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4 Bessel and Airy Functions
4.1
Function Overview
Airy functions: The MATLAB-function w = airy(k,z) returns the Airy function Ai(z), Bi(z) and its first derivative in dependence of the parameter “k”. The
input variable “z” could be an arbitrary complex array. The SPECFUNPHYS-class
airyrel returns in addition the modulus and the phase of the Airy function
and its first derivative. In addition, the scorer function “Gi” and “Hi” and its first
derivative will be covered.
Bessel functions: MATLAB comes with the Bessel functions J ν (z), Y ν (z),
H
(1)
ν (z), H
(2)
ν (z), I ν (z), and K ν (z), e.g., besselj(nu, z). The argument
“z” could be an arbitrary complex array, but the order “ν” has to be real. The
SPECFUNPHYS-class bessel covers the same Bessel functions as MATLAB,
e.g., bessel(’J’, nu, z), and in addition the spherical Bessel functions
j ν (z), y ν (z), h
(1)
ν (z), h
(2)
ν (z), i
(1)
ν (z), i
(2)
ν (z), k ν (z), and the Kelvin functions
ber ν (z), bei ν (z), ker ν (z), and kei ν (z). In contrast to the standard MATLAB
functions the order “ν” could be as well a complex array.
4.2
Airy Functions and Related Functions
The Airy functions “Ai” and “Bi” are independent solutions of the Airy differential
equation.
d 2 w
dz 2 − zw = 0
.
(4.1)
“Ai” and “Bi” are entire functions. An important feature, for applications in both
classical and quantum physics, is associated with turning points. The Airy function
changes from an oscillatory function for negative arguments to a monotonic function
for positive arguments. The same holds for the Scorer function Gi. This phenomenon
is called Stokes phenomenon.
Example: Stokes and Anti-Stokes Lines
The phrase Stokes and Anti-Stokes lines is differently used in literature. Mathematicians tend to call anti-Stoke line what physicists call Stokes line and vice versa. In
this example I will stick to the physicist’s convention.
For large values of x the Airy function can be approximated by
Ai(x) ≈
exp
−
2
3 x 3/2
2
√
πx 1/4 , and
(4.2a)
Ai(−x) ≈
sin
2
3 x 3/2 +
1
4 π
√
πx 1/4
.
(4.2b)
4 Bessel and Airy Functions
4.1
Function Overview
Airy functions: The MATLAB-function w = airy(k,z) returns the Airy function Ai(z), Bi(z) and its first derivative in dependence of the parameter “k”. The
input variable “z” could be an arbitrary complex array. The SPECFUNPHYS-class
airyrel returns in addition the modulus and the phase of the Airy function
and its first derivative. In addition, the scorer function “Gi” and “Hi” and its first
derivative will be covered.
Bessel functions: MATLAB comes with the Bessel functions J ν (z), Y ν (z),
H
(1)
ν (z), H
(2)
ν (z), I ν (z), and K ν (z), e.g., besselj(nu, z). The argument
“z” could be an arbitrary complex array, but the order “ν” has to be real. The
SPECFUNPHYS-class bessel covers the same Bessel functions as MATLAB,
e.g., bessel(’J’, nu, z), and in addition the spherical Bessel functions
j ν (z), y ν (z), h
(1)
ν (z), h
(2)
ν (z), i
(1)
ν (z), i
(2)
ν (z), k ν (z), and the Kelvin functions
ber ν (z), bei ν (z), ker ν (z), and kei ν (z). In contrast to the standard MATLAB
functions the order “ν” could be as well a complex array.
4.2
Airy Functions and Related Functions
The Airy functions “Ai” and “Bi” are independent solutions of the Airy differential
equation.
d 2 w
dz 2 − zw = 0
.
(4.1)
“Ai” and “Bi” are entire functions. An important feature, for applications in both
classical and quantum physics, is associated with turning points. The Airy function
changes from an oscillatory function for negative arguments to a monotonic function
for positive arguments. The same holds for the Scorer function Gi. This phenomenon
is called Stokes phenomenon.
Example: Stokes and Anti-Stokes Lines
The phrase Stokes and Anti-Stokes lines is differently used in literature. Mathematicians tend to call anti-Stoke line what physicists call Stokes line and vice versa. In
this example I will stick to the physicist’s convention.
For large values of x the Airy function can be approximated by
Ai(x) ≈
exp
−
2
3 x 3/2
2
√
πx 1/4 , and
(4.2a)
Ai(−x) ≈
sin
2
3 x 3/2 +
1
4 π
√
πx 1/4
.
(4.2b)