I have forgotten

• https://me.yahoo.com
COST (GBP)
15.00
0.00
0

Kelvin

viewed 2790 times and licensed 54 times
Function used at calculating asymptotic expansions.
Controller: CodeCogs

C++

Overview

This module contains components which calculate different types of Kelvin functions.

Bei

 doubleBei( double x )
This function uses two methods of approximation. If the argument falls below the MIN limit it uses the polynomial approximation :
$\mathrm{Bei\,}&space;x&space;=&space;\sum_{n&space;\geq&space;0}&space;\frac&space;{&space;(-1)^n&space;(\frac{1}{2}x)^{4n+2}}&space;{[(2n+1)!]^2}$

otherwise it calculates its asymptotic expansion :
$\mathrm{Bei\,}&space;x&space;=&space;\frac&space;{&space;\mathrm{e}^{&space;\frac{x}{\sqrt{2}}&space;}&space;}{&space;\sqrt{2\pi&space;x}&space;}&space;\left(&space;f(x)&space;\sin&space;\alpha&space;+&space;g(x)&space;\cos&space;\alpha&space;\right)&space;+&space;\frac{&space;\mathrm{Ker\,}&space;x}{\pi}$

where

$\alpha&space;=&space;\frac{x}{\sqrt{2}}&space;-&space;\frac{\pi}{8}$
$f(x)&space;\sim&space;1&space;+&space;\sum_{n&space;\geq&space;1}&space;\frac{1&space;\cdot&space;9&space;\cdot&space;\ldots&space;\cdot&space;(2n&space;-&space;1)^2}{n!(8x)^n}&space;\cos&space;\left(&space;\frac{n\pi}{4}&space;\right)$
$g(x)&space;\sim&space;\sum_{n&space;\geq&space;1}&space;\frac{1&space;\cdot&space;9&space;\cdot&space;\ldots&space;\cdot&space;(2n&space;-&space;1)^2}{n!(8x)^n}&space;\sin&space;\left(&space;\frac{n\pi}{4}&space;\right)$

Parameters

 x The value at which the function is to be evaluated.

Authors

Lucian Bentea
Source Code

Source code is available when you buy a Commercial licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.

Ber

 doubleBer( double x )
This function uses two methods of approximation. If the argument falls below the MIN limit it uses the polynomial approximation :
$\mathrm{Ber\,}&space;x&space;=&space;1&space;+&space;\sum_{n&space;\geq&space;1}&space;\frac&space;{&space;(-1)^n&space;(\frac{1}{2}x)^{4n}}&space;{[(2n)!]^2}$

otherwise it calculates its asymptotic expansion :
$\mathrm{Bei\,}&space;x&space;=&space;\frac&space;{&space;\mathrm{e}^{&space;\frac{x}{\sqrt{2}}&space;}&space;}{&space;\sqrt{2\pi&space;x}&space;}&space;\left(&space;f(x)&space;\cos&space;\alpha&space;+&space;g(x)&space;\sin&space;\alpha&space;\right)&space;-&space;\frac{&space;\mathrm{Kei\,}&space;x}{\pi}$

where

$\alpha&space;=&space;\frac{x}{\sqrt{2}}&space;-&space;\frac{\pi}{8}$
$f(x)&space;\sim&space;1&space;+&space;\sum_{n&space;\geq&space;1}&space;\frac{1&space;\cdot&space;9&space;\cdot&space;\ldots&space;\cdot&space;(2n&space;-&space;1)^2}{n!(8x)^n}&space;\cos&space;\left(&space;\frac{n\pi}{4}&space;\right)$
$g(x)&space;\sim&space;\sum_{n&space;\geq&space;1}&space;\frac{1&space;\cdot&space;9&space;\cdot&space;\ldots&space;\cdot&space;(2n&space;-&space;1)^2}{n!(8x)^n}&space;\sin&space;\left(&space;\frac{n\pi}{4}&space;\right)$

Parameters

 x The value at which the function is to be evaluated.

Authors

Lucian Bentea
Source Code

Source code is available when you buy a Commercial licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.

Kei

 doubleKei( double x )
This function uses two methods of approximation. If the argument falls below the MIN limit it uses the polynomial approximation :
$\mathrm{Kei\,}&space;x&space;=&space;-&space;\left(\ln&space;\frac{x}{2}&space;+&space;\gamma&space;\right)&space;\mathrm{Bei\,}x&space;-&space;\frac{\pi}{4}&space;\mathrm{Ber\,&space;}x&space;+&space; \sum_{n&space;\geq&space;0}&space;(-1)^n&space;\frac{\mathrm{H}_{2n+1}}{[(2n+1)!]^2}&space;&space;\left(&space;\frac{x}{2}\right)^{4n+2}$
where
$\gamma&space;\approx&space;0.577215664...&space;\quad&space;\mbox{(the&space;Euler-Mascheroni&space;constant)}&space;\quad&space;\mbox{and}&space;\quad&space; \mathrm{H}_n&space;=&space;\sum_{k=1}^n&space;\frac{1}{k}$

otherwise it calculates its asymptotic expansion :
$\mathrm{Kei\,}&space;x&space;=&space;\sqrt{\frac{\pi}{2x}}&space;\mathrm{e}^\frac{-x}{\sqrt{2}}&space;\left(-f(x)&space;\sin&space;\beta&space;-&space;g(x)&space;\cos&space;\beta&space;\right)$

where

$\beta&space;=&space;\frac{x}{\sqrt{2}}&space;+&space;\frac{\pi}{8}$
$f(x)&space;\sim&space;1&space;+&space;\sum_{n&space;\geq&space;1}&space;(-1)^n&space;\frac{1&space;\cdot&space;9&space;\cdot&space;\ldots&space;\cdot&space;(2n&space;-&space;1)^2}{n!(8x)^n}&space;\cos&space;\left(&space;\frac{n\pi}{4}&space;\right)$
$g(x)&space;\sim&space;\sum_{n&space;\geq&space;1}&space;(-1)^n&space;\frac{1&space;\cdot&space;9&space;\cdot&space;\ldots&space;\cdot&space;(2n&space;-&space;1)^2}{n!(8x)^n}&space;\sin&space;\left(&space;\frac{n\pi}{4}&space;\right)$

Parameters

 x The value at which the function is to be evaluated.

Authors

Lucian Bentea
Source Code

Source code is available when you buy a Commercial licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.

Ker

 doubleKer( double x )
This function uses two methods of approximation. If the argument falls below the MIN limit it uses the polynomial approximation :
$\mathrm{Ker\,}&space;x&space;=&space;-&space;\left(\ln&space;\frac{x}{2}&space;+&space;\gamma&space;\right)&space;\mathrm{Ber\,}x&space;-&space;\frac{\pi}{4}&space;\mathrm{Bei\,&space;}x&space;+&space; \sum_{n&space;\geq&space;0}&space;(-1)^n&space;\frac{\mathrm{H}_{2n}}{[(2n)!]^2}&space;&space;\left(&space;\frac{x}{2}\right)^{4n}$
where
$\gamma&space;\approx&space;0.577215664...&space;\quad&space;\mbox{(the&space;Euler-Mascheroni&space;constant)}&space;\quad&space;\mbox{and}&space;\quad&space; \mathrm{H}_n&space;=&space;\sum_{k=1}^n&space;\frac{1}{k}$

otherwise it calculates its asymptotic expansion :
$\mathrm{Ker\,}&space;x&space;=&space;\sqrt{\frac{\pi}{2x}}&space;\mathrm{e}^\frac{-x}{\sqrt{2}}&space;\left(f(x)&space;\cos&space;\beta&space;-&space;g(x)&space;\sin&space;\beta&space;\right)$

where

$\beta&space;=&space;\frac{x}{\sqrt{2}}&space;+&space;\frac{\pi}{8}$
$f(x)&space;\sim&space;1&space;+&space;\sum_{n&space;\geq&space;1}&space;(-1)^n&space;\frac{1&space;\cdot&space;9&space;\cdot&space;\ldots&space;\cdot&space;(2n&space;-&space;1)^2}{n!(8x)^n}&space;\cos&space;\left(&space;\frac{n\pi}{4}&space;\right)$
$g(x)&space;\sim&space;\sum_{n&space;\geq&space;1}&space;(-1)^n&space;\frac{1&space;\cdot&space;9&space;\cdot&space;\ldots&space;\cdot&space;(2n&space;-&space;1)^2}{n!(8x)^n}&space;\sin&space;\left(&space;\frac{n\pi}{4}&space;\right)$

Parameters

 x The value at which the function is to be evaluated.

Authors

Lucian Bentea
Source Code

Source code is available when you buy a Commercial licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.

DBei

 doubledBei( double x )
This function uses two methods of approximation. If the argument falls below the MIN limit it uses the polynomial resulted by differentiating the approximation polynomial of the Bei function. Otherwise it calculates its asymptotic expansion :

$\mathrm{Bei}'\,x&space;=&space;M&space;\sin&space;\left(&space;\theta&space;-&space;\frac{\pi}{4}&space;\right)$

where

$M&space;=&space;\frac{&space;\mathrm{e}^{&space;\frac{x}{\sqrt{2}}}&space;}{\sqrt{2\pi&space;x}}&space;\left(&space;1+\frac{1}{8\sqrt{2}x}&space;+&space;\frac{1}{256x^2}&space;-&space;\frac{399}{6144\sqrt{2}x^3}&space;+&space;\mathrm{O}&space;\left(&space;\frac{1}{x^4}&space;\right)&space;\right)$

$\theta&space;=&space;\frac{x}{\sqrt{2}}&space;-&space;\frac{\pi}{8}&space;-&space;\frac{1}{8\sqrt{2}x}&space;-&space;\frac{1}{16x^2}&space;-&space;\frac{25}{384\sqrt{2}x^3}&space;+&space;\mathrm{O}&space;\left(&space;\frac{1}{x^5}&space;\right)$

Parameters

 x The value at which the derivative is to be evaluated.

Authors

Lucian Bentea
Source Code

Source code is available when you buy a Commercial licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.

DBer

 doubledBer( double x )
This function uses two methods of approximation. If the argument falls below the MIN limit it uses the polynomial resulted by differentiating the approximation polynomial of the Ber function. Otherwise it calculates its asymptotic expansion :

$\mathrm{Ber}'\,x&space;=&space;M&space;\cos&space;\left(&space;\theta&space;-&space;\frac{\pi}{4}&space;\right)$

where

$M&space;=&space;\frac{&space;\mathrm{e}^{&space;\frac{x}{\sqrt{2}}}&space;}{\sqrt{2\pi&space;x}}&space;\left(&space;1+\frac{1}{8\sqrt{2}x}&space;+&space;\frac{1}{256x^2}&space;-&space;\frac{399}{6144\sqrt{2}x^3}&space;+&space;\mathrm{O}&space;\left(&space;\frac{1}{x^4}&space;\right)&space;\right)$

$\theta&space;=&space;\frac{x}{\sqrt{2}}&space;-&space;\frac{\pi}{8}&space;-&space;\frac{1}{8\sqrt{2}x}&space;-&space;\frac{1}{16x^2}&space;-&space;\frac{25}{384\sqrt{2}x^3}&space;+&space;\mathrm{O}&space;\left(&space;\frac{1}{x^5}&space;\right)$

Parameters

 x The value at which the derivative is to be evaluated.

Authors

Lucian Bentea
Source Code

Source code is available when you buy a Commercial licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.

DKei

 doubledKei( double x )
This function uses two methods of approximation. If the argument falls below the MIN limit it uses the polynomial resulted by differentiating the approximation polynomial of the Kei function. Otherwise it calculates its asymptotic expansion :

$\mathrm{Kei}'\,x&space;=&space;N&space;\sin&space;\left(&space;\phi&space;-&space;\frac{\pi}{4}&space;\right)$

where

$N&space;=&space;\sqrt{\frac{\pi}{2x}}&space;\mathrm{e}^{\frac{-x}{\sqrt{2}}}&space;\left(&space;1-\frac{1}{8\sqrt{2}x}&space;+&space;\frac{1}{256x^2}&space;+&space;\frac{399}{6144\sqrt{2}x^3}&space;+&space;\mathrm{O}&space;\left(&space;\frac{1}{x^4}&space;\right)&space;\right)$

$\phi&space;=&space;-\frac{x}{\sqrt{2}}&space;-&space;\frac{\pi}{8}&space;+&space;\frac{1}{8\sqrt{2}x}&space;-&space;\frac{1}{16x^2}&space;+&space;\frac{25}{384\sqrt{2}x^3}&space;+&space;\mathrm{O}&space;\left(&space;\frac{1}{x^5}&space;\right)$

Parameters

 x The value at which the derivative is to be evaluated.

Authors

Lucian Bentea
Source Code

Source code is available when you buy a Commercial licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.

DKer

 doubledKer( double x )
This function uses two methods of approximation. If the argument falls below the MIN limit it uses the polynomial resulted by differentiating the approximation polynomial of the Ker function. Otherwise it calculates its asymptotic expansion :

$\mathrm{Ker}'\,x&space;=&space;N&space;\cos&space;\left(&space;\phi&space;-&space;\frac{\pi}{4}&space;\right)$

where

$N&space;=&space;\sqrt{\frac{\pi}{2x}}&space;\mathrm{e}^{\frac{-x}{\sqrt{2}}}&space;\left(&space;1-\frac{1}{8\sqrt{2}x}&space;+&space;\frac{1}{256x^2}&space;+&space;\frac{399}{6144\sqrt{2}x^3}&space;+&space;\mathrm{O}&space;\left(&space;\frac{1}{x^4}&space;\right)&space;\right)$

$\phi&space;=&space;-\frac{x}{\sqrt{2}}&space;-&space;\frac{\pi}{8}&space;+&space;\frac{1}{8\sqrt{2}x}&space;-&space;\frac{1}{16x^2}&space;+&space;\frac{25}{384\sqrt{2}x^3}&space;+&space;\mathrm{O}&space;\left(&space;\frac{1}{x^5}&space;\right)$

Parameters

 x The value at which the derivative is to be evaluated.

Authors

Lucian Bentea
Source Code

Source code is available when you buy a Commercial licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.