Presentation on theme: "Prepared by Dr. Taha MAhdy"— Presentation transcript:
1 Prepared by Dr. Taha MAhdy Complex AnalysisPrepared byDr. 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 + ibWe call a the real part of the complex number c and refer to b as the imaginary part of c.
10 Complex Variables A Complex Variable can assume any complex value We use z to represent a complex variable.z = x + jyWe 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 angle θ 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
14 The Polar Representation Note that r > 0 and that we havetanθ = 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.
16 EULER’S FORMULAEuler’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: