Complex Analysis

1. Complex Analysis Prepared by Dr. Taha MAhdy

2. Complex analysis importance • 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,

3. What is a complex number • It is a solution for the equation

4. The Algebra of Complex Numbers • 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.

5. Addition , subtraction, multiplication

6. Complex conjugate • The complex conjugate is: • Note that

7. Complex conjugate

8. Division is defiened in terms of conjugate of the denominator

9. Graphical representation of complex number

10. Complex Variables • 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.

11. Very Important complex transformations It appears that complex numbers are not so “imaginary” after all;

12. The Polar Representation • 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

13. The Polar Representation

14. The Polar Representation • 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.

15. THE ARGUMENT OF Z

16. EULER’S FORMULA • 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:

17. Note the similarity

19. EULER’S FORMULA

20. EULER’S FORM • These relationships allow us to write a complex number in complex exponential form or more commonly polar form. This is given by

21. EULER’S FORM operations

22. EULER’S FORM operations

23. EULER’S FORM operations

24. DE MOIVRE’S THEOREM

25. Assignment • Solve the problems of the chapter