Kohlrausch (FsODE)

FsODE

Documentation for 0.0.2

Kohlrausch Type

Namespace: Altaxo.Calc
Assembly: AltaxoCore.Redist.dll
Base Type: obj

The Kohlrausch function in the frequency domain.

Constructors

Constructor	Description
`Kohlrausch()` Full Usage: `Kohlrausch()`

Static members

Static member	Description
`Kohlrausch.GetBetaLnWOfMaximum(beta)` Full Usage: `Kohlrausch.GetBetaLnWOfMaximum(beta)` Parameters: beta : `float` - The beta parameter. Returns: `float` The maximum location (beta*ln(w)) of the imaginary part.	Returns (approximately) the location of the maximum of the imaginary part of the Fourier transformed Kohlrausch function. beta : `float` The beta parameter. Returns: `float` The maximum location (beta*ln(w)) of the imaginary part.
`Kohlrausch.GetLnWOfMaximum(beta)` Full Usage: `Kohlrausch.GetLnWOfMaximum(beta)` Parameters: beta : `float` - The beta parameter. Returns: `float` The maximum location ln(w) of the imaginary part.	Returns (approximately) the location of the maximum of the imaginary part of the Fourier transformed Kohlrausch function. beta : `float` The beta parameter. Returns: `float` The maximum location ln(w) of the imaginary part.
`Kohlrausch.GetMaximumImaginaryPart(beta)` Full Usage: `Kohlrausch.GetMaximumImaginaryPart(beta)` Parameters: beta : `float` - Returns: `float`	Returns (approximately) the maximum value of the imaginary part of the Fourier transformed Kohlrausch function. beta : `float` Returns: `float`
`Kohlrausch.GetWOfMaximum(beta)` Full Usage: `Kohlrausch.GetWOfMaximum(beta)` Parameters: beta : `float` - The beta parameter. Returns: `float` The maximum location ln(w) of the imaginary part.	Returns (approximately) the location of the maximum of the imaginary part of the Fourier transformed Kohlrausch function. beta : `float` The beta parameter. Returns: `float` The maximum location ln(w) of the imaginary part.
`Kohlrausch.Im(beta, w)` Full Usage: `Kohlrausch.Im(beta, w)` Parameters: beta : `float` - Beta parameter (0..1). w : `float` - Circular frequency. Returns: `float` Real part of the Fourier transformed derivative of the Kohlrausch function, or double.NaN if the function can not be evaluated (can happen for beta smaller than 1/64).	Imaginary part of the Fourier transformed derivative of the Kohlrausch function for all frequencies. In dependence on the parameters, either a series expansion (accuracy ca. 1E-14) or a bivariate Akima spline (accuracy 1E-4) is used. This is the imaginary part of the Fourier transform (in Mathematica notation): Im[Integrate[-D[Exp[-t^beta],t]*Exp[-I w t],{t, 0, Infinity}]]. For beta smaller than one, the return value is always negative. Wanted! Who can contribute to a more accurate interpolation between the points? (The points itself are calculated with an accuracy of 1E-18.) beta : `float` Beta parameter (0..1). w : `float` Circular frequency. Returns: `float` Real part of the Fourier transformed derivative of the Kohlrausch function, or double.NaN if the function can not be evaluated (can happen for beta smaller than 1/64).
`Kohlrausch.Im1(beta, z)` Full Usage: `Kohlrausch.Im1(beta, z)` Parameters: beta : `float` - Beta parameter. z : `float` - Circular frequency. Returns: `float` Imaginary part of the Fourier transformed derivative of the Kohlrausch function for high frequencies, or double.NaN if the series not converges.	Imaginary part of the Fourier transformed derivative of the Kohlrausch function for high frequencies. This is the imaginary part of the Fourier transform (in Mathematica notation): Im[Integrate[D[Exp[-t^beta],t]*Exp[-I w t],{t, 0, Infinity}]]. The sign of the return value here is positive!. beta : `float` Beta parameter. z : `float` Circular frequency. Returns: `float` Imaginary part of the Fourier transformed derivative of the Kohlrausch function for high frequencies, or double.NaN if the series not converges.
`Kohlrausch.Im1OldStyle(beta, z)` Full Usage: `Kohlrausch.Im1OldStyle(beta, z)` Parameters: beta : `float` - Beta parameter. z : `float` - Circular frequency. Returns: `float` Imaginary part of the Fourier transformed derivative of the Kohlrausch function for high frequencies, or double.NaN if the series not converges.	Imaginary part of the Fourier transformed derivative of the Kohlrausch function for high frequencies. This is the imaginary part of the Fourier transform (in Mathematica notation): Im[Integrate[D[Exp[-t^beta],t]*Exp[-I w t],{t, 0, Infinity}]]. The sign of the return value here is positive!. beta : `float` Beta parameter. z : `float` Circular frequency. Returns: `float` Imaginary part of the Fourier transformed derivative of the Kohlrausch function for high frequencies, or double.NaN if the series not converges.
`Kohlrausch.Im2(beta, z)` Full Usage: `Kohlrausch.Im2(beta, z)` Parameters: beta : `float` - Beta parameter. z : `float` - Circular frequency. Returns: `float` Imaginary part of the Fourier transformed derivative of the Kohlrausch function for high frequencies, or double.NaN if the series not converges.Imaginary part of the Fourier transformed derivative of the Kohlrausch function for high frequencies, or double.NaN if the series not converges.	Imaginary part of the Fourier transformed derivative of the Kohlrausch function for low frequencies. This is the imaginary part of the Fourier transform (in Mathematica notation): Im[Integrate[D[Exp[-t^beta],t]*Exp[-I w t],{t, 0, Infinity}]]. The sign of the return value here is positive!. beta : `float` Beta parameter. z : `float` Circular frequency. Returns: `float` Imaginary part of the Fourier transformed derivative of the Kohlrausch function for high frequencies, or double.NaN if the series not converges.Imaginary part of the Fourier transformed derivative of the Kohlrausch function for high frequencies, or double.NaN if the series not converges.
`Kohlrausch.Im2SmallBeta(beta, z)` Full Usage: `Kohlrausch.Im2SmallBeta(beta, z)` Parameters: beta : `float` - Beta parameter. z : `float` - Circular frequency. Returns: `float` Imaginary part of the Fourier transformed derivative of the Kohlrausch function for low frequencies, or double.NaN if the series not converges.Imaginary part of the Fourier transformed derivative of the Kohlrausch function for high frequencies, or double.NaN if the series not converges.	Imaginary part of the Fourier transformed derivative of the Kohlrausch function for low frequencies, and beta<=1/20.. This is the imaginary part of the Fourier transform (in Mathematica notation): Im[Integrate[D[Exp[-t^beta],t]*Exp[-I w t],{t, 0, Infinity}]]. The sign of the return value here is positive!. beta : `float` Beta parameter. z : `float` Circular frequency. Returns: `float` Imaginary part of the Fourier transformed derivative of the Kohlrausch function for low frequencies, or double.NaN if the series not converges.Imaginary part of the Fourier transformed derivative of the Kohlrausch function for high frequencies, or double.NaN if the series not converges.
`Kohlrausch.Re(beta, w)` Full Usage: `Kohlrausch.Re(beta, w)` Parameters: beta : `float` - Beta parameter (0..1). w : `float` - Circular frequency. Returns: `float` Real part of the Fourier transformed derivative of the Kohlrausch function, or double.NaN if the function can not be evaluated (can happen for beta smaller than 1/64).	Real part of the Fourier transformed derivative of the Kohlrausch function for all frequencies. In dependence on the parameters, either a series expansion (accuracy ca. 1E-14) or a bivariate akima spline (accuracy 1E-4) is used. This is the real part of the Fourier transform (in Mathematica notation): Re[Integrate[-D[Exp[-t^beta],t]*Exp[-I w t],{t, 0, Infinity}]]. For beta smaller than one, the return value is always positive. Wanted! Who can contribute to a more accurate interpolation between the points? (The points itself are calculated with an accuracy of 1E-18.) beta : `float` Beta parameter (0..1). w : `float` Circular frequency. Returns: `float` Real part of the Fourier transformed derivative of the Kohlrausch function, or double.NaN if the function can not be evaluated (can happen for beta smaller than 1/64).
`Kohlrausch.Re1(beta, z)` Full Usage: `Kohlrausch.Re1(beta, z)` Parameters: beta : `float` - Beta parameter. z : `float` - Circular frequency. Returns: `float` Real part of the Fourier transformed derivative of the Kohlrausch function for high frequencies, or double.NaN if the series not converges.	Real part of the Fourier transformed derivative of the Kohlrausch function for high frequencies. This is the real part of the Fourier transform (in Mathematica notation): Re[Integrate[-D[Exp[-t^beta],t]*Exp[-I w t],{t, 0, Infinity}]]. For beta smaller than one, the return value is always positive. beta : `float` Beta parameter. z : `float` Circular frequency. Returns: `float` Real part of the Fourier transformed derivative of the Kohlrausch function for high frequencies, or double.NaN if the series not converges.
`Kohlrausch.Re1OldStyle(beta, z)` Full Usage: `Kohlrausch.Re1OldStyle(beta, z)` Parameters: beta : `float` - Beta parameter. z : `float` - Circular frequency. Returns: `float` Real part of the Fourier transformed derivative of the Kohlrausch function for high frequencies, or double.NaN if the series not converges.	Real part of the Fourier transformed derivative of the Kohlrausch function for high frequencies. beta : `float` Beta parameter. z : `float` Circular frequency. Returns: `float` Real part of the Fourier transformed derivative of the Kohlrausch function for high frequencies, or double.NaN if the series not converges.
`Kohlrausch.Re2(beta, z)` Full Usage: `Kohlrausch.Re2(beta, z)` Parameters: beta : `float` - Beta parameter. z : `float` - Circular frequency, must be much lesser than one. Returns: `float` Real part of the Fourier transformed derivative of the Kohlrausch function for low frequencies, or double.NaN if the series not converges.	Real part of the Fourier transformed derivative of the Kohlrausch function for low frequencies. This is the real part of the Fourier transform (in Mathematica notation): Re[Integrate[-D[Exp[-t^beta],t]*Exp[-I w t],{t, 0, Infinity}]]. For beta smaller than one, the return value is always positive. beta : `float` Beta parameter. z : `float` Circular frequency, must be much lesser than one. Returns: `float` Real part of the Fourier transformed derivative of the Kohlrausch function for low frequencies, or double.NaN if the series not converges.
`Kohlrausch.Re2SmallBeta(beta, z)` Full Usage: `Kohlrausch.Re2SmallBeta(beta, z)` Parameters: beta : `float` - Beta parameter. z : `float` - Circular frequency, must be much lesser than one. Returns: `float` Real part of the Fourier transformed derivative of the Kohlrausch function for low frequencies, or double.NaN if the series not converges.	Real part of the Fourier transformed derivative of the Kohlrausch function for low frequencies, and beta<=1/20.. This is the real part of the Fourier transform (in Mathematica notation): Re[Integrate[-D[Exp[-t^beta],t]*Exp[-I w t],{t, 0, Infinity}]]. For beta smaller than one, the return value is always positive. beta : `float` Beta parameter. z : `float` Circular frequency, must be much lesser than one. Returns: `float` Real part of the Fourier transformed derivative of the Kohlrausch function for low frequencies, or double.NaN if the series not converges.
`Kohlrausch.ReIm(beta, w)` Full Usage: `Kohlrausch.ReIm(beta, w)` Parameters: beta : `float` - Beta parameter (0..1). w : `float` - Circular frequency. Returns: `Complex` Fourier transformed derivative of the Kohlrausch function, or Complex.NaN if the function can not be evaluated (can happen for beta smaller than 1/64).	Fourier transformed derivative of the Kohlrausch function for all frequencies. In dependence on the parameters, either a series expansion (accuracy ca. 1E-14) or a bivariate Akima spline (accuracy 1E-4) is used. This is the imaginary part of the Fourier transform (in Mathematica notation): Integrate[-D[Exp[-t^beta],t]*Exp[-I w t],{t, 0, Infinity}]. For beta smaller than one, this is a retardation function, so the real part is positive and the imaginary part is negative. Wanted! Who can contribute to a more accurate interpolation between the points? (The points itself are calculated with an accuracy of 1E-18.) beta : `float` Beta parameter (0..1). w : `float` Circular frequency. Returns: `Complex` Fourier transformed derivative of the Kohlrausch function, or Complex.NaN if the function can not be evaluated (can happen for beta smaller than 1/64).