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maths › special › bessel ›

Kelvin

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Function used at calculating asymptotic expansions.

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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)
ExcelReal cc_f1 (Real x)
This function is available as a Microsoft Excel add-in.
double f2 (double x)
ExcelReal cc_f2 (Real x)
This function is available as a Microsoft Excel add-in.
double g1 (double x)
ExcelReal cc_g1 (Real x)
This function is available as a Microsoft Excel add-in.
double g2 (double x)
ExcelReal cc_g2 (Real x)
This function is available as a Microsoft Excel add-in.
double M (double x)
ExcelReal cc_M (Real x)
This function is available as a Microsoft Excel add-in.
double theta (double x)
ExcelReal cc_theta (Real x)
This function is available as a Microsoft Excel add-in.
double N (double x)
ExcelReal cc_N (Real x)
This function is available as a Microsoft Excel add-in.
double phi (double x)
ExcelReal cc_phi (Real x)
This function is available as a Microsoft Excel add-in.
double Bei (double x)
Evaluates the Bei function.
ExcelReal cc_Bei (Real x)
This function is available as a Microsoft Excel add-in.
double Ber (double x)
Evaluates the Ber function.
ExcelReal cc_Ber (Real x)
This function is available as a Microsoft Excel add-in.
double Kei (double x)
Evaluates the Kei function.
ExcelReal cc_Kei (Real x)
This function is available as a Microsoft Excel add-in.
double Ker (double x)
Evaluates the Ker function.
ExcelReal 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.
ExcelReal 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.
ExcelReal 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.
ExcelReal 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.
ExcelReal cc_dKer (Real x)
This function is available as a Microsoft Excel add-in.

Function Documentation

Bei Calculator
  
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doubleBei( doublex )
This function uses two methods of approximation. If the argument falls below the MIN limit it uses the polynomial approximation :
(1)
\displaystyle \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)
\displaystyle \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)
\displaystyle \alpha = \frac{x}{\sqrt{2}} - \frac{\pi}{8}
(4)
\displaystyle 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)
\displaystyle 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:
xThe value at which the function is to be evaluated.
Authors:
Lucian Bentea
Source Code:

You do not own any licences for this module.
To view or download source code you must get a GPL licence or buy a commercial licence.

buy now     get GPL     add to cart

For advanced download and development options Register with CodeCogs. Already a Member, then Login.


Ber Calculator
  
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doubleBer( doublex )
This function uses two methods of approximation. If the argument falls below the MIN limit it uses the polynomial approximation :
(6)
\displaystyle \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)
\displaystyle \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)
\displaystyle \alpha = \frac{x}{\sqrt{2}} - \frac{\pi}{8}
(9)
\displaystyle 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)
\displaystyle 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:
xThe value at which the function is to be evaluated.
Authors:
Lucian Bentea
Source Code:

You do not own any licences for this module.
To view or download source code you must get a GPL licence or buy a commercial licence.

buy now     get GPL     add to cart

For advanced download and development options Register with CodeCogs. Already a Member, then Login.


Kei Calculator
  
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doubleKei( doublex )
This function uses two methods of approximation. If the argument falls below the MIN limit it uses the polynomial approximation :
(11)
\displaystyle \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)
\displaystyle \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)
\displaystyle \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)
\displaystyle \beta = \frac{x}{\sqrt{2}} + \frac{\pi}{8}
(15)
\displaystyle 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)
\displaystyle 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:
xThe value at which the function is to be evaluated.
Authors:
Lucian Bentea
Source Code:

You do not own any licences for this module.
To view or download source code you must get a GPL licence or buy a commercial licence.

buy now     get GPL     add to cart

For advanced download and development options Register with CodeCogs. Already a Member, then Login.


Ker Calculator
  
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doubleKer( doublex )
This function uses two methods of approximation. If the argument falls below the MIN limit it uses the polynomial approximation :
(17)
\displaystyle \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)
\displaystyle \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)
\displaystyle \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)
\displaystyle \beta = \frac{x}{\sqrt{2}} + \frac{\pi}{8}
(21)
\displaystyle 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)
\displaystyle 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:
xThe value at which the function is to be evaluated.
Authors:
Lucian Bentea
Source Code:

You do not own any licences for this module.
To view or download source code you must get a GPL licence or buy a commercial licence.

buy now     get GPL     add to cart

For advanced download and development options Register with CodeCogs. Already a Member, then Login.


D Bei Calculator
  
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doubledBei( doublex )
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)
\displaystyle \mathrm{Bei}'\,x = M \sin \left( \theta - \frac{\pi}{4} \right)

where

(24)
\displaystyle 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)
\displaystyle \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:
xThe value at which the derivative is to be evaluated.
Authors:
Lucian Bentea
Source Code:

You do not own any licences for this module.
To view or download source code you must get a GPL licence or buy a commercial licence.

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D Ber Calculator
  
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doubledBer( doublex )
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)
\displaystyle \mathrm{Ber}'\,x = M \cos \left( \theta - \frac{\pi}{4} \right)

where

(27)
\displaystyle 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)
\displaystyle \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:
xThe value at which the derivative is to be evaluated.
Authors:
Lucian Bentea
Source Code:

You do not own any licences for this module.
To view or download source code you must get a GPL licence or buy a commercial licence.

buy now     get GPL     add to cart

For advanced download and development options Register with CodeCogs. Already a Member, then Login.


D Kei Calculator
  
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doubledKei( doublex )
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)
\displaystyle \mathrm{Kei}'\,x = N \sin \left( \phi - \frac{\pi}{4} \right)

where

(30)
\displaystyle 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)
\displaystyle \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:
xThe value at which the derivative is to be evaluated.
Authors:
Lucian Bentea
Source Code:

You do not own any licences for this module.
To view or download source code you must get a GPL licence or buy a commercial licence.

buy now     get GPL     add to cart

For advanced download and development options Register with CodeCogs. Already a Member, then Login.


D Ker Calculator
  
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doubledKer( doublex )
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)
\displaystyle \mathrm{Ker}'\,x = N \cos \left( \phi - \frac{\pi}{4} \right)

where

(33)
\displaystyle 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)
\displaystyle \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:
xThe value at which the derivative is to be evaluated.
Authors:
Lucian Bentea
Source Code:

You do not own any licences for this module.
To view or download source code you must get a GPL licence or buy a commercial licence.

buy now     get GPL     add to cart

For advanced download and development options Register with CodeCogs. Already a Member, then Login.


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