SpecialFunctions Class Reference

This class is subject to change. Programs using this class may require changes with a new version of VirtualLab.
Class containing special (named) functions like the Laguerre, Bessel, Legendre, ... functions. The methods are just static methods, they are not derived from BaseFunction class. More...

Static Public Member Functions
static VectorD	ConvertToEllipticalCoordinatesForIncePolynomials (VectorD rhoC, double f)
	Transformation of coordinate system, from Cartesian to Elliptic More...
static DoubleComplexVector	EvenInceCoefficient (int p, int m, double ellipticity)
	Calculation of Coefficient Series of the Even Ince Polynomials (Cpm) More...
static Complex	FirstKindBesselFunction (double n, Complex z)
	Calculates the Bessel function of the first kind. More...
static Complex	FirstKindHankelFunction (double order, Complex coordinate)
	Calculates the Hankel function of the first kind. More...
static double	FirstKindSphericalBesselDerivative (int order, Complex coordinate)
	Calculates the derivative of the Bessel function of the first kind. More...
static Complex	FirstKindSphericalBesselFunction (int order, Complex coordinate)
	Calculates the Bessel function of the first kind for spherical coordinates. More...
static Complex	FirstKindSphericalHankelDerivative (Complex coordinate, int order)
	Calculates the derivative of the Hankel function of the first kind. More...
static Complex	FirstKindSphericalHankelFunction (int order, Complex coordinate)
	Calculates the Hankel function of the first kind for spherical coordinates. More...
static double	HermitePolynomial (int order, double position)
	Calculates the Hermite Polynomial of order m at position x. More...
static Complex	IncePoly (int p, Complex eta, DoubleComplexVector Coef, bool evenPolynomial)
	Calculation of Ince Polynomials on one point More...
static double	LaguerrePolynomial (int p, int l, double x)
	Calculates the Laguerre Polynomial. More...
static double	LegendrePolynomial0 (int order, double coordinate)
	Method to calculate the first kind Legendre polynomial More...
static double	LegendrePolynomial1 (int order, double coordinate)
	Method to calculate the second kind Legendre polynomial More...
static double	LegendrePolynomial2 (int order, double coordinate)
	Method to calculate the third kind Legendre polynomial More...
static DoubleComplexVector	OddInceCoefficient (int p, int m, double ellipticity)
	Calculation of Coefficient Series of the Odd Ince Polynomials (Spm) More...
static Complex	SecondKindBesselFunction (double n, Complex z)
	Calculates the Bessel function of the second kind. More...
static Complex	SecondKindSphericalBesselFunction (int order, Complex coordinate)
	Calculates the Bessel function of the second kind for spherical coordinates. More...
static void	SetTruncationFactorBessel (double truncationAccuracy)
	Sets the truncation factor for truncated calculation of Bessel functions. More...

Detailed Description

This class is subject to change. Programs using this class may require changes with a new version of VirtualLab.
Class containing special (named) functions like the Laguerre, Bessel, Legendre, ... functions. The methods are just static methods, they are not derived from BaseFunction class.

Member Function Documentation

ConvertToEllipticalCoordinatesForIncePolynomials()

static VectorD ConvertToEllipticalCoordinatesForIncePolynomials ( VectorD  rhoC, double  f  )

static

Transformation of coordinate system, from Cartesian to Elliptic

Parameters

rhoC	VectorD, position vector in Cartesian CS, rhoC = (x, y)
f	double, semi-focal separation at the waist plane of the Ince Gaussian beam

Returns

VectorD, position vector in Elliptic CS, rhoE = (xi, eta)

EvenInceCoefficient()

static DoubleComplexVector EvenInceCoefficient ( int  p, int  m, double  ellipticity  )

static

Calculation of Coefficient Series of the Even Ince Polynomials (Cpm)

Parameters

p	int, order of the Ince Gaussian beam
m	int, degree of the Ince Gaussian beam, m and p must have same parity
ellipticity	ellipticity parameter

Returns

DoubleComplexVector, coefficient series - A()

FirstKindBesselFunction()

static Complex FirstKindBesselFunction ( double  n, Complex  z  )

static

Calculates the Bessel function of the first kind.

Parameters

n	The order (half-integer) to evaluate.
z	The argument of the Bessel Function

Returns

The value of the Bessel function for the given input parameters.

FirstKindHankelFunction()

static Complex FirstKindHankelFunction ( double  order, Complex  coordinate  )

static

Calculates the Hankel function of the first kind.

Parameters

order	The order (non-integer) to evaluate.
coordinate	The argument of the Hankel Function

Returns

The value of the Hankel function for the given input parameters.

FirstKindSphericalBesselDerivative()

static double FirstKindSphericalBesselDerivative ( int  order, Complex  coordinate  )

static

Calculates the derivative of the Bessel function of the first kind.

Parameters

order	The order to evaluate.
coordinate	The argument of the Bessel Function

Returns

The derivative of the Bessel function for the given input parameters.

FirstKindSphericalBesselFunction()

static Complex FirstKindSphericalBesselFunction ( int  order, Complex  coordinate  )

static

Calculates the Bessel function of the first kind for spherical coordinates.

Parameters

order	The order to evaluate.
coordinate	The argument of the Bessel Function in spherical coordinates.

Returns

The value of the Bessel function for the given input parameters.

FirstKindSphericalHankelDerivative()

static Complex FirstKindSphericalHankelDerivative ( Complex  coordinate, int  order  )

static

Calculates the derivative of the Hankel function of the first kind.

Parameters

order	The order to evaluate.
coordinate	The argument of the Hankel Function

Returns

The derivative of the Hankel function for the given input parameters.

FirstKindSphericalHankelFunction()

static Complex FirstKindSphericalHankelFunction ( int  order, Complex  coordinate  )

static

Calculates the Hankel function of the first kind for spherical coordinates.

Parameters

order	The order to evaluate.
coordinate	The argument of the Hankel Function in spherical coordinates.

Returns

The value of the Hankel function for the given input parameters.

HermitePolynomial()

static double HermitePolynomial ( int  order, double  position  )

static

Calculates the Hermite Polynomial of order m at position x.

Parameters

order	The order of the polynomial to evaluate.
position	the coordinate to evaluate.

Returns

The value of the Hermite polynomial for the given input parameters.

IncePoly()

static Complex IncePoly ( int  p, Complex  eta, DoubleComplexVector  Coef, bool  evenPolynomial  )

static

Calculation of Ince Polynomials on one point

Parameters

p	int, order of the Ince Gaussian beam
eta	double, angular elliptic position variable
evenPolynomial	Whether even Ince polynomials (true) are used, or odd Ince polynomials (false).
Coef	DoubleComplexVector, coefficient series of Ince polynomials

Returns

DoubleComplexVector, coefficient series - A()

LaguerrePolynomial()

static double LaguerrePolynomial ( int  p, int  l, double  x  )

static

Calculates the Laguerre Polynomial.

Parameters

p	The radial order.
l	The angular order.
x	the coordinate to evaluate.

Returns

The value of the Laguerre polynomial for the given input parameters.

LegendrePolynomial0()

static double LegendrePolynomial0 ( int  order, double  coordinate  )

static

Method to calculate the first kind Legendre polynomial

Parameters

order	the order of the polynomial to evaluate
coordinate	the coordinate to evaluate

Returns

The value of the Legendre polynomial for the given input parameters.

LegendrePolynomial1()

static double LegendrePolynomial1 ( int  order, double  coordinate  )

static

Method to calculate the second kind Legendre polynomial

Parameters

order	the order of the polynomial to evaluate
coordinate	the coordinate to evaluate

Returns

the value of the Legendre polynomial for the given input parameters

LegendrePolynomial2()

static double LegendrePolynomial2 ( int  order, double  coordinate  )

static

Method to calculate the third kind Legendre polynomial

Parameters

order	the order of the polynomial to evaluate
coordinate	the coordinate to evaluate

Returns

the value of the Legendre polynomial for the given input parameters

OddInceCoefficient()

static DoubleComplexVector OddInceCoefficient ( int  p, int  m, double  ellipticity  )

static

Calculation of Coefficient Series of the Odd Ince Polynomials (Spm)

Parameters

p	order of the Ince Gaussian beam
m	degree of the Ince Gaussian beam, m and p must have same parity
ellipticity	ellipticity parameter

Returns

DoubleComplexVector, coefficient series - B()

SecondKindBesselFunction()

static Complex SecondKindBesselFunction ( double  n, Complex  z  )

static

Calculates the Bessel function of the second kind.

Parameters

n	The order (half-integer) to evaluate.
z	The argument of the Bessel Function

Returns

The value of the Bessel function for the given input parameters.

SecondKindSphericalBesselFunction()

static Complex SecondKindSphericalBesselFunction ( int  order, Complex  coordinate  )

static

Calculates the Bessel function of the second kind for spherical coordinates.

Parameters

order	The order to evaluate.
coordinate	The argument of the Bessel Function in spherical coordinates.

Returns

The value of the Bessel function for the given input parameters.

SetTruncationFactorBessel()

static void SetTruncationFactorBessel ( double  truncationAccuracy )

static

Sets the truncation factor for truncated calculation of Bessel functions.

Parameters

truncationAccuracy

The truncation accuracy \(a\). For that the truncation factor \(f = 10^{-a}\) is calculated.