Complex Numbers

A complex number is a number, z, which can be put in the form a + bi, where a and b are real numbers and i is called the imaginary unit, where i² = −1. In this expression, a is called the real part, Re(z), and b the imaginary part, Im(z), of the complex number. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b).  

Fig.1 An illustration of the complex plane

Basic Calculus:

1. About i

, , , .

2. Addition and Subtraction

3. Multiplication and Division

Euler's formula

Fig.2 Euler's formula for a general angle

Example 1:

Prove 

Proof:

Meanwhile, we know that 

Example 2:

Prove  and 

Proof:

Therefore,  and .

Exercise:

 and 

Maclaurin series

                                                                        maxima code: taylor(1/(1-x),x,0,5);

                                 maxima code: taylor(log(1+x),x,0,5);

Hint:  or 

Hint:  or