Faddeeva


Faddeeva(z)

Computes the Faddeeva function, or Kramp function, which is a function on the complex numbers involving an integral of $ e^{t^2} $, making it closely related to the Dawson and Erf functions. The real and imaginary parts are called the Voigt functions or Voigt profiles. The function is scaled by $ e^{-z^2} $, which gives it favorable numeric properties that help to avoid numeric overflow.

The Faddeeva function is given by

$ w(z) = e^{-z^2} \left( 1 - { 2 \over\sqrt{-\pi}} \int_0^z e^{t^2} dt \right) $

where «z» is a complex number, as is $ \sqrt(-\pi) $.

This is mathematically equivalent to the expression

Exp(-z^2) * (1 - Erf(-1j*z))

which reveals its relationship to the Erf function; however, when using the expression based on Erf, extremely large values that overflow the largest numeric values that can be represented in Analytica will occur quickly as «z» obtains even moderate size. In these cases, the Faddeeva can sometimes be utilized to avoid numeric overflow. The Dawson function is used for similar reasons in similar situations (and the Dawson function is usually easier to apply, especially if you are dealing with real-valued problems).

History

Introduced in Analytica 5.0.

See Also

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