Euler’s equation

e^ix =  cos x + isin x

The most remarkable and mysterious discoveries in mathematics is Euler’s equation. It shows the relationship between the trigonometric function and complex exponential function.

Let us compare cos x + sin x with e^x. do notice that, cos x + sin x is almost same to Taylor series of e^x; the series in the terms are exactly same except signs.  As input x grows, the exponential function e^x increases exponentially. But what this exponential function have to do with periodic functions, cos x and sin x?

To find out this strange relationship between the exponential function and sum of 2 periodic functions, mathematicians had tried hardly. Finally Leonhard Euler completed this relation by bringing the imaginary number, I into the above Taylor series; e^ix instead of e^x and cos x+ I sin x instead of cos x + sin x.

Now, we find out e^ix =  cos x + isin x, this is Euler’s equation.

Meaning of Euler’s equation

Graph of e^ix on the complex plane

When the graph of e^ix is projected to the complex plane, the function e^ix is tracing on the unit circle. It is the periodic function with the period 2π.

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

In electrical engineering and physics this complex exponential forms are frequently used. For example, in Fourier analysis, a periodic signal can be represented the sum of sine and cosine functions and the movements of a mass attached to a string is also sinusoidal. For simpler computation, this function can be replaced with the complex exponential forms.

Euler’s identity

If we substitute the value of x = π Euler’s equation, then we will get:

This equation is called Euler identity showing the link between five fundamental mathematical constants; 0, 1, e, I and π.

Log function is only defined for the domain x > 0. But to define the logarithm of negative x, Euler’s identity allows by converting exponent to logarithm form.

e^iπ = -1

iπ= ln(-1)

If we substitute x = π/2 to Euler’s equation, then we get:

Then raise both sides to the power i:

This equation tells us, i is actually a real number not imaginary.

• What is Euler's equation?
• Explain Euler's identity?