Euler's Formula is . It is named after the 18th-century mathematician Leonhard Euler.
Euler's formula is a fundamental tool used when solving problems involving complex numbers and/or trigonometry. Euler's formula replaces "cis", and is a superior notation, as it encapsulates several nice properties:
De Moivre's Theorem
De Moivre's Theorem states that for any real number and integer , .
Sine/Cosine Angle Addition Formulas
Start with , and apply Euler's forumla both sides:
Expanding the right side gives
Comparing the real and imaginary terms of these expressions gives the sine and cosine angle-addition formulas:
Geometry on the complex plane
Other nice properties
A special, and quite fascinating, consequence of Euler's formula is the identity , which relates five of the most fundamental numbers in all of mathematics: e, i, pi, 0, and 1.
The proof of Euler's formula can be shown using the technique from calculus known as Taylor series.
We have the following Taylor series:
The key step now is to let and plug it into the series for . The result is Euler's formula above.
Define . Then ,
; we know , so we get , therefore .