Page 20 - Calculus with Complex Numbers

and second that

Iz2 + 81k: 8 - Iz2I= 8 - 4 = 4.
The left-hand inequality is proved similarly.

I . I 2 Exte nsion to 3 term s (or m ore)
W e give the inequalities for 3 terms. The generalization to more terms is left to the
reader.
Inequality 6

Proof Observe that

by repeated application of lnequality 1.

Inequality 7 I.4 + B + C I k: I .4 I - I B I - IC I.

Proof Observe that

= I A + B + CI + I B I+ ICI.

N otes
W enever delined ei0 or provedthatthe laws of indices hold for complex exponents .
A rigorous treatment of this material would dehne c: cos z, sin z by theirM aclaurin
series

and prove their properties by manipulation of these series.