On Eulers Identity exp^iθ

Three interesting numbers in Maths are π, the ratio of a circle's circumference to its diameter, Euler's number e, approx 2.71818, and i, √-1. Amazingly, e^iπ = -1.

This comes from his identity, e^iθ = cos(θ) + i sin(θ). Here are some remarks on it and a circular plot on the Argand plane.

You can also use his identity for expressions for cos(θ) and sin(θ) in terms of e^iθ, as is shown here.

See also, my pages explaining a use of e^π, which start here.

Leonhard Euler is credited with finding the identity (and many other concepts in mathematics) published in his 1748 book Introductio im analysin infinitorum. In fact, the British Mathematician Roger Coates, who worked with Isaac Newton, had a geometric demonstration in 1714 that iθ = ln (cos(θ) + i sin(θ)). This becomes Euler's formula if both sides are raised to the power of e!

Often the identity is demonstrated using power series of exp, sin and cos. Here is a shorter method using differentiation.

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