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Function used at calculating asymptotic expansions.

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Interface

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

using namespace Maths::Special::Bessel;

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 Ber(double);
double Bei(double);
double Ker(double);
double Kei(double);
double dBer(double);
double dBei(double);
double dKer(double);
double dKei(double);
double Bei(double x)
double Ber(double x)
double Kei(double x)
double Ker(double x)
double dBei(double x)
double dBer(double x)
double dKei(double x)
double dKer(double x)

Overview

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

FUNCTION

f1

Interactive Calculator

Result –

FUNCTION

f2

Interactive Calculator

Result –

FUNCTION

g1

Interactive Calculator

Result –

FUNCTION

g2

Interactive Calculator

Result –

FUNCTION

M

Interactive Calculator

Result –

FUNCTION

theta

Interactive Calculator

Result –

FUNCTION

N

Interactive Calculator

Result –

FUNCTION

phi

Interactive Calculator

Result –

FUNCTION

Ber

FUNCTION

Bei

FUNCTION

Ker

FUNCTION

Kei

FUNCTION

dBer

FUNCTION

dBei

FUNCTION

dKer

FUNCTION

dKei

FUNCTION

Bei

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

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

(1)

otherwise it calculates its asymptotic expansion :

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

(2)

where

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

(3)

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

(4)

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

(5)

Parameters

The value at which the function is to be evaluated.

Author

Lucian Bentea

Interactive Calculator

Result –

FUNCTION

Ber

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

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

(6)

otherwise it calculates its asymptotic expansion :

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

(7)

where

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

(8)

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

(9)

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

(10)

Parameters

The value at which the function is to be evaluated.

Author

Lucian Bentea

Interactive Calculator

Result –

FUNCTION

Kei

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

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

(11)

where

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

(12)

otherwise it calculates its asymptotic expansion :

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

(13)

where

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

(14)

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

(15)

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

(16)

Parameters

The value at which the function is to be evaluated.

Author

Lucian Bentea

Interactive Calculator

Result –

FUNCTION

Ker

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

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

(17)

where

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

(18)

otherwise it calculates its asymptotic expansion :

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

(19)

where

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

(20)

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

(21)

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

(22)

Parameters

The value at which the function is to be evaluated.

Author

Lucian Bentea

Interactive Calculator

Result –

FUNCTION

dBei

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 :

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

(23)

where

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

(24)

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

(25)

Parameters

The value at which the derivative is to be evaluated.

Author

Lucian Bentea

Interactive Calculator

Result –

FUNCTION

dBer

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 :

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

(26)

where

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

(27)

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

(28)

Parameters

The value at which the derivative is to be evaluated.

Author

Lucian Bentea

Interactive Calculator

Result –

FUNCTION

dKei

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 :

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

(29)

where

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

(30)

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

(31)

Parameters

The value at which the derivative is to be evaluated.

Author

Lucian Bentea

Interactive Calculator

Result –

FUNCTION

dKer

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 :

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

(32)

where

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

(33)

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

(34)

Parameters

The value at which the derivative is to be evaluated.

Author

Lucian Bentea

Interactive Calculator

Result –