Lecture Notes in Differential Equations - Bruce E. Shapiro

155
Theorem 18.2. Euler’s Formula
e iθ = cos θ + i sin θ (18.18)
Proof. Use the fact that
⎫ ⎧
i 2 = −1
i 4k+1 ⎫
= i
i 3 = i(i 2 ) = −i
⎪⎬ ⎪⎨ i 4k+2 = −1
⎪⎬
=⇒
i 4 = i(i 3 ) = i(−i) = 1 i 4k+3 = −i
⎪⎭ ⎪⎩
⎪⎭
i 5 = i(i 4 ) = i
i 4k+4 = 1
in the formula’s for a Taylor Series of e iθ :
for all k = 0, 1, 2, . . .
(18.19)
e iθ = 1 + (iθ) + (iθ)2
2!
= 1 + iθ + i2 θ 2
=
2!
+ (iθ)3
3!
+ i3 θ 3
3!
+ i4 θ 4
4!
+ (iθ)4
4!
+ i5 θ 5
5!
+ (iθ)5
5!
+ · · · (18.20)
+ · · · (18.21)
= 1 + iθ − θ2
2! + −iθ3 3! + θ4
4! + iθ5 5! + · · · (18.22)
even powers of θ
{ }} {
(1 − θ2
2! + θ4
4! − θ6
6! + · · · )
+i
(θ − θ3
)
3! + θ5
5! − θ7
7! + · · · } {{ }
odd powers of θ
(18.23)
= cos θ + i sin θ (18.24)
where the last step follows because we have used the Taylor series for sin θ
and cos θ.
Theorem 18.3. If z = a + ib where a and b are real numbers than
e z = e a+ib = e a (cos θ + i sin θ) (18.25)
Theorem 18.4. If z = x + iy where x, y are real, then
where θ = Phase(z) and r = |z|.
z = r(cos θ + i sin θ) = re iθ (18.26)
Proof. By definition θ = Phase(z) is the angle between the x axis and the