30
2 Error Functions and Fresnel Integrals
The class [obj, ergU, ergV, ergH] = voigt(x,t) computes the
Voigt functions and the line broadening function based on Eq. (2.15). The input
variables “x, t” have to be on the same size or scalars with “x” real and “t” positive
definite. The object “obj” has the properties “resU” (value of Voigt function U),
“resV” (Voigt function V), “resH” (line broadening function), “info” (computational
method), “inx” (numerical Input x), “int” (numerical Input t), “a” (H function
argument), and “u” (H function argument). “ergU,” “ergV,” and “ergH” are the
values of the corresponding functions in doubles. As an example, Fig. 2.3 was
computed via
%%
tl=[0.1,1,2.5,5,10];
% Data
x = linspace(0,15);
x = x(:);
ergUpl = [];
ergVpl = [];
for t=tl
% computation
[obj, ergU, ergV, ergH] = voigt(x,t);
ergUpl = [ergUpl,ergU];
ergVpl = [ergVpl,ergV];
end
%%
Visualization
subplot(1,2,1)
plot(x,ergUpl)
xlabel(’x’),ylabel(’U’)
subplot(1,2,2)
plot(x,ergVpl)
xlabel(’x’),ylabel(’V’)
2.4
Fresnel Integrals
The Fresnel integrals are entire functions in the complex area and arise in the
approximative description of the near-field Fresnel diffraction phenomena. The
Fresnel integrals [1] are defined as
F F (z) =
∞
z
exp
1
2
πit
2
dt
(2.17a)
C(z) =
z
0
cos
1
2
πt
2
dt
(2.17b)
S(z) =
z
0
sin
1
2
πt
2
dt.
(2.17c)
C(z) is called the Fresnel cosine integral function and S(z) the Fresnel sine integral
function. In the literature, e.g., in [3], there are minor differences in the definition,
which might necessitate to scale the functional argument and the integral value. The
definition above follows [1]. The Fresnel integrals are closely related to the error