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Complex Numbers

Page history last edited by CAI, Sheng 11 years, 6 months ago

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 (ab) 

Fig.1 An illustration of the complex plane

 

Basic Calculus:

1. About i

FormulaFormulaFormulaFormula.

Formula

 

2. Addition and Subtraction

Formula

Formula

 

3. Multiplication and Division

Formula

Formula

 

 

Euler's formula

Formula

 

Fig.2 Euler's formula for a general angle

 

Example 1:

Prove Formula

 

Proof:

Formula

Formula

Formula

Formula

Formula

 

Meanwhile, we know that 

Formula

Formula

 

Example 2:

Prove Formula and Formula

 

Proof:

Formula

Formula

Formula

Formula

Therefore, Formula and Formula.

 

Exercise:

Formula and Formula

 

Maclaurin series

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

 

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

 

Formula

 

Formula

Hint: Formula or Formula

 

Formula

Hint: Formula or Formula

 

 

 

 

 

 

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