Maths › Special › Bessel ›

Kelvin

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Function used at calculating asymptotic expansions.

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Contents

Interface
Function Documentation
Page Comments

Group Description

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

Interface

#include <codecogs/maths/special/bessel/kelvin.h>

using namespace Maths::Special::Bessel;

	double	`f1` (double x)
	Real	cc_f1 (Real x) This function is available as a Microsoft Excel add-in.
	double	`f2` (double x)
	Real	cc_f2 (Real x) This function is available as a Microsoft Excel add-in.
	double	`g1` (double x)
	Real	cc_g1 (Real x) This function is available as a Microsoft Excel add-in.
	double	`g2` (double x)
	Real	cc_g2 (Real x) This function is available as a Microsoft Excel add-in.
	double	`M` (double x)
	Real	cc_M (Real x) This function is available as a Microsoft Excel add-in.
	double	`theta` (double x)
	Real	cc_theta (Real x) This function is available as a Microsoft Excel add-in.
	double	`N` (double x)
	Real	cc_N (Real x) This function is available as a Microsoft Excel add-in.
	double	`phi` (double x)
	Real	cc_phi (Real x) This function is available as a Microsoft Excel add-in.
	double	`Bei` (double x) Evaluates the Bei function.
	Real	cc_Bei (Real x) This function is available as a Microsoft Excel add-in.
	double	`Ber` (double x) Evaluates the Ber function.
	Real	cc_Ber (Real x) This function is available as a Microsoft Excel add-in.
	double	`Kei` (double x) Evaluates the Kei function.
	Real	cc_Kei (Real x) This function is available as a Microsoft Excel add-in.
	double	`Ker` (double x) Evaluates the Ker function.
	Real	cc_Ker (Real x) This function is available as a Microsoft Excel add-in.
	double	`dBei` (double x) Evaluates the derivative of the Bei function.
	Real	cc_dBei (Real x) This function is available as a Microsoft Excel add-in.
	double	`dBer` (double x) Evaluates the derivative of the Ber function.
	Real	cc_dBer (Real x) This function is available as a Microsoft Excel add-in.
	double	`dKei` (double x) Evaluates the derivative of the Kei function.
	Real	cc_dKei (Real x) This function is available as a Microsoft Excel add-in.
	double	`dKer` (double x) Evaluates the derivative of the Ker function.
	Real	cc_dKer (Real x) This function is available as a Microsoft Excel add-in.

Function Documentation

Bei Calculator

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doubleBei( double x )

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

(1)

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

otherwise it calculates its asymptotic expansion :

(2)

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

where

(3)

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

(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)

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

Parameters:

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

Authors:

Lucian Bentea

Source Code:

To view or download source code you need either a GPL or Commercial Licence.

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Ber Calculator

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doubleBer( double x )

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

(6)

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

otherwise it calculates its asymptotic expansion :

(7)

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

where

(8)

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

(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)

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

Parameters:

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

Authors:

Lucian Bentea

Source Code:

To view or download source code you need either a GPL or Commercial Licence.

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

Kei Calculator

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doubleKei( double x )

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

(11)

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

where

(12)

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

otherwise it calculates its asymptotic expansion :

(13)

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

where

(14)

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

(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)

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

Parameters:

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

Authors:

Lucian Bentea

Source Code:

To view or download source code you need either a GPL or Commercial Licence.

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

Ker Calculator

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doubleKer( double x )

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

(17)

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

where

(18)

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

otherwise it calculates its asymptotic expansion :

(19)

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

where

(20)

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

(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)

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

Parameters:

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

Authors:

Lucian Bentea

Source Code:

To view or download source code you need either a GPL or Commercial Licence.

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

D Bei Calculator

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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 :

(23)

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

where

(24)

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

(25)

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

Parameters:

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

Authors:

Lucian Bentea

Source Code:

To view or download source code you need either a GPL or Commercial Licence.

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

D Ber Calculator

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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 :

(26)

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

where

(27)

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

(28)

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

Parameters:

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

Authors:

Lucian Bentea

Source Code:

To view or download source code you need either a GPL or Commercial Licence.

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D Kei Calculator

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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 :

(29)

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

where

(30)

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

(31)

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

Parameters:

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

Authors:

Lucian Bentea

Source Code:

To view or download source code you need either a GPL or Commercial Licence.

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

D Ker Calculator

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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 :

(32)

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

where

(33)

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

(34)

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

Parameters:

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

Authors:

Lucian Bentea

Source Code:

To view or download source code you need either a GPL or Commercial Licence.

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

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Last Modified: 26 Jan 08 @ 22:09 Page Rendered: 2010-03-10 11:25:38