculator written in C; however, I have not been able to find an easy way to programmatically compute an approximation to ...
4
votes
1answer
502 views

Efficiently calculating the logarithmic integral with complex argument

My number theory library of choice doesn't implement the logarithmic integral for complex values. I thought that I might take a crack at coding it, but I thought I'd ask here first for algorithmic ...
2
votes
1answer
537 views

Elliptic integrals with parameter outside 0<m<1

I'm attempting to implement an equation (for calculating magnetic forces between coils, eqs (22–24) in the linked paper) that requires the use of elliptic integrals. Unfortunately these equations ...
11
votes
1answer
1k views

Iterative refinement algorithm for computing exp(x) with arbitrary precision

I'm working on a multiple-precision library. I'd like to make it possible for users to ask for higher precision answers for results already computed at a fixed precision. My $\mathrm{sqrt}(x)$ can ...
8
votes
2answers
2k views

How to evaluate Riemann Zeta function

How do I evaluate this function for given $s$? $$\zeta(s) = \sum_{n=1}^\infty \frac1{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$$
7
votes
2answers
737 views

Minimum of the Gamma Function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$?

The $\Gamma (x)$ function has just one minimum for $x>0$ . This result uses some properties of the gamma function: $\Gamma ^{\prime \prime }(x)>0$ and $\Gamma (x)>0$ for all $x>0$ $\Gamma (1)=\Gamma ...
5
votes
3answers
2k views

How to accurately calculate the error function erf(x) with a computer?

I am looking for an accurate algorithm to calculate the error function I have tried using [this formula] (http://stackoverflow.com/a/457805) (Handbook of Mathematical Functions, formula ...