Series Expansion of the Complex Logarithm
For the case x є К, |x | < 1, it follows from Taylor’s theorem that ln(1 + x) has the series representation
ln(1 + x) = ^(-1)*-1 xk /k. (III.18)
I will now address the issue of whether this series representation carries over if we replace x by i ■ x because this will yield a useful approximation of exp(i ■ x),
which plays a key role in proving central limit theorems for dependent random variables. See Chapter 7.
If (III.18) carries over we can write, for arbitrary integers m,
log(1 + i ■ x) = ^2(—1)k-1ikxk/k + i ■ mn
= ( — 1)2k-1i 2kx 2k / (2k)
+ (-1)2k-1-1i2k-1 x2k-1/(2k – 1) + i ■ mn
= J2 (—1)k-1 x 2k/(2k)
+ i J2(—1)k-1 x2k-1/(2k – 1) + i ■ mn. (III.19)
On the other hand, it follows from (III.17) that 12
log(1 + i ■ x) = ln(1 + x2) + i ■ [arctan(x) + mn].
Therefore, we need to verify that, for x є К, |x | < 1,
-In(1 + x 2) = J2 (-1)k-1 x 2k/(2k)
(1 + i ■ x) exp(-x2/2 + r(x)), where |r(x)| <|x |3 .
arctan(x) = (_1)k-1 x2*-1/(2* – 1). (Ш.21)
Equation (Ш.20) follows from (III.18) by replacing x with x2. Equation (III.21) follows from
~Y (-1)k-1 x2k-1/(2k – 1) dx
and the facts that arctan (0) = 0 and darctan(x) 1
dx 1 + x 2
Therefore, the series representation (III.19) is true.
In probability and statistics we encounter complex integrals mainly in the form of characteristic functions, which for absolutely continuous random variables are integrals over complex-valued functions with real-valued arguments. Such functions take the form
f (x) = q>(x) + i ■ ф (x), x є К,
where ф and ф are real-valued functions on R. Therefore, we may define the (Lebesgue) integral off over an interval [a, b] simply as
b b b
j f (x )dx = j y>(x)dx + i ■ j ф (x)dx
a a a
provided of course that the latter two integrals are defined. Similarly, if ц is a probability measure on the Borel sets in Rk and Re( f (x)) and Im( f (x)) are Borel-measurable-real functions on Rk, then
f Jx )d„.(x) = f Re(/<x ))dx(x) + . f M</<x l)^),
provided that the latter two integrals are defined.
Integrals of complex-valued functions of complex variables are much trickier, though. See, for example, Ahlfors (1966). However, these types of integrals have limited applicability in econometrics and are therefore not discussed here.