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.

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

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

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

$$re^{\theta i} = \cos (\theta ) + i \sin (\theta )$$

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