1550251515-Classical_Complex_Analysis__Gonzalez_

472 Chapter^7

4. Let ][' = l:;=l ArCr. where the cycles Cr are closed contours. Prove
   that
   n

ilr(a) = LArilcr(a)

r=l

and show that equation (1) in problem 3 holds for r, assuming that

bk cf. I'*.

5. Let f be analytic in a region G and C a closed contour homotopic to

a point in G. Show that

j f(z)dz = 0

c

*6. Prove Jordan's inequality 28/'rr:::; sin():::;() for 0:::; ():::; 1Ji7r, and derive
the inequality
28 1
cos8>1-- -7r' O<()< - -2 -7r

Hint: Consider the behavior of g(8) = si~O (8-:/= 0), g(O) = 1, in the
interval 0 :::; () :5! 7r.

• 
  
  
  
  7. Show that
  
  
  
  

f

00

sinx^2 dx = f

00

cosx^2 dx = ~ (Fresnel's integrals)

.fo lo 2v2

by integrating f(z) = ei=

2

along the boundary of the triangle with

vertices at (0, 0), (R, 0), and (R, R), R > 0. Assume that

1

00

e -2x2 d x= --..fo

o 2../2

8. Let f be a continuous function in a region G and let 'Y be a rectifiable

arc with graph in G. Show that for every e > 0 there exists a piecewise

smooth arc r with I'* CG and with the same endpoints as"(, such that

9. If f is analytic in a convex open set A, show that there exists a function

F(z) analytic in A such that F'(z) = f(z). Hint: Fix a EA and let

F(z) = (L) 1= f(()d(

for any z EA, L being the line segment [a, z]. For any z 0 in A (z 0 -:/= z),

note that the triangle with vertices a, z 0 , and z lies in A. Then fix z 0

and compute lim[F(z) - F(zo)]/(z - z 0 ) as z -t z 0 •