The Complex Exponential Function

image972 Подпись: (III.10)

Recall that, for real-valuedx the exponential function ex, also denoted by exp(x), has the series representation

image974 Подпись: (III.11)

The property ex+y = ex ey corresponds to the equality

k=0

^ (x + y)k

k=0 k!

= E 1 Z(k)xk—mym

k! m

k= 0 m = 0

Ж k xk—m ym

Подпись: a+і b df
Подпись: e image978

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

Подпись: (2„ + 1)! ’cose») _E (-12_.^, Srn(b) _£ (-1>, + . (Ш.13)

and thus (III.12) becomes

Подпись: (III.14)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:

Подпись: sin(a) sin(b) sin(a) cos(b) cos(a) sin(b) cos(a) cos(b) sin(a + b) cos(a + b) sin(a - b) cos(a - b)[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.

image982

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).

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