Complex analysis
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Complex Analysis. Prepared by Dr. Taha MAhdy. 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 .

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Complex Analysis

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Complex Analysis

Prepared by

Dr. Taha MAhdy


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,


What is a complex number

  • It is a solution for the equation


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.


Addition , subtraction, multiplication


Complex conjugate

  • The complex conjugate is:

  • Note that


Complex conjugate


Division is defiened in terms of conjugate of the denominator


Graphical representation of complex number


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.


Very Important complex transformations

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


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


The Polar Representation


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.


THE ARGUMENT OF Z


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:


Note the similarity


EULER’S FORMULA


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


EULER’S FORM operations


EULER’S FORM operations


EULER’S FORM operations


DE MOIVRE’S THEOREM


Assignment

  • Solve the problems of the chapter


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