Algebra • Topic 22

Euler's Formula

"The most remarkable formula in mathematics." Euler's formula bridges calculus, trigonometry, and complex algebra.

The Formula

Theorem (Euler's Formula)

For any real number

x

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

When $x = π$ , we get Euler's Identity: $e^{iπ} + 1 = 0$ .

Exponentials vs Trig

This allows us to express sine and cosine in terms of exponentials:

$cos x = \frac{e ^{i x} + e ^{- i x}}{2}$
$sin x = \frac{e ^{i x} - e ^{- i x}}{2 i}$

Applications: Linearization

Euler's formula simplifies trigonometric identities by turning products into sums.

Example (Proving Identities)

Show

sin (2 x) = 2 sin x cos x

Proof. $2 sin x cos x = 2 (\frac{e ^{i x} - e ^{- i x}}{2 i}) (\frac{e ^{i x} + e ^{- i x}}{2})$ $= \frac{1}{2 i} (e^{2 i x} + e^{0} - e^{0} - e^{- 2 i x})$ $= \frac{e ^{2 i x} - e ^{- 2 i x}}{2 i} = sin (2 x)$ . ∎

Exponential Form of Complex Numbers

z = r e^{i θ}

This notation makes differentiation and integration of complex functions natural.

$\frac{d}{d t} e^{i t} = i e^{i t}$ .

Practice Problems

Exercise (Problem 1)

Express

cos^{3} x

in terms of

cos (n x)

using Euler's formula. (Hint: Write $cos x = \frac{u + 1/ u}{2}$ where $u = e^{i x}$ , then cube it)

Exercise (Problem 2)

Evaluate the sum

\sum_{k = 0}^{n} cos (k x)

by considering the real part of geometric series

\sum e^{ik x}

Exercise (Problem 3)

Find the value of

i^{i}

(using the principal branch of the logarithm).