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Maths › Special › Bessel ›

Kelvin

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Function used at calculating asymptotic expansions.

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Contents

Interface
Bei
Ber
Kei
Ker
DBei
DBer
DKei
DKer
Page Comments

Interface

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 :

$\mathrm{Bei\,} x = \sum_{n \geq 0} \frac { (-1)^n (\frac{1}{2}x)^{4n+2}} {[(2n+1)!]^2}$

(2)

otherwise it calculates its asymptotic expansion :

$\mathrm{Bei\,} x = \frac { \mathrm{e}^{ \frac{x}{\sqrt{2}} } }{ \sqrt{2\pi x} } \left( f(x) \sin \alpha + g(x) \cos \alpha \right) + \frac{ \mathrm{Ker\,} x}{\pi}$

(3)

where

$\alpha = \frac{x}{\sqrt{2}} - \frac{\pi}{8}$

(4)

$f(x) \sim 1 + \sum_{n \geq 1} \frac{1 \cdot 9 \cdot \ldots \cdot (2n - 1)^2}{n!(8x)^n} \cos \left( \frac{n\pi}{4} \right)$

(5)

$g(x) \sim \sum_{n \geq 1} \frac{1 \cdot 9 \cdot \ldots \cdot (2n - 1)^2}{n!(8x)^n} \sin \left( \frac{n\pi}{4} \right)$

(6)

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.

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Ber

doubleBer( double x )

This function uses two methods of approximation. If the argument falls below the MIN limit it uses the polynomial approximation :

$\mathrm{Ber\,} x = 1 + \sum_{n \geq 1} \frac { (-1)^n (\frac{1}{2}x)^{4n}} {[(2n)!]^2}$

(7)

otherwise it calculates its asymptotic expansion :

$\mathrm{Bei\,} x = \frac { \mathrm{e}^{ \frac{x}{\sqrt{2}} } }{ \sqrt{2\pi x} } \left( f(x) \cos \alpha + g(x) \sin \alpha \right) - \frac{ \mathrm{Kei\,} x}{\pi}$

(8)

where

$\alpha = \frac{x}{\sqrt{2}} - \frac{\pi}{8}$

(9)

$f(x) \sim 1 + \sum_{n \geq 1} \frac{1 \cdot 9 \cdot \ldots \cdot (2n - 1)^2}{n!(8x)^n} \cos \left( \frac{n\pi}{4} \right)$

(10)

$g(x) \sim \sum_{n \geq 1} \frac{1 \cdot 9 \cdot \ldots \cdot (2n - 1)^2}{n!(8x)^n} \sin \left( \frac{n\pi}{4} \right)$

(11)

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.

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Kei

doubleKei( double x )

This function uses two methods of approximation. If the argument falls below the MIN limit it uses the polynomial approximation :

$\mathrm{Kei\,} x = - \left(\ln \frac{x}{2} + \gamma \right) \mathrm{Bei\,}x - \frac{\pi}{4} \mathrm{Ber\, }x + \sum_{n \geq 0} (-1)^n \frac{\mathrm{H}_{2n+1}}{[(2n+1)!]^2} \left( \frac{x}{2}\right)^{4n+2}$

(12)

where

$\gamma \approx 0.577215664... \quad \mbox{(the Euler-Mascheroni constant)} \quad \mbox{and} \quad \mathrm{H}_n = \sum_{k=1}^n \frac{1}{k}$

(13)

otherwise it calculates its asymptotic expansion :

$\mathrm{Kei\,} x = \sqrt{\frac{\pi}{2x}} \mathrm{e}^\frac{-x}{\sqrt{2}} \left(-f(x) \sin \beta - g(x) \cos \beta \right)$

(14)

where

$\beta = \frac{x}{\sqrt{2}} + \frac{\pi}{8}$

(15)

$f(x) \sim 1 + \sum_{n \geq 1} (-1)^n \frac{1 \cdot 9 \cdot \ldots \cdot (2n - 1)^2}{n!(8x)^n} \cos \left( \frac{n\pi}{4} \right)$

(16)

$g(x) \sim \sum_{n \geq 1} (-1)^n \frac{1 \cdot 9 \cdot \ldots \cdot (2n - 1)^2}{n!(8x)^n} \sin \left( \frac{n\pi}{4} \right)$

(17)

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.

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Ker

doubleKer( double x )

This function uses two methods of approximation. If the argument falls below the MIN limit it uses the polynomial approximation :

$\mathrm{Ker\,} x = - \left(\ln \frac{x}{2} + \gamma \right) \mathrm{Ber\,}x - \frac{\pi}{4} \mathrm{Bei\, }x + \sum_{n \geq 0} (-1)^n \frac{\mathrm{H}_{2n}}{[(2n)!]^2} \left( \frac{x}{2}\right)^{4n}$

(18)

where

$\gamma \approx 0.577215664... \quad \mbox{(the Euler-Mascheroni constant)} \quad \mbox{and} \quad \mathrm{H}_n = \sum_{k=1}^n \frac{1}{k}$

(19)

otherwise it calculates its asymptotic expansion :

$\mathrm{Ker\,} x = \sqrt{\frac{\pi}{2x}} \mathrm{e}^\frac{-x}{\sqrt{2}} \left(f(x) \cos \beta - g(x) \sin \beta \right)$

(20)

where

$\beta = \frac{x}{\sqrt{2}} + \frac{\pi}{8}$

(21)

$f(x) \sim 1 + \sum_{n \geq 1} (-1)^n \frac{1 \cdot 9 \cdot \ldots \cdot (2n - 1)^2}{n!(8x)^n} \cos \left( \frac{n\pi}{4} \right)$

(22)

$g(x) \sim \sum_{n \geq 1} (-1)^n \frac{1 \cdot 9 \cdot \ldots \cdot (2n - 1)^2}{n!(8x)^n} \sin \left( \frac{n\pi}{4} \right)$

(23)

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.

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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 :

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

(24)

where

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

(25)

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

(26)

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.

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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 :

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

(27)

where

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

(28)

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

(29)

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.

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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 :

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

(30)

where

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

(31)

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

(32)

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.

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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 :

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

(33)

where

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

(34)

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

(35)

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.

double	`f1` (double x)
double	`f2` (double x)
double	`g1` (double x)
double	`g2` (double x)
double	`M` (double x)
double	`theta` (double x)
double	`N` (double x)
double	`phi` (double x)
double	`Bei` (double x) Evaluates the Bei function.
double	`Ber` (double x) Evaluates the Ber function.
double	`Kei` (double x) Evaluates the Kei function.
double	`Ker` (double x) Evaluates the Ker function.
double	`dBei` (double x) Evaluates the derivative of the Bei function.
double	`dBer` (double x) Evaluates the derivative of the Ber function.
double	`dKei` (double x) Evaluates the derivative of the Kei function.
double	`dKer` (double x) Evaluates the derivative of the Ker function.