Website owner: James Miller
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 eiα . Let z = reiθ be the vector OA in Fig. 5. Then the product
zeiα = reiθeiα = rei(θ+α)
is the vector OB shown in the figure. Multiplication of a vector z by eiα amounts to rotating z counterclockwise by an angle α. We can consider eiα as an operator which acts on z to produce this rotation.
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