Solved Problems on Complex Analysis - Desklib

Solution 1: We need to find all values of c for which
.
Since,
.
Now
Therefore is not possible.
Now let , then
Let , then
Let , then
Similarly for
Hence, for , where C is the set of complex number of the form , a is
any real number and .
Solution 2:
2(a): .Consider C1. Along AB, and x varies from -1 to 1.
Since and . So
Along BAC: and y varies from 0 to 1. Let . Since
, where and .
So,
Hence,
2(b): .Consider C2. Along PR, and y varies from 0
to 1.
Since and . So

Along RS: and x varies from -1 to 1.
Since .
So,
Along SQ: and y varies from 1 to 0.
Since and . So
Hence,
2(c): .Consider C1. Along AB, and x varies from -1 to 1.
Since and . So
Along BAC: and y varies from 0 to 1. Let . Since
, where and .
So,
Hence,
2(d): .Consider C2. Along PR, and y varies from 0
to 1.
Since and . So
Along RS: and x varies from -1 to 1.
Since .
So,
Along SQ: and y varies from 1 to 0.
Since and . So
Hence,