1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 493

4. Evaluate fc zP(l - z)q dz, where C: z = reit, 0 :::; t :::; 271", 1· > 1, and

p, q are integers. Consider all possible cases.

5. Let C: z = eit, 0:::; t :::; 271", and let m and n be integers.
   (a) Show that

ifm= n

if m# n

(b) Assuming that n ;::: 0, use part (a) to prove

2 ~i j z(z+1r(z+1)ndz= (~)

2

+ (~)

2

+···+ (:)

2

c

( c) Also, show that

~ J z( z + 1 re z + 1) n dz = ~ J z(2 + z + zr dz

27rz 27rz

c c

= _1 J (z + 1)

2

n dz= (2n)

27ri zn+l n

c

thus deriving the identity

(S. Minsker [22])

6. The Legendre polynomial Pn(z) is given by Rodrigues's formula,

Pn(z) = ___!__, Dn(z^2 - It, D = dd

2nn. z

(a) Use (7.21-2) to show that

1 J ( (^2 - 1) n d(

Pn(z) = 2 7ri 2 n(( _ z)n+l

c+

where C is any simple closed contour around z.

(b) By taking c+ to be the circle ( - z = reit, 0 :::; t < 27r, with

r = lz^2 - 11112 , derive the Laplace formula

Pn(z) = -1111' (z + Vz2=l cos tr dt

7r 0

(c) Use the last formula to show that Po(z) = 1, P1(z) = z, P2(z) =

(^1) M3z^2 - 1), and P 3 (z) = 1/z(5z^3 - 3z).