28
2 Error Functions and Fresnel Integrals
0
0.1
0.2
0.3
0.4
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0.8
0.9
1
x
0
1
2
3
4
5
6
7
8
9
10
Dt
0
0.1
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1
Fig. 2.2 Color coded concentration C/C 0 for diffusion in a semi-finite medium. Horizontal, the
space coordinate x, and vertical, the scaled time D · t
and is the solution of the differential equation
dF (x)
dx
+ 2x F (x) = 1
(2.12)
with initial condition F (0) = 1. The Dawson integral is closely related to the error
function
F (z) = −
i
√
π
2
exp(−z
2 ) erf(iz),
(2.13)
and its computation is based on this equation.
The SPECFUNPHYS class [obj, erg] = dawsonC(z) computes the Dawson integral or function in the complex plane. “z” could be an arbitrary complex
array. The computation is based on Eq. (2.13). The object “obj” comes with the
properties, value (function value), info (information about computational method),
and inz (numerical input z), and “erg” is the computational value in doubles. The
class dawsonC supports the same methods as erfComp.
2.3
Voigt Profile
The Voigt profile is a spectral line profile resulting from the convolution of a
Gaussian profile and a Lorentzian profile. The Gaussian profile is due to Doppler