A calculus of the absurd

15.4 The exponential form of a complex number

Consider the Maclaurin series for \(\cos (x)\), \(i\sin (x)\) and \(e^{ix}\). What happens when we add \(\cos (x) + i\sin (x)\)? Well a fair amount of algebra to start with! After that, however, we do get an interesting result, though.

\begin{align*} \cos (x) + i\sin (x) &= 1 - \frac {x^2}{2!} + \frac {x^4}{4!} - \frac {x^6}{6!} + \frac {ix}{1!} - \frac {ix^3}{3!} + \frac {ix^5}{5!} - \frac {ix^7}{7!} \\ &= 1 + \frac {ix}{1!} - \frac {x^2}{2!} - \frac {ix^3}{3!} + \frac {x^4}{4!} + \frac {ix^5}{5!} - \frac {x^6}{6!} - \frac {ix^7}{7!} \\ &= 1 + \frac {ix}{1!} + \frac {(ix)^2}{2!} + \frac {(ix)^3}{3!} + \frac {(ix)^4}{4!} + \frac {(ix)^5}{5!} + \frac {(ix)^6}{6!} + \frac {(ix)^7}{7!} \\ &= e^{ix} \end{align*}

This means that 111111 This is sometimes referred to as "Euler’s formula"

\begin{equation} \label {Euler's formula} e^{ix} = \cos (x) + i \sin (x) \end{equation}

Which provides a link between trigonometry and complex numbers! This turns out to be very useful in proving trig identities.

For any complex number, where \(r\) is the modulus and \(\theta \) is the argument, we have

\begin{equation} re^{\theta i} = \cos (\theta ) + i \sin (\theta ) \end{equation}

We can use this to write other complex numbers, such as \(1 + i\) in the form \(re^{i \theta }\).