Introduction to the Mathematical and Statistical Foundations of Econometrics
The Complex Exponential Function
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Recall that, for real-valuedx the exponential function ex, also denoted by exp(x), has the series representation
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The property ex+y = ex ey corresponds to the equality
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k=0
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The first equality in (Ш.11) results from the binomial expansion, and the last equality follows easily by rearranging the summation. It is easy to see that (Ш.11) also holds for complex-valued x and y. Therefore, we can define the complex exponential function by the series expansion (Ш.10):
(III.12)
Moreover, it follows from Taylor’s theorem that
о / nm i2m со / nm u2„ +1
cose») _E (-12_.^, Srn(b) _£ (-1>, + . (Ш.13)
and thus (III.12) becomes
ga+ib _ ga [cos(b) + i • sin(b)].
Setting a _ 0, we find that the latter equality yields the following expressions for the cosines and sine in terms of the complex exponential function:
gi •» _|_ g-i - b gi -b g-i'b
cos(b) _----------- +------ . sin(b) _ ——--------------- .
2 2 • i
These expressions are handy in recovering the sine-cosine formulas:
[cos(a - b) - cos(a + b)]/2 [sin(a + b) + sin(a - b)]/2 [sin(a + b) - sin(a - b)]/2 [cos(a + b) + cos(a - b)]/2 sin(a) cos(b) + cos(a) sin(b) cos(a) cos(b) - sin(a) sin(b) sin(a) cos(b) - cos(a) sin(b) cos(a) cos(b) + sin(a) sin(b).
Moreover, it follows from (III.14) that, for natural numbers n,
gl nb _ [cos(b) + i • sin(b) ]n _ cos(n • b) + i • sin(n • b). (III.15)
This result is known as DeMoivre’s formula. It also holds for real numbers n, as we will see in Section III.3.
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Finally, note that any complex number z _ a + i •b can be expressed as
where у є [0, 1] is such that 2ж<р _ arccos(a/Va2 + b2) _ arcsin(b/
Va2 + b2).
