Category Archives: Complex Analysis

Euler’s famous formula

A hand waving proof of Euler’s formula e^{i\theta}=\cos\theta +i\sin\theta starts from the following Taylor series:

\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots

\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots

e^{x}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots

Plugging in i\theta in this series and separating the real and imaginary parts yields the formula.

Here is a sketch for a more formal proof:

1. Consider the initial value problem \frac{dw}{dz}=w; w(0)=1. We solve it by formally setting

w=a_0+a_1z+a_2z^2+\cdots

\frac{dw}{dz}=a_1+2a_2z+\cdots

Evidently to satisfy the differential equation we must have a_{n-1}=na_n and the initial condition gives a_0=1. It follows by induction that a_n=1/n!.

2. The solution w is now redesignated as e^z=1+z+\frac{z^2}{2!}+\frac{z^3}{3!}+\cdots.

3. We note that this power series has an infinite radius of convergence as (n!)^{1/n}\rightarrow\infty. By the existence and uniqueness theorem of differential equations, it is certain that this is the only solution of the given IVP.

4. The trigonometric functions are now defined by \cos z=\frac{e^{iz}+e^{-iz}}{2} and \sin z=\frac{e^{iz}-e^{-iz}}{2i} and it immediately follows that e^{iz}=\cos z+i\sin z. We thus have Euler’s formula.

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