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Chapter 8 Complex Numbers & 3D Plots

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Complex Numbers

Complex numbers are numbers with both a real and an imaginary component.

General form: operations:

c=a+b i

where a and b are both real, and i is .

A complex number can be represented as a rectangular coordinates.

A complex number can also be represented as a vector of length z and angle q where

z= √(a2+b2) and q=tan-1(b/a).

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Complex Numbers2

Arithmetic operations:

Complex addition

complex subtraction

complex multiplication

complex division

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Complex Numbers2

Relational operators:

Equal operator: ==

Not equal operator: ~=

Absolute value:

For a complex number z=x+yi, we define the

absolute value |z| as being the distance from z to 0 in the complex plane.

|z| = √(x2 + y2)

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Some Supporting Functions

Function Descriptions

conj(z) Returns the complex conjugate of a number z.

real(z) Returns the real portion of the complex number z.

imag(z) Returns the imaginary portion of the complex number z.

isreal(z) Returns true (1) if no element of array z has an

imaginary component.

abs(z) Returns the magnitude of the complex number z.

angle(z) Retunes the angle of the complex number z.

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Example 8.1

The quadratic equation

1. State the problem:

Write a program to solve for the roots of a

quadratic equation, ax2+bx+c=0.

2. Define the inputs and outputs:

Inputs will be the coefficients a, b, and c.

Outputs will be the roots of equations

3. Define the solution:

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Example 8.12

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Plotting Complex Data

Consider the function

y(t) = e–0.2t (cos t + i sin t)

If this function is plotted with the function plot, only

the real part will be plotted with a warning message:

Imaginary parts of complex X and/or Y

arguments ignored.

If both the real and imaginary parts of the function are of interest, then both parts are needed to be plotted.

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Plotting Complex Data2

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Plotting Complex Data3

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Plotting Complex Data4

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Plotting Complex Data4

Alternatively, the real(y) can be plotted versus imag(y) or showing abs(y) versus angle(y).

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Multidimensional Arrays

Multidimensional arrays are based on the first 2D as building block, called a page.

Some useful functions:.

ones(n, m, p)

zeros(n, m. p)

rand(n, m, p)

ndims(x)

size(x)

length(x)

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Multidimensional Arrays2

Some examples.

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Three-Dimensional Plots

The plot creates a 2D chart.

Consider the function:

x(t) = e–0.2t cos t

y(t) = e–0.2t sin t

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Three-Dimensional Plots2

A 3D line plot can be created by plot3.

Consider the same function:

x(t) = e–0.2t cos t

y(t) = e–0.2t sin t

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Three-Dimensional Surfaces

Surface, mesh, and contour plots are convenient ways to represent data that is a function of two variables, f(x,y,z).

Arrays x, y, and z must have the identical size.

The x-y grid can be created by meshgrid.

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Three-Dimensional Surfaces2

Consider the 3D sinc function as a folded sheet

z(x, y) = sin(x + y)/(x + y)

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Three-Dimensional Surfaces3

Consider the 3D sinc function as a rain drop.

z(x, y) = sin(sqrt(x2+y2))/sqrt(x2+y2)

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Three-Dimensional Surfaces4

Draw the 3D sinc function by using surf.

z(x, y) = sin(sqrt(x2+y2))/sqrt(x2+y2)

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Three-Dimensional Surfaces4

Draw the 3D sinc function by using contour.

z(x, y) = sin(sqrt(x2+y2))/sqrt(x2+y2)

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Three-Dimensional Surfaces5

Draw a 3D sphere function.

x=r*cos(q)*cos(f)

y=r*cos(q)*sin(f)

z=r*sin(q)

q=[–p, p]

f=[–p/2, p/2]

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Three-Dimensional Surfaces6

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Three-Dimensional Surfaces7

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Homework Assignment #15

8.5 Exercises

Page 345: 8-1, 8-2, 8-14, 8-15

This assignment is for your reference.