Page 31 - Calculus with Complex Numbers

Ffgure 2 . / 0

N otes
W e have not actually delined ez sin z, cos z, log z for complex z. W e have merely
assumed that these functions can be delined and that they continue to have
the properties they possess in the real domain. For example laws of indices
laws of logarithms trigonometric identities. A rigorous treatment would deline
ez sin z cos z log z from their M aclaurin series and derive their properties from
these series. The function log z would be delined as the inverse function of ez .
The functions we have drawn complex graphs of are all conformal mappings
in the sense that curves which intersect at an angle 0 in the z-plane transform to
curves which intersect at the same angle 0 in the w-plane. Observe that in evel'y
case the grid lines .x = constant y = constant in the z-plane transform to curves
which intersect orthogonally in the w-plane. This conformal property is crucial in
applications to lluid dynamics.

Exam ples

Prove that for all Izl= 2

2 :% Iz - 41:i 6.

Prove that for all IzI= 3
8 .::2 + 1 10

V s z + 2 1q -,j- .
z
Prove that for all Izl= 4

3 z+ ï 5
Y -'
i 3
lj Y c
-