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Section 2.5 Euler's Formula

As soon as we allow complex numbers into our mathematics, we also need to understand how to extend the definition of familiar functions of real-valued arguments to complex-valued arguments. Most of these extensions are based on Euler's formula.

Definition 2.3. Euler's formula.

The formula

\begin{equation} e^{i\theta}=\cos\theta + i\sin\theta ,\tag{2.5.1} \end{equation}

called Euler's formula, defines the exponential of a pure imaginary number in terms of the sine and cosine of a real number.

Frequently used formulas.

We can rearrange Euler's formula and its complex conjugate

\begin{equation} e^{-i\theta}=\cos\theta - i\sin\theta\tag{2.5.2} \end{equation}

to find expressions for \(\sin{\theta}\) and \(\cos{\theta}\) in terms of complex exponentials:

\begin{align} \cos{\theta}\amp = \frac{1}{2}(e^{i\theta}+e^{-i\theta})\tag{2.5.3}\\ \sin{\theta}\amp = \frac{1}{2i}(e^{i\theta}-e^{-i\theta})\tag{2.5.4} \end{align}

You will use these expressions for sine and cosine frequently. Make sure you can recognize the right-hand sides of these equations.

Proofs.

Euler's formula can be "proved" in two ways:

  1. Expand the left-hand and right-hand sides of Euler's formula (2.5.1) in terms of known power series expansions. Compare equal powers.

  2. Show that both the left-hand and right-hand sides of Euler's formula (2.5.1) are solutions of the same second order linear differential equation with constant coefficients. Since only two solutions of a second order linear equation are linearly independent, write

    \begin{equation} e^{i\theta}=A\cos\theta + B\sin\theta\tag{2.5.5} \end{equation}

    and choose boundary conditions and known properties of the exponential function to show that \(A=1\) and \(B=i\text{.}\)

Note: If you only know properties of the exponential function for real numbers, then Euler's formula and the "proofs" above are not really proofs, rather they are the definition of the exponential for a pure imaginary argument. Technically, this happens through a process called analytic continuation.