236 Chapter 8Complex numbers
A more advanced application of complex numbers is in the extension of the concepts
of variable and function to include complex variables and functions of a complex
variable.
4Thus, if xand yare real variables thenz 1 = 1 x 1 + 1 iyis a complex variable, and
f(z)is a function of the complex variable z. For example, the function
f(z) 1 = 1 z
21 + 1 z 1 + 11
can be written as
f(z) 1 = 1 (x 1 + 1 iy)
21 + 1 (x 1 + 1 iy) 1 + 111 = 1 (x
21 − 1 y
21 + 1 x 1 + 1 1) 1 + 1 i(2xy 1 + 1 y)
= 1 g(x, y) 1 + 1 ih(x, y)
whereg(x, 1 y)andh(x, 1 y), the real and imaginary parts off(z), are real functions of
xand y. The properties off(z)as a function of the single complex variable zare more
general than the properties of real functions. For many purposes in the physical
sciences only one such function is of importance, and is discussed in the following
section.
0 Exercise 29
8.5 Euler’s formula
It is known from the theory of functions of a complex variable that the exponential
functione
z, where zis a complex number, can be expanded in the familiar infinite
series
(8.32)
If zis the imaginary numberz 1 = 1 iθthen
The real and imaginary parts of this function are the series expansions of the
trigonometric functionscos 1 θandsin 1 θ, respectively (see Section 7.6), so that
e
iθ1 = 1 cos 1 θ 1 + 1 i 1 sin 1 θ (8.33)
=−
!
!
−
!
+−
!
!
1 −
246 35
246 357θθθ
θ
θθθ
i
77!
ei
ii
iθθ
θθ
=+ +
!
!
1 +
23
23()
() ()
ez
zz
z=+ +
!
!
1 +
23
234The first discussion of functions of a complex variable appeared in a letter by Gauss to Bessel in 1811, together
with a description of the geometric interpretation of complex numbers. The theory was developed independently
by Cauchy from about 1814. Cauchy was the most prolific mathematician of the 19th century. He swamped the
weekly bulletin of the Paris Academy of Sciences, Comptes Rendus, forcing it to introduce a rule, still in force,
restricting publications to four pages.