1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

2 6 CHAPTER 1 • COMPl,EX NUMBERS

e" = (-1, 0) = - 1

Y~A= (0, l ) = i

, ,,-,;i = (! 2' -11) 2 =! 2 + ,qi 2

\ ;-e'°"=ellz=(l,0)= I

.._~-+-~--1~~-+---x

l ,•'\' = .-it= ("i 2' -"i\ v = .Ji 2 - .Ji 2 i

Figure 1.1 2 The location of eilJ for various values of 8.

If 8 is a real number, e^18 will be located somewhere on the circle with radius

1 centered at the origin. This assertion is easy to verify because

(1-33)

Figure 1.12 illustrates the location of the points ei^8 for various values of 8.
Note that, when 8=1f, we get ei" = (cos1f, sin1f) = (-1, 0) = -1, so

(1-34)

Euler was the first to djscover this relationship; it is referred to as E uler's
identity. It has been labeled by many mathematicians as the most amazing
relation in analysis- and with good reason. Symbols with a rich history are
miraculously woven together-the constant 7r used by Hippocrates as early as
400 s.c.; e, the base of the natural logarithms; the basic concepts of addition ( +)
and equality (=);the foundational whole numbers 0 and 1; and i, the number
that is the central focus of this book.
Euler's formula (1-32) is of tremendous use in establishing important alge-
braic and geometric properties of complex numbers. You will see shortly that it
enables you to multiply complex numbers with great ease. It also allows you to
express a polar form of the complex number z in a more compact way. Recall

that if r = lzl and 8 E arg z, then z = r(cos8 + isin8). Using E uler's formula

we can now write z in its exponential form:

(1-35)

• EXAMPLE 1.12 With .reference to Example 1.10, with z = -J3 -i, we

have z = 2 ei(-^5 "/^6 ).