Exponential Form of Complex Numbers

Another day in our complex lives. You already know about complex numbers a little, the polar form of it as well. What then is the need to have another form of complex numbers? Well, you will have to go and ask Euler about that (he is probably grinning at us from up there). But for now, another form of Complex Numbers is inevitable, i.e., the Exponential Form of Complex Numbers.

Exponential Form of Complex Numbers derivation

A complex number is fundamentally expressed as \(z=a+ib\) where \(a\) and \(b\) are real-valued constants and \(b≠0\). We also know of another form that involves the argument of a complex number as well, i.e., the Polar form of a Complex Number.

Recall that the Polar form of a complex number whose argument is \(\theta\) is given as follows:

$$z=r(\cos\theta +i\sin\theta)$$

where \(r\) is the modulus of the complex number: \(r=\sqrt{a^{2}+b^{2}}\).

There is a more compact way we can write this; in an exponential form. But where does the exponential form come from? The answer is Euler's Formula.

Euler's Formula

It is to no one's surprise that we encounter Leonhard Euler, here as well, like in almost every other branch of mathematics. There is a very elegant equation, that encapsulates exponential functions, complex numbers and trigonometric functions, all in one formula. It is known as Euler's Formula or Euler's Identity.

The formula goes as follows:

$$e^{i\theta}=\cos\theta+i\sin\theta$$

The proof of this equation is unfortunately beyond the current scope of this article. The Right-hand Side of the equation is very familiar if you observe it closely. It is nothing but an integral part of the Polar form of a Complex number.

Polar form and Euler's formula

And now we can substitute for Euler's formula in the Polar form to get our Exponential form of a complex number. So substituting \(e^{i\theta}=\cos\theta+i\sin\theta\) in \(z=r(\cos\theta+i\sin\theta)\):

$$z=re^{i\theta}$$

which is what we were after. Now we have a formula that converts a complex number in simple form to an exponential form.

Note that \(\cos\theta+i\sin\theta\) is often abbreviated as \(\rm{cis}\,\theta\) for convenience and to avoid clutter.

This form can also be extended to powers of complex numbers, such as \(z^{3}=r^{3}e^{i3\theta}\), \(z^{4}=r^{4}e^{i4\theta}\) and so on. In general: \(z^{n}=r^{n}e^{in\theta}\).

Exponential form to Rectangular Form

Every so often, one might desire the rectangular form of a complex number instead of the Exponential form. We can convert one into another by comparing the two forms, as follows:

$$z=a+i b \hspace{5mm} \ \text{and} \ \hspace{5mm} z=r(\cos \theta+i \sin \theta)=r e^{i \theta}$$

Comparing the RHS of both of the above equations,

$$a=r \cos \theta \hspace{5mm} \ \text{and} \ \hspace{5mm} b=r \sin \theta$$

where \(r\) is the magnitude of \(z\) so we have

$$\cos \theta=\frac{a}{\sqrt{a^{2}+b^{2}}} \hspace{5mm} \ \text{and} \ \hspace{5mm} \sin \theta=\frac{b}{\sqrt{a^{2}+b^{2}}}$$

Substituting for \(\sin \theta\) and \(\cos \theta\) in the exponential form:

$$z=r e^{i \theta} \Rightarrow e^{i \theta}=\frac{a}{\sqrt{a^{2}+b^{2}}}+\frac{i b}{\sqrt{a^{2}+b^{2}}}$$

Hence, one can convert a given complex number in exponential form to rectangular form using the above formula.

Exponential Form of Complex Numbers Calculation

To give a view of how the exponential form of a complex number is represented on a complex plane, we need to plot a graph. For a complex number \(z=re^{i\theta}\), the complex number will originate from the origin and incline at an angle \(\theta\) with the positive \(x-\)axis.

Fig. 1: A complex number on an Argand plane.

The exponential form is a very concise way of writing complex numbers, and it is also very useful since it displays the argument and magnitude of the complex number.

One important thing to note about the complex numbers in this form is that a complex number of the form \(z=a+ib\) can be written in not one, but several exponential forms. This is because the argument \(\theta\) belongs to the interval \((0,2\pi]\) and the function can attain the same value for numerous arguments. For instance, \(\tan \frac{\pi}{4}=\tan \frac{5\pi}{4}\), which also implies that \(re^{\frac{\pi i}{4}}=re^{\frac{5\pi i}{4}}\).

For rectangular coordinates, there is only one form it can take at a time. That is why, for exponential forms, to avoid confusion, we only take into account the Principal argument of a complex number.

Follow the steps below to convert a complex number into an Exponential form:

1. From the given \(z=a+ib\), find the magnitude of \(z\): \(r=\sqrt{a^2+b^2}\)

2. Now calculate the principal argument of the complex number: \(\tan\theta=\frac{b}{a}\)

3. Thus, we now have the exponential form as \(z=re^{i\theta}\)

Exponential Form of Complex Numbers examples