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700 CHAPTER 20 Complex Integration

5. f (z)dz can be written as a sum of two real line integrals. To do this, suppose γ is
γ
defined on [a,b], and write

f (z) = f (x + iy) = u(x, y) + iv(x, y) and dz = (x (t) + iy (t))dt.
Then
b

f (z)dz = [u(x(t), y(t)) + iv(x(t), y(t))][x (t) + iy (t)]dt
γ a

= udx − v dy + i v dx + udy. (20.1)
γ γ
6. Let γ be a smooth curve defined on [a,b], and let f be continuous on γ . Suppose | f (z)|≤
M for all z on γ , and let L be the length of γ . Then

f (z)dz ≤ ML.

γ

b
g(x)dx ≤ M(b − a) if |g(x)|≤ M for

a
This is the complex version of the inequality
a ≤ x ≤ b for real integrals.
SECTION 20.1 PROBLEMS

In each of Problems 1 through 15, evaluate f (z)dz. 9. f (z) =−i cos(z);γ is any smooth curve from 0 to
γ
2 + i.
2
1. f (z) = 1;γ(t) = t − it for 1 ≤ t ≤ 3.
2
10. f (z) =|z| ;γ is the line segment from −4to i.
2. f (z) = z − iz;γ is the quarter circle about the origin 3 2
2
from2to2i. 11. f (z) = (z − i) ;γ(t) = t − it for 0 ≤ t ≤ 2.
iz
3. f (z) = Re(z);γ is the line segment from 1 to 2 + i. 12. f (z) = e ;γ is any smooth curve from −2to −4 − i.
4. f (z) = 1/z;γ is the part of the half circle of radius 4 13. f (z) = iz;γ is the line segment from 0 to −4 + 3i.
about the origin from 4i to −4i. 14. f (z) = Im(z);γ is the circle of radius 4 about the
5. f (z) = z − 1;γ is any piecewise smooth curve from origin (oriented positively).
2i to 1 − 4i. 15. f (z) =|z| ;γ is the line segment from −i to 1.
2
2
2
6. f (z) = iz ;γ is the line segment from 1 + 2i to 3 + i. 16. Find a bound for | cos(z )dz| if γ is the circle of
γ
radius 4 about the origin.
7. f (z) = sin(2z);γ is the line segment from −i to −4i.

2
8. f (z) = 1 + z ;γ is the part of the circle of radius 3 17. Find a bound for | (1/(1 + z))dz| if γ is the line
segment from 2 + i to 4 + 2i.
about the origin from −3i to 3i.
20.2 Cauchy’s Theorem
Cauchy’s theorem is the cornerstone of complex integration theory. We need some terminology
and preparation for its statement.
If γ is a continuous, simple, closed curve in the plane, then γ separates the plane into three
parts: the graph of the curve itself, a bounded open set called the interior of γ , and an unbounded
open set called the exterior of the curve (Figure 20.5). This is the Jordan curve theorem, which
we will assume.

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October 14, 2010 15:32 THM/NEIL Page-700 27410_20_ch20_p695-714