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Dawson function
From LaTeX CAS translator demo
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P.V. denotes the Cauchy principal value, and we restrict ourselves to real . can be related to the Dawson function as follows. Inside a principal value integral, we can treat as a generalized function or distribution, and use the Fourier representation
With , we use the exponential representation of and complete the square with respect to to find
We can shift the integral over to the real axis, and it gives . Thus
We complete the square with respect to and obtain
We change variables to :
The integral can be performed as a contour integral around a rectangle in the complex plane. Taking the imaginary part of the result gives
where is the Dawson function as defined above.
The Hilbert transform of is also related to the Dawson function. We see this with the technique of differentiating inside the integral sign. Let
Introduce
The nth derivative is
We thus find
The derivatives are performed first, then the result evaluated at . A change of variable also gives . Since , we can write where and are polynomials. For example, . Alternatively, can be calculated using the recurrence relation (for )
libcerf, numeric C library for complex error functions, provides a function voigt(x, sigma, gamma) with approximately 13–14 digits precision. It is based on the Faddeeva function as implemented in the MIT Faddeeva Package