Create account or Sign in. Basic Properties of Complex Numbers 1. Topological Properties of the Complex Numbers 1. Elementary Complex Functions 1. Complex Differentiable Functions 2. The Cauchy-Riemann Theorem 2. Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3. They would rather have math. Also note that the functions defined in cmath always return a complex number, even if the answer can be expressed as a real number in which case the complex number has an imaginary part of zero.

A note on branch cuts: They are curves along which the given function fails to be continuous. They are a necessary feature of many complex functions. It is assumed that if you need to compute with complex functions, you will understand about branch cuts. Consult almost any not too elementary book on complex variables for enlightenment.

For information of the proper choice of branch cuts for numerical purposes, a good reference should be the following:. In Iserles, A. Clarendon Press pp— Note On platforms with hardware and system-level support for signed zeros, functions involving branch cuts are continuous on both sides of the branch cut: the sign of the zero distinguishes one side of the branch cut from the other.

Note The modulus absolute value of a complex number x can be computed using the built-in abs function. New in version 2. Table of Contents 9. Conversions to and from polar coordinates 9. Power and logarithmic functions 9.

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Trigonometric functions 9. Hyperbolic functions 9. Classification functions 9.

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Constants Previous topic 9. Quick search. I will explain basic mathematical operation for complex number, but you would hardly have chance to do this kind of calculation by hands.

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In most case you may use various software to do this calculation. So in this section, I would like you to understand the meaning of each operation.

## Intro to the imaginary numbers

One of the easiest way to understand the meaning of these operation would be to present it on a coordinate system and try to visualize it. Let's first look into the addition of two complex number. Let's first look at the symbolic process. You would notice that this is exactly the same result as summation of two vectors [a, b], [c, d] and graphical presentation is also same as vector summation as shown below. Unlike addition and substraction, multiplication of two complex number is totally different from the multiplication of two vector You can refer to 'Vector' section for the details of vector multiplication.

The graphical interpretation of complex number multiplication is as follows. If I put this into simple words, when two complex number are multiplied, the angle of the resulting number is the summation of the two complex number and the magnitude absolute number of the resulting number is multiplication of the magnitue of the two complex numbers.

## cmath — Mathematical functions for complex numbers — Python documentation

Note : You may realize that complex number multiplication become much easier if you represent the vectors in polar form. As I mentioned above, one of aspect of complex number multiplication is adding the angle of the two complex numbers. It means that the role of c2 in this case is to rotate c1 or to shift phase of c2. This is very important property and you would see this kind of operation in many area in engineering. I want to introduce a theorem that you can easily understand and is very useful to you.

It is De Moivre's theorem saying as follows. What does this mean? Let's look at an example as shown below. The question is simple. It is to get the power of 2 for a complex number. How would you get this? You can try following. I would not show you how to calculate this. You can do it with junior high school math. If you remember what you read in previous section about 'multiplication of complex numbers'. I said the multiplication of two complex numbers can be obtained just by going through follow steps.

Why this theorem is important?

## Calculus/Complex numbers

It is important because it is a very handy tool for you. Let's say we have to get the power of for a complex number. How would you get the answer for this? The first thing you can think of would be to try as follows.

Can you do this? You would do it if you have around one full day to kill. What if you made a mistake when you do this around 50 times and didn't realize it? You would notice that this kind of brutal multiplication would not be a practical method for this case. With the theorem, you can get the solution to power of for the complex number in single step as shown below. But there is a precondition for using this theorem. In most case, people would not give you a complex in trigonometric form and ask you to get the power of n for it.

So you have to know first on how to convert a complex number into trigonometric form. See Representation of a complex number section and try to get familiar to all of those presentation and how to convert one form to another form. Euler Equation. One of the most famous form of complex number which is used in engineering would be what we call 'Euler form' or 'Euler Equation'. It is represented as follows. I am not going to explain on how this form is derived and how to prove this.

You may goole on this subject if you are interested