Theory of Statistics - George Mason University

0.0 Some Basic Mathematical Concepts 655
Other examples are the exponentiation operation, the functions log(·),
sin(·), and so on. The standard way of defining the elementary functions is
through analytic continuation of a Taylor series expansion into IC. For example,
for z ∈ IC, we may define exp(z) in terms of the convergent series
e z = 1 + z + z2 z3 z4
+ + + · · · (0.0.66)
2! 3! 4!
The ratio test can be used to show that this is convergent over all IC (exercise).
The definition of ez can be used to define the exponentiation operation z z2
1
in general.
Complex Conjugates
For z = x + iy ∈ IC, we define define the complex conjugate as x − iy, and
denote it as z. We define the modulus of z as √ zz, and denote it as |z|. It is
clear that |z| is real and nonnegative, and it corresponds to the absolute value
of x if z = x + 0i.
We have some simple relationships for complex conjugates:
Theorem 0.0.16
For all z, z1, z2 ∈ IC, we have
• z = z,
• z1 + z2 = z1 + z2,
• z1z2 = z1z2.
Proof. Exercise.
Euler’s Formula
One of the most useful facts is given in Euler’s formula, for a real number x:
e ix = cos(x) + i sin(x). (0.0.67)
This relationship can be derived in a number of ways. A straightforward
method is to expand e ix in a Taylor series about 0 and then reorder the
terms:
e ix = 1 + ix + (ix)2 (ix)3 (ix)4
+ + + · · ·
2! 3! 4!
= 1 − x2
x4 x6
+ + + · · · + i x −
2! 4! 6! x3
x5 x7
+ − + · · ·
3! 5! 7!
= cos(x) + i sin(x).
We can do this because the series are absolutely convergent for x ∈ IR.
Euler’s formula has a number of applications and special cases. For examples,
Theory of Statistics c○2000–2013 James E. Gentle