- 345 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Complex Analysis' - sierra

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

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.

Graphical representation of complex number denominator

Complex Variables denominator

- 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 denominator

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

The Polar Representation denominator

- 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 denominator

The Polar Representation denominator

- 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 denominator

EULER’S FORMULA denominator

- 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 denominator

EULER’S FORMULA denominator

EULER’S FORM denominator

- 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 denominator

EULER’S FORM operations denominator

EULER’S FORM operations denominator

DE MOIVRE’S THEOREM denominator

Assignment denominator

- Solve the problems of the chapter

Download Presentation

Connecting to Server..