Page 315 - Schaum's Outline of Differential Equations

298 GAMMA AND BESSEL FUNCTIONS [CHAP. 30

30.5. Prove that T(l) = 1.
Using Eq. (30.1), we find that

30.6. Prove that if p = n, a positive integer, then Y(n + !) = «!.
The proof is by induction. First we consider n= 1. Using Problem 30.4 with p= 1 and then Problem 30.5,
we have

Next we assume that T(n + 1) = n\ holds for n = k and then try to prove its validity for n = k + 1:

(Problem 30.4 with p = k + 1)
(from the induction hypothesis)

Thus, T(n + 1) = n\ is true by induction.
Note that we can now use this equality to define 0!; that is,

30.7. Prove that Y(p + k + 1) = (p + k)(p + k - 1) • • • (p + 2)(p + l)r(p + 1).
Using Problem 30.4 repeatedly, where first/) is replaced by p + k, then by p + k— 1, etc., we obtain

30.8. Express as a gamma function.

112
2
Let z = x ; hence x = z and dx = —z Il2 dz. Substituting these values into the integral and noting that as x goes
from 0 to °° so does z, we have

The last equality follows from Eq. (30.1), with the dummy variable x replaced by z and with P -

30.9. Use the method of Frobenius to find one solution of Bessel's equation of order/?:

Substituting Eqs. (28.2) through (28.4) into Bessel's equation and simplifying, we find that