Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

z

− 10 − 5

− 0. 5

0. 5

1

Figure 25.11 The functions Ai(z) (full line) and Bi(z) (broken line) for real

z.

Then we have

Ai(0) =

1

2 πi

∫ 0

∞

e−se^4 πi/^3 (3s)−^2 /^3 ds+

1

2 πi

∫∞

0

e−se^2 πi/^3 (3s)−^2 /^3 ds

=

3 −^2 /^3

2 πi

∫∞

0

e−s(−e^4 πi/^3 +e^2 πi/^3 )s−^2 /^3 ds

=

√

3 i 3 −^2 /^3

2 πi

∫∞

0

e−ss−^2 /^3 ds

=

3 −^1 /^6

2 π

Γ(^13 ),

where we have used the standard integral defining the gamma function in the last line.

Finally in this subsection we should mention that the Airy functions and their

derivatives are closely related to Bessel functions of orders±^13 and±^23 and that