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Complex Analysis

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Complex Analysis

Prepared by

Dr. Taha MAhdy

- Complex analysis has not only transformed the world of mathematics, but surprisingly, we find its application in many areas of physics and engineering.
- For example, we can use complex numbers to describe the behavior of the electromagnetic field.
- In atomic systems, which are described by quantum mechanics, complex numbers and complex functions play a central role,

- It is a solution for the equation

- More general complex numbers can be written down. In fact, using real numbers a and b we can form a complex number:
c = a + ib

- We call a the real part of the complex number c and refer to b as the imaginary part of c.

- The complex conjugate is:

- Note that

- A Complex Variable can assume any complex value
- We use z to represent a complex variable.
z = x + jy

- We can graph complex numbers in the x-y plane, which we sometimes call the complex plane or the z plane.
- We also keep track of the angleθ that this vector makes with the real axis.

It appears that complex numbers are not so “imaginary” after all;

- Let z = x + iy is the Cartesian representation of a complex number.
- To write down the polar representation, we begin with the definition of the polar coordinates (r,θ ):
x = r cosθ ; y = r sinθ

- Note that r > 0 and that we have
- tanθ = y / x as a means to convert between polar and Cartesian representations.
- The value of θ for a given complex number is called the argument of z or arg z.

- Euler’s formula allows us to write the expression cosθ + i sinθ in terms of a complex exponential.
- This is easy to see using a Taylor series expansion.
- First let’s write out a few terms in the well-known Taylor expansions of the trigonometric functions cos and sin:

- These relationships allow us to write a complex number in complex exponential form or more commonly polar form. This is given by

- Solve the problems of the chapter