1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 485

10. Show that the substitution z = ei^9 , 0 :::::; () :::::; 211", transforms the real
    integral

into the complex integral

1

(^2) 11" _dB
0 5 + 4cosB
·j dz
-i 2z^2 + 5z + 2
c
where C: z = ei^9 , 0 :::::; ():::::; 211", and use (2) to evaluate (1).

11. Use the method of problem 10 to show that:
    {211" ·o

(a) lo e-iOee' d(} = 211"

[^2 11" d(} 211"a

(b) lo (a+ bcosB)2 = (a2 - b2)3/2 (a> b > 0)

1

11"/2 d(} 11"

(~ =

o a+sin^2 B 2.ja(a+l)

(a > 0)

( d) r cos'2(} d(} 11"a^2

lo 1 - 2a cos(} + a^2 = 1 - a2 (lal < 1)

( e) 111" 1 + cos (} d(} = ~ y'7r

0 1 + cos^2 (}^2

(f) [271" cosnB d(} = (-lr 2~e-na

lo cosh a+ cos(} smh a

(a > 0)

1

11" (2n)'

(g) o sin^2 n(}dB=7r (2nnf ) 2 (n=l,2, ... )

(1)

(2)

* 12. Suppose that f is analytic in the upper half-plane Im z 2'.: 0, and such

that lf(z)I < Alzl-m (A > 0, m > 0). Prove that

f(z) = ~ j+oo f(t) dt (t real)

27ri -oo t - z

for any z such that Im z > 0.

13. Show that

l lc+ioo ezt

(PV)-. -dz

211"i c-ioo z

where c > 0 and t is a real variable, represents the Heaviside function

fort> 0

fort= 0

fort< 0