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Euler's Formula

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Equation

(Eq1)	e iθ = cos θ + i sin θ	Euler's formula
(Eq2)	z = r(cos θ + i sin θ) = re iθ	complex number represented by a point in polar coordinates

Nomenclature

a	real number
b	real number
i	√  −1   , the imaginary unit
z	complex number

Details
Consider the complex number z lying on the unit circle as shown:



Writing z in polar coordinates, and using the fact that for the unit circle, r = 1:

z = f (θ) = cos θ + i sin θ

It turns out that there is a particularly compact way of rewriting f (θ) using complex exponentials. Taking the derivative of f, treating i like any other constant but using the fact that i² = −1:

f '(θ) = −sin θ + i cos θ = i cos θ + i² sin θ

Factoring out an i gives:

f '(θ) = i(cos θ + i sin θ) = i * f (θ)

The only real-valued function whose derivative is proportional to the function itself is the exponential function. In other words, if:

g'(x) = k*g(x)

then:

g(x) = Ce^kx

for some constant C. If it is assumed that a similar result holds for complex-valued functions, then:

f '(θ) = i*f (θ)

so:

f (x) = Ce^iθ

for some constant C. To find C, θ = 0 is substituted. Now:

f (0) = Ce^i*0 = C

and since:

f (0) = cos 0 + i sin 0 = 1

Then C = 1. Therefore:

f (θ) = e^iθ.

Thus, Euler's formula is obtained:

(Eq1)

e iθ = cos θ + i sin θ

The Euler formula allows the complex number represented by the point with polar coordinates (r, θ) to be written in the following form:

(Eq2)

z = r(cos θ + i sin θ) = re iθ

Similarly, since cos(−θ) = cos θ and sin (−θ) = −sin θ:

re^−iθ = r (cos (−θ) + i sin(−θ)) = r(cos θ − i sin θ)

Among its many benefits, the polar form of complex numbers makes finding powers and roots of complex numbers much easier. Using the polar form, z = re ^iθ, for a complex number, any power of z may be found as follows:

z ^p = (re ^iθ) ^p = r ^pe ^ipθ