Applications of Euler’s Formula
Complex Multiplication, Exponentiation, & Trigonometry
In a previous article, I laid out how one might end up with, and make sense of, the following equation:
In this article, a sort of continuation, I will be discussing some applications of this formula. Mainly how it allows us to manipulate complex numbers in newfound ways.
Polar Form of Complex Numbers
A complex number z is one of the form z=x+yi, where x and y are real numbers and i is the square root of -1. Since it has two parts, real and imaginary, plotting them requires 2 axes, unlike the real numbers which only require a single axis. The plane formed by the real and imaginary axes is called the complex plane.
One can quickly see that a complex number graphed on the complex plane is very similar to a vector graphed on a 2D plane. The x coordinate of the vector is the real part and the y coordinate the imaginary part. But, just like vectors, we don’t have to represent complex numbers in terms of their x and y coordinates. We can represent them by their magnitude and angle:
Basic knowledge of trigonometry tells us that when given a vector with magnitude r and an angle θ, the x component is r cos(θ) and the y component is r sin(θ). As such we can represent a complex number in polar form like so:
This equation serves as a bridge between the rectangular and polar forms of complex numbers. Simply plug in r and θ and the real and imaginary parts reveal themselves. But wait, doesn’t this expression seem familiar…
By recognizing Euler’s formula in the expression, we were able to reduce the polar form of a complex number to a simple and elegant expression:
Converting from polar to rectangular is as simple as plugging θ into Euler’s formula and multiplying by r. Converting from rectangular to polar is as simple as finding the magnitude and angle of the number:
This makes manipulating complex numbers in polar form much less of a hassle. But why even represent complex numbers in polar form to begin with? Well as it turns out, many operations involving complex numbers are much easier to perform when they are in polar form…
Multiplication of Complex Numbers
Multiplying a complex number by a real number is simple enough, just distribute the real number to both the real and imaginary parts of the complex number. However, due to having two terms, multiplying 2 complex numbers together in rectangular form is a bit more tricky:
However, thanks to the formula for polar complex numbers we derived earlier, we can see that a simpler method for multiplying complex numbers exists:
From the derivation we can see that to multiply two complex numbers, you multiply their magnitudes and add their angles. We can convert this back to rectangular form if we expand using Euler’s formula:
**Side Note: Division is the same thing as multiplication by an inverse (a/b=a*(1/b)). So if you want to divide two complex numbers simply divide their magnitudes and subtract their angles.
Exponentiation of Complex Numbers
Exponentiation of a complex number in rectangular form is a tedious, even soul sucking, task to do by hand and should be avoided at all costs. Moreover, it really only makes sense for whole number powers:
Thankfully there exists a simpler method. One that, like before, relies on the polar form of complex numbers:
From the derivation we can see that to raise a complex number to the nth power, raise its magnitude to the nth power and multiply its angle by n. We can convert this back to rectangular form if we expand using Euler’s formula:
This is called De Moivre’s Theorem, and works for any real n, not just integers.
**Side Note: taking the nth root of a complex number is the same thing as raising it to the power of 1/n (ex. ∛x=x^(1/3)). So to take the nth root simply find the nth root of the magnitude and divide the angle by n. Also notice that there are several solutions to taking the nth root of a complex number due to how the division works out with different multiples of 2π. Read more here.
Exponentiation by Complex Numbers
We’ve covered how to raise a complex number to a real power, but what about raising a real number to a complex one? Well we already have an expression for raising e to an imaginary number (i.e Euler’s formula) so let’s use that to extend imaginary exponentiation to any base:
Let’s unpack this. The first line just restates the namesake of this article, Euler’s formula. In line 3 we plug in x ln b into Euler’s formula. In line 4 we notice that this expression also equals bⁱˣ. This because the function ln x is the inverse to eˣ, meaning we can cancel them out and bring b down. Putting these two equalities together we get:
Okay now we have imaginary exponentiation but what about complex exponentiation? How can we add the real part? Well its quite simple really, For any number b raised to the power of x+iy:
Here’s an example:
**Side Note: Exponentiation of Complex Numbers by Complex Numbers
In the last two sections I showed you how Euler’s formula allows us to raise complex numbers to real powers and real numbers to complex powers. But what about raising complex numbers to complex powers?
Well as it turns out there are actually an infinite amount of solutions to a complex number raised to another complex number and moreover the math behind it is less than friendly (as you can imagine). And so I leave it to you the reader to explore the topic yourself, should you choose to do so.
Complex Angles and Trigonometry
Part 1: Hyperbolic Sine and Cosine
There is one last neat consequence of Euler’s formula, and that is the introduction of complex valued angles and corresponding trigonometric functions to boot.
Let’s start by creating a modified version of Euler’s formula, one with a negative x value/angle:
Line 1 just restates Euler’s formula. In line 3 we plug in -x into Euler’s formula. In line 4 we use the properties of cosine (cos -x = cos x) and sine (sin -x = -sin x) to simplify the expression.
Notice that this equation is the same as Euler’s formula except the imaginary part is negative. When you take a complex number and negate its imaginary part that is called its complex conjugate.
Taking Euler’s formula we can subtract it from its conjugate form to reach a new definition of sine:
Dividing both sides by 2i in the last line leaves us with this:
The same can be done with cosine:
Dividing both sides by 2 produces:
Whats special about these forms of the functions is that we can plug i into them and they’ll actually produce an answer! See for yourself:
First we plug ix into the sine expression (line 3) and multiply both the numerator and denominator by i, to move i to the numerator. By simplifying the i² terms (which equal -1) we end up with:
Notice that I introduced a new function on the right side: sinh x. This function is used in a variety of mathematics and is formally known as the hyperbolic sine function and factors out the i term. We’ll make use of it to simplify some expressions later on.
The same can, of course, be done for cosine:
Just plug ix into the cosine expression simplify the i² terms and:
This new function is called the hyperbolic cosine. Notice that from this equation, the cosine of any imaginary number is always real. There are no imaginary terms in the expression. Neat huh?
Part 2: Putting it together
Now that we have expressions for plugging purely imaginary angles/values into sine (i sinh x) and cosine (cosh x) we can now solve for the complex valued version using a bit of trigonometry:
Line 1 just states the angle addition identity of sine. In line 3 we plug a complex number a+bi into the formula. In line 4 we simplify the equation using the hyperbolic functions we defined earlier.
The same can be done for cosine:
With these two equations we can plug any complex angle into the sine and cosine functions and get an answer. Here’s an example:
Summary
To review, we covered:
- The polar form of a complex number.
- The multiplication of 2 complex numbers.
- The exponentiation of a complex number by a real one.
- The exponentiation of a real number by a complex one.
- Complex valued angles/trigonometric functions.
