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Frequently used formulas

1. Euler’s formula. The formula e^{i θ} = cos θ + i sin θ . Interesting special cases are those in
which θ = π and 2π for which e^{πi} = -1 and e^{2πi} = 1, respectively.

2. Expressions for a complex number in polar coordinates.

z = x + iy = r(cos θ + i sin θ)

z = x + iy = re^{i θ}

3. For complex numbers z_{1} = r_{1} (cos θ_{1} + i sin θ_{1}) and z_{2} = r_{2} (cos θ_{2} +
i sin θ_{2}):

z_{1}z_{2} = r_{1}r_{2} [cos (θ_{1} + θ_{2}) + i sin (θ_{1} + θ_{2})]

z_{1}z_{2} ... z_{n} = r_{1}r_{2} ... z_{n} [cos (θ_{1} + θ_{2} + ... + θ_{n}) + i sin (θ_{1} + θ_{2} + ... + θ_{n})]

z_{1}/z_{2} = ( r_{1}/r_{2})_{ }[cos (θ_{1} - θ_{2}) + i sin (θ_{1} - θ_{2})]

z^{n} = [ r(cos θ + i sin θ)]^{n} = r^{n} (cos nθ + i sin n)

ln z = ln r + iθ

e^{z} = e ^{x + iy } = e^{x }(cos y + i sin y)

a^{z} = e ^{z ln a} where a is either real or complex

e^{πi} = -1

e^{2πi} = 1

4. For a complex variable z = x + iy

e^{iz } = cos z + i sin z

e^{ -iz} = cos z - i sin z

5. For a real variable x

e^{ix } = cos x + i sin x

e^{ -ix} = cos x - i sin x

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

ze^{iα} = re^{iθ}e^{iα} = re^{i(θ+α)}

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

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