24
2 Error Functions and Fresnel Integrals
Voigt profile: The SPECFUNPHYS class voigt computes the Voigt functions and
the line broadening function.
Fresnel integral: The SPECFUNPHYS class fresnel evaluates various Fresnel
integral functions in the complex domain.
2.2
Error Functions
2.2.1 Fundamental Equations and Computation
The error function erf, Fig. 2.1, and complementary error function erfc [1, 3]
equal
erf(z) =
2
√
π
z
0
exp(−t
2 )dt
(2.1a)
erfc(z) =
2
√
π
∞
z
exp(−t
2 )dt .
(2.1b)
3
2
1
imag
0
0
-1
-3
200
-2
400
-1
real
-2
0
600
1
800
2
-3
1000
3
1200
1400
1600
1800
3
2
1
imag
0
0
-1
-3
-2
1
-1
real
-2
0
2
1
2
-3
3
3
4
5
Fig. 2.1 Absolute value of the error function erf(z) in the complex plane. Please note, for z =
±3i, the absolute value | erf(z)| is of the order 10 9 , which is hidden due to the finite plot resolution.
On the right-hand side, the maximum plot value of | erf(z)| is restricted to 5 to uncover the structure
of the lower values