68
4 Bessel and Airy Functions
Table 4.1 Overview of the first input variable “wh” of the class airyrel
WH
EXPLANATION
Ai
Airy function Ai(z), computed via airy(0,z,scaled).
dAi
First derivative of Ai(z): airy(1,z,scaled).
Bi
Airy function Bi(z): airy(2,z,scaled).
dBi
First derivative of Bi(z): airy(3,z,scaled)
M
Modulus of Ai(x), Bi(x). (Restricted to real values.)
N
Modulus of the first derivative of Ai(x), Bi(x). (Restricted to real values.)
Theta
Phase of Ai(x), Bi(x).
Phi
Phase of the first derivative of Ai(x), Bi(x).
Gi
Scorer function Gi(z).
dGi
First derivative of the Scorer function Gi(z).
Hi
Scorer function H i(z).
dH
First derivative of the Scorer function H i(z).
Airy functions are related to the modified Bessel functions via
Ai(z) =
1
π
z/3K 1/3 (ζ )
(4.3)
Bi(z) =
z/3(I 1/3 (ζ ) + I −1/3 (ζ ))
(4.4)
with ζ = 2/3z 3/2 .
The SPECFUNPHYS-Class Airyrel
The SPECFUNPHYS-class airyrel, [obj, result] = airyrel(wh, z,
scale), returns the value of the Airy functions and related functions. The input
variable “wh” decides which function will be evaluated at the positions “z”. All
possibilities are listed in Table 4.1. “z” is an arbitrary complex array. The optional
input variable “scale” has exactly the same meaning as for the corresponding
MATLAB-functions. For “Ai”, “Bi” and its first derivative airyrel rely on the
MATLAB-functions airy. The output variables are the object “obj” with the
properties “value” (the corresponding function values) “z” the input variable “z”,
and “info” with information about the function in quest and its computation.
The following example shows the computation of the left-hand side of Fig. 4.1.
x = [linspace(-18,0,1000),linspace(0.1,5,25)];
Ai = airyrel(’Ai’,x).value;
Bi = airyrel(’Bi’,x).value;
M = airyrel(’M’,x(1:1000)).value;
%%
subplot(1,2,1)
plot(x,Ai,x,Bi),shg, axis tight
ylim([-1,1]), hold on, shg
%%