Page 122 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

112 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
y

r

x

FIGURE 7.1: Polar representation of a complex variable z

Addition gives

e iθ + e −iθ
cos θ = = cosh(iθ) (7.7)
2
and subtraction gives

e iθ − e −iθ
sin θ = =−i sinh(iθ) (7.8)
2i

Note that

1 x+iy −x−iy 1 x −x
cosh z = e + e = e [cos y + i sin y] + e [cos y − i sin y]
2 2
x
x
e + e −x e − e −x
= cos y + i sin y
2 2
= cosh x cos y + i sinh x sin y (7.9)

The reader may show that

sinh z = sinh x cos y + i cosh x sin y. (7.10)

Trigonometric functions are defined in the usual way:

iz
iz
e − e −iz e + e −iz sin z
sin z = cos z = tan z = (7.11)
2i 2 cos z
Two complex numbers are equal if and only if their real parts are equal and their imaginary
parts are equal.