Euler's equation is one of most remarkable and mysterious discoveries in Mathematics. Euler's equation (formula) shows a deep relationship between the trigonometric function and complex exponential function.

First, take a look the Taylor series representation of exponential function, and trigonometric functions, sine, and cosine, .

Les't compare with . Notice is almost identical to Taylor series of ; all terms in the series are exactly same except signs. As you know, the exponential function, increases exponentially as input *x* grows. But, what does this exponential function have to do with periodic (oscillating) functions, and ?

Mathematicians had tried to figure out this weird relationship between the exponential function and the sum of 2 oscillating functions. Finally, Leonhard Euler completed this relation by bringing the imaginary number, into the above Taylor series; instead of and instead of .

Now, we find out equals to , which is known as Euler's Equation.

We know that the exponential function, is increasing exponentially as x grows. But, what does the function look like?

The following images show the graph of the complex exponential function, , by plotting the Taylor series of in the 3D complex space (x - real - imaginary axis). Surprisingly, it is a spiral spring (coil) shape, rotating around a unit circle. And, when it is projected to the real number (top view) and imaginary number axis (side view), it becomes a trigonometric function, respectively cosine and sine.

This graphical representation of the complex exponential function, clearly shows the relation to the trigonometric function; the real number part of is cosine and the imaginary part of is sine function with the period of in radians.

This is an interactive OpenGL application to draw the complex exponential function, in 3D.

You can download the source codes and binary:

euler.zip (*Updated 2021-04-08*)

Use the mouse cursor to rotate the view and use 'd' key to change drawing modes.

When the graph of is projected to the complex plane, the function is tracing on the unit circle. It is a periodic function with the period .

It means that raising to an imaginary power produces the complex number with the angle *x* in radians. This polar form of is very convenient to represent rotating objects or periodic signals because it can represent a point in the complex plane with only single term instead of two terms, . Plus, it simplifies the mathematics when used in multiplication, for example, .

The complex exponential forms are frequently used in electrical engineering and physics. For example, a periodic signal can be represented the sum of sine and cosine functions in Fourier analysis, and the movement of a mass attached to a string is also sinusodial. These sinusodial functions can be replaced with the complex exponential forms for simpler computation.

We can extend this polar form representation for any complex number.

The magnitude (norm), |z| of the complex number is a scalar value and can be rewrtten by using the laws of exponent and logarithm:

The multiplication of two complex numbers implies a rotation in 2D space.

Take a look at the following figure showing 2 complex numbers on the unit circle of the complex plane.

When we compare these two complex numbers, we notice that the sum of angles in the imaginary exponential form equals to the product of two complex numbers, . It tells us multiplying by performs rotating with angle .

We can extend it to any arbitary complex number. In order to rotate a complex number *z = x + iy* with a certain angle , we simply multiply to the number. Then, the rotated result *z'* becomes .

If we re-write it as a matrix form by omitting , it becomes a 2x2 rotation matrix that we are familiar with.

If we substitute the value into Euler's equation, then we get:

This equation is called Euler Identity showing the link between 5 fundamental mathematical constants; 0, 1, , , and .

Logarithmic function is only defined for the domain *x* > 0. But, Euler Identity allows to define the logarithm of negative *x* by converting exponent to logarithm form:

If we substitute to Euler's equation, then we get:

Then, raise both sides to the power :

The above equation tells us that is actually a real number (not an imaginary number).

This is a proof using calculus. Let's start with the right-hand side only and its differentiate is:

Modify the right-hand side of the above equation:

Move *t* to left-hand side, then apply integral:

To find constant C, substitute *x* = 0:

Now, we know C = 0, so the above equation is:

Finally, convert this logarithm form to the exponential form: