Euler's Equation
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.
Discovery of Euler's Equation
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, i into the above Taylor series; instead of and instead of .
Now, we find out equals to , which is known as Euler's Equation.
Graph of 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.
Use the mouse cursor to change the view and use 'd' key to change drawing modes.
Meaning of Euler's Equation
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.
The magnitude (norm), |z| of the complex number is a scalar value and can be rewrtten by using the laws of exponent and logarithm:
2D Rotation with Euler's Equation
The multiplication of two complex numbers implies a rotation in 2D space.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 Φ.
We can extend it to any arbitary complex number. In order to rotate a complex number z = x + iy with a certain angle Φ, we simply multiply to the number. Then, the rotated result z' becomes .
If we re-write it as a matrix form, it becomes a 2x2 rotation matrix that we are familiar with.
Euler Identity
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).
Proof of Euler's Equation
This is a proof using calculus. Let's start with the right-hand side only and its differentiate is:Modify 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:
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