A First Course in Complex Analysis · Problem 8

Consequences of Cauchy's Theorem — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · complex-integration

Problem

Compute

\int_{\gamma} f

for the following functions

f

and paths

\gamma

: (a)

f(z) = z^2

and

\gamma(t) = t + it^2

0 \leq t \leq 1

. (b)

f(z) = z

and

\gamma

is the semicircle from

1

through

i

-1

. (c)

f(z) = \exp(z)

and

\gamma

is the line segment from

0

to a point

z_0

. (d)

f(z) = |z|^2

and

\gamma

is the line segment from

2

3 + i

. (e)

f(z) = z + \frac{1}{z}

and

\gamma

is parametrized by

\gamma(t)

0 \leq t \leq 1

, and satisfies

\operatorname{Im}\gamma(t) > 0

\gamma(0) = -4 + i

, and

\gamma(1) = 6 + 2i

. (f)

f(z) = \sin(z)

and

\gamma

is some piecewise smooth path from

i

\pi