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Frequently used formulas

1. Euler’s formula. The formula ei θ = cos θ + i sin θ . Interesting special cases are those in which θ = π and 2π for which eπi = -1 and e2πi = 1, respectively.

2. Expressions for a complex number in polar coordinates.

z = x + iy = r(cos θ + i sin θ)

z = x + iy = rei θ

3.     For complex numbers   z1 = r1 (cos θ1 + i sin θ1) and z2 = r2 (cos θ2 + i sin θ2):

z1z2 = r1r2 [cos (θ1 + θ2) + i sin (θ1 + θ2)]

z1z2 ... zn = r1r2 ... zn [cos (θ1 + θ2 + ... + θn) + i sin (θ1 + θ2 + ... + θn)]

z1/z2 = ( r1/r2) [cos (θ1 - θ2) + i sin (θ1 - θ2)]

zn = [ r(cos θ + i sin θ)]n = rn (cos nθ + i sin n)

ln z = ln r + iθ

ez = e x + iy = ex (cos y + i sin y)

az = e z ln a    where a is either real or complex

eπi = -1

e2πi = 1

4. For a complex variable z = x + iy

eiz = cos z + i sin z

e -iz = cos z - i sin z

5. For a real variable x

eix = cos x + i sin x

e -ix = cos x - i sin x

Geometric meaning of multiplication of a vector z by e . Let z = re be the vector OA in Fig. 5. Then the product

ze = ree = rei(θ+α)

is the vector OB shown in the figure. Multiplication of a vector z by e amounts to rotating z counterclockwise by an angle α. We can consider e as an operator which acts on z to produce this rotation.