Before we start talking about complex exponential, I’m going to explain the purpose of this story or posting. We’ll look around the concept of complex exponential function and how to use it to prepare the study of circuit theory. I know everyone(it means maybe…) who is the beginner of electric circuit get difficulty to solve the differential equation and use the complex plane. Hope this story would be helpful for you guys.
In electric engineering, we would meet a variety of “complex exponential” functions frequently. Then what’s the meaning of complex exponential? Why we use complex exponential form and use complex exponential signals?
Before going into the mainstream, let’s take a look normal exponential. Exponential function is used in various places.
One of the notable property of exponential is it’s derivative is itself.
Because of this property, exponential is used to solve the differential equation as a form of solution set. When we are going to study circuit theory or electric circuit, we often meet the differential equation problem. In the next time, I’m going to show you how to solve the DE(differential equation).
Now finally I’d like to demonstrate the meaning of complex exponential only in terms of electric engineering. I did the efforts to write legibly and simply, but it can be not enough for you.
1. Conversion property between addition and multiplication
One of the important property of exponential is to convert between addition and multiplication. In this post, we are going to focus on this property.
We’ll talk about conversion property of exponential both in real number line and in complex plane.
(1) Real Number Line
Real number is countable number in real world. Real numbers lie on the 1 dimension axis called x-axis. They have only magnitude. In other words, we can mapping all real numbers to a number line.
How to explain addition and multiplication over number line? Put ‘x’ to number line and imagine what to do to add ‘x’ to ‘1’. Leave the point of x alone and just sliding the axis. We can shift the axis to the left side one point and then the position of x becomes ‘x+1’. Since we consider addition not as the operator needs two input but as the system which can be defined as ‘+1’, systematic and geometric interpretation is possible in number line. Therefore addition along number line means sliding the axis. If you want to add then slide the axis to the left-side as much as the magnitude of the number of multiply and if you want to subtract then slide the axis to the right-side.
Likewise how to explain the multiplication over number line? Imagine the multiplication ‘x’ by ‘a’. We can move the point of ‘x’ to the point of ‘ax’ while leaving ‘x’ alone by stretching the axis ‘a’ times. ‘x 2’ means the reduction of the axis 2 times and ‘x 0.5’ means the expansion of the axis 2 times. Please refer to the following video for understanding what I mean. It explains the mechanism of addition and multiplication using the axis well.
(2) Conversion property in real number line.
By the followed property of exponential we can use the exponential function to convert between addition and multiplication. The following image show the mechanism of the conversion. You can see that the equation to about addition is transformed to the equation to about multiplication in exponential form. Therefore addition is equal to the multiplication over exponential of x. Notice that you should use exponential form as a system or a function.
What does it mean? Remember addition is exposed to sliding or shifting the axis(real number line) and multiplication is exposed to stretching the axis. In sum, sliding the axis is equal to stretching the axis over exponential form. Of course, any other exponential function which has the other base is OK. Both are different only in how much is the axis stretched.
(3) Complex plane
In contrast to real number line, complex consists of 2 axis. One is real number line and the other is imaginary number line. Since they lie on 2 Dimensional plane, complex numbers have magnitude and phase. Just think about polar coordinate.
What is the difference between real number line and complex plane? There are only two way of operating in real number line, sliding and stretching. But we can rotating operation in complex plane. Rotation means modify the phase of complex number keeping the magnitude of it. Imagine the mechanism of rotation. So we have to stretching the plane and rotating the plane to multiply complex number to complex number since multiplication would change both the magnitude and the phase. In other words, multiplication in complex plane is displayed the combination stretching and rotation.
For instance, imaginary number i means 90 degree rotation in complex plane. And square of i means 180 degree rotation. In fact, imaginary number doesn’t reveal in real world. The reason is we live in only real axis(1 D number system).
2. Euler’s identity
Based on the previous knowledge, let’s focus on exponential function in complex plane. Exponential has the same functionality in both 1 D an 2 D. As you know, it means the conversion between addition and multiplication. So it is very clear that complex exponential change the mechanism of sliding the plane to the mechanism of stretching and rotating the plane.
The point is the distance between two points is the same.
Therefore Euler’s identity means addition to i*pi is equal to multiplication by exponential form of it. Moreover multiplication by exp(i*pi) is the 180 degree rotation in unit circle. The following equation is Euler’s identity.
3. Euler’s equation
Euler’s equation is just the expansion of Euler’s identity for anonymous variable.
By dealing with complex number, we can use the magnitude and phase of numbers. And exp(i*pi) means the 180 degree rotation along the unit circle. Then we conclude that exp(i*x) means the rotation along the unit circle by deduction.
Complex exponential (exp(i*x))is the rotating function of the phase x. See the followed image. Rotation during the time interval project the cosine and sine shadow in real time plane and imaginary time plane. It develops cosine function in real axis.(It also develops sine function in imaginary axis.) In real world, cosine is just periodic function, however complex exponential in complex plane implies the rotation.
Finally the problem is simple when modifying cosine function to complex exponential or putting it into complex plane. “Change the problem and just solve the circle problem.”
