4.2 Airy Functions and Related Functions
69
plot(x(1:1000),M,x(1:1000), -M), shg
grid on
The Modulus and the Phase
With respect to moduli, M(x), N(x), and phases, Θ(x), Φ(x), we follow the
definition of [3], which differs slightly from the definition of [1]. With x ∈ R −
the following equations hold:
Ai(x) = M(x) sin Θ(x)
(4.5a)
Bi(x) = M(x) cos Θ(x)
(4.5b)
M(x) =
Ai(x) 2 + Bi(x) 2
(4.5c)
Θ(x) = arctan
Ai(x)
Bi(x)
,
(4.5d)
d
dx
Ai(x) = N(x) sin Φ(x)
(4.5e)
d
dx
Bi(x) = N(x) cos Φ(x)
(4.5f)
N(x) =
Ai (x) 2 + Bi (x) 2
(4.5g)
Φ(x) = arctan
Ai (x)
Bi (x)
.
(4.5h)
The computation will be based on the MATLAB function airy(k,x).
4.2.1 The Scorer Functions
The solutions of the differential equation
d 2 w
dz 2 − zw =
1
π
(4.6)
are the Scorer functions Gi(z) and H i(z).
Computational Aspects
The Scorer functions and similar their first derivative follow the connection formula
H i(z) + Gi(z) = Bi(z).
(4.7)
Unfortunately in some areas in complex space H i and −Gi are so close to each
other that adding them will lead to a tremendous loss of significance. Thus the
connection formula (4.7) has to be carefully controlled.