Module # ONE

Complex Numbers

using

MatLab

By Professor Mohamad H. Hassoun

Department of Electrical & Computer Engineering

Wayne Satate University

Detroit, MI 48202

October 22, 1996

Contents:

I. What Are Complex Numbers and Why Study Them?

II. Complex Algebra

III. Polar Form

IV. Exponential Form

V. Trigonometric & Hyperbolic Functions & Logarithms

I. What Are Complex Numbers and Why Study Them?

Definition

A complex number z is an ordered pair (x, y) of real numbers x and y written as z = x + i*y [or as (0, 1)] where by definition i = (-1)^1/2 (or i² = -1) is the imaginary unit i = 0 + i*1.0). x is called the real part of z and y is called the imaginary part of z. In circuit theory, j is used to replace i in order not to confuse the imaginary unit with the symbol for electric current. i (and j) are already defined for you in MatLab:

0 + 1.0000i

0 + 1.0000i

0 + 1.0000i

-1.0000 + 0.0000i

The real and imaginary parts of a complex number are evaluated as follows:

-2.5000 + 3.0000i

-2.5000

3

Where do Complex Numbers Come From

Complex numbers arise from solving for the roots of polynomial equations (polynomials appear in the solution of linear differential equations and in solving for the eigenvalues of a matrix). First to use complex numbers was the Italian mathemetician Girolamo Cardano (1501-1576). The term "complex number" was introduced by the German mathematician Carl F. Gauss.

Examples #1: Solve for the roots of the equation x² -10x + 40 =0:

solve('x^2-10*x+40=0')

[5+i*15^(1/2)]

[5-i*15^(1/2)]

numeric(ans) % give numerical answer

5.0000 + 3.8730i

5.0000 - 3.8730i

Example # 2: Find the eigenvalues of matrix A, where A is given by

A = [.5,.5,1,-1;1,-1,-1,-1;-1,-.25,1,1;1,-1,-8,1]

0.5000 0.5000 1.0000 -1.0000

1.0000 -1.0000 -1.0000 -1.0000

-1.0000 -0.2500 1.0000 1.0000

1.0000 -1.0000 -8.0000 1.0000

The eigenvalues are

1.4307 + 3.0894i

1.4307 - 3.0894i

-1.3205

-0.0408

of which two are complex numbers.

NOTE: In fact these eigenvalues are the roots of the fourth degree characteristic polynomial of A, namely |xI - A| = 0, as seen below:

B = sym('[x,0,0,0;0,x,0,0;0,0,x,0;0,0,0,x]')

[x,0,0,0]

[0,x,0,0]

[0,0,x,0]

[0,0,0,x]

p = determ(symsub(B,sym(A)))

x^4-3/2*x^3+31/4*x^2+125/8*x+5/8

4 3 2

x - 3/2 x + 31/4 x + 125/8 x + 5/8

Verification (solving for the roots of p = 0):

numeric(solve(p, 'x'))

1.4307 + 3.0894i

1.4307 - 3.0894i

-0.0408

-1.3205

It is an interesting fact that if a polynomial with real valued coefficients has a complex root z₁ = a + ib, then it must also have z₂ = a - ib as its root. z₂ is called the complex conjugate of z₁. The conj comman computes the conjugate of a complex number

1.0000 - 5.0000i

1.0000 + 5.0000i

II. Complex Algebra

The following are examples of addition, subtraction, multiplication and division of complex numbers. We illustrate these operations with the two complex numbers 4 -5i and 2 + 3i.

z1=4-5i, z2=2+i*3

4.0000 - 5.0000i

z2 =

2.0000 + 3.0000i

3*z2 % multiplication by a real number c: c*(a1 + ib1) = (c*a1) + i(c*b1)

6.0000 + 9.0000i

z1+z2 % addition: (a1 + ib1) + (a2 + ib2) = (a1+a2) + i(b1+b2)

6.0000 - 2.0000i

z1-z2 % addition: (a1 + ib1) - (a2 + ib2) = (a1-a2) + i(b1-b2)

2.0000 - 8.0000i

0 -33.0000i

z1*z2 % multiplication: (a1 + ib1)(a2 + ib2) = (a1a2 - b1b2) + i(a1b2 + b1a2)

23.0000 + 2.0000i

z1*conj(z1) % multiplication by the conjugate: (a1 + ib1)(a1 - ib1) = (a1a1 + b1b1)

41

z1/z2 % division: (a1 + ib1)/(a2 + ib2) = [(a1a2 + b1b2) +i(a1b2 + b1a2)]/(a2a2 + b2b2)

-0.5385 - 1.6923i

8

-9.0000 -40.0000i

2.2807 - 1.0962i

-1.0000 + 0.0000i

0.0000 - 1.0000i

1.0000 - 0.0000i

0.0000 + 1.0000i

III. Polar Form

The manipulation of complex algebriac expressions may be made easier by employing polar representation of complex numbers. The polar form of a complex number is obtained by viewing a complex number z as a vector (with magnitude r and angle q) in the complex plane: a plane where the x-axis represents the real part of z and the y axis represents the imaginary part of z. Then, one may use the polar-to-rectangular transformation: x = r cos q and y = r sin q. Hence, we may now write z = x + iy = r(cos q + i sin q). The inverse transformation from rectangular to polar form is given by: r = (x² + y²)^-1/2 and q =arc tan(y/x), where q is measured in radians and positive in the counterclockwise sense. MatLab has two convenient commants for computing the magnitude (abs) and angle (angle) of a complex number.

4.0000 - 5.0000i

abs(z1), angle(z1)

6.4031

ans =

-0.8961

This is the explicit way of computing the angle of z1

-0.8961

which we can express in degrees as

-51.3402

Therefore, z1 = 4 - i5 may be represented in the complex plane as a vector originating at the origion and extending 6.4031 units in a dirstion making an angle of 51.3402 degrees w.r.t. the x-axis in the clockwise direction (because of the negative sign!).

IV. Exponential Form

The exponential form of a complex number is a convenient way of expressing a complex number which renders it amenable to analytic manipulation. In the following, we utilize MatLab's symbolic manipulation features to show that a complex number z with magnitude r and angle q may be expressed as the product of r and the exponential term eⁱ^q; i.e., z = reⁱ^q

The idea here is to show that z = r(cos q + i sin q) = reⁱ^q, or eⁱ^q = cos q + i sin q. This last expression is known as Euler's (pronounced Oiler's) formula. Let us express eⁱ^q as a power series using Taylor's series expansion at zero:

taylor('exp(i*x)') % we use x in place of q

1+i*x-1/2*x^2+(-1/6*i)*x^3+1/24*x^4+(1/120*i)*x^5+O(x^6)

2 3 4 5 6

1 + i x - 1/2 x - 1/6 i x + 1/24 x + 1/120 i x + O(x )

The above expansion may be rearranged as: [1 -1/2 x² + 1/24 x⁴ + O(x⁶)] + i [x - 1/6 x³ + 1/120 x⁵ + O(x⁷)] which is equal to cos(x) + i sin(x) since the power series representations for cos(x) and sin(x) are, respectively,

pretty(taylor('cos(x)'))

2 4 6

1 - 1/2 x + 1/24 x + O(x )

and

pretty(taylor('sin(x)'))

3 5 6

x - 1/6 x + 1/120 x + O(x )

This completes the proof that eⁱ^q = cos q + i sin q, and therefore r(cos q + i sin q) = reⁱ^q.

Multiplication & Division in Polar/Exponential Form

The exponential form allows us to derive simple expressions for computing the magnitude and angle of the product or ratio of two complex numbers given in polar form. Given z1 = r1eⁱ^q1 and z2 = r2eⁱ^q2, then z1z2 = r1r2 r1eⁱ⁽^{q1 +}^q2) = r1r2[cos (q1 + q2) + i sin (q1+ q2)] (and z1/z2 = (r1/r2) eⁱ⁽^q1
-^q2) = (r1/r2)[cos (q1 - q2) + i sin (q1- q2)]. One can also easily show that zⁿ = (reⁱ^q)ⁿ = rⁿ e^{i n} = rⁿ[cos nq + i sin nq]. Here are some examples:

exp(i*pi) % should give us cos(p) + i sin(p) = -1

-1.0000 + 0.0000i

1.0000 - 0.0000i

0.0000 + 1.0000i

z1=2*exp(i*pi/4)

1.4142 + 1.4142i

compare to

2*(cos(pi/4) + i*sin(pi/4))

1.4142 + 1.4142i

We can retain the symbolic (exponential) representation of z1 by using the command

z1='2*exp(i*pi/4)'

2*exp(i*pi/4)

z2='-exp(i*pi/3)'

-exp(i*pi/3)

z1z2 = symmul(z1,z2);

numeric(z1z2) % symbolic multiplication

0.5176 - 1.9319i

2.0000

-1.3090

Verification: z1*z2 = (2)*(-1)*exp[i(p/4-p/3)] = -2*exp(i7p/12) = 2*exp(ip)*exp(i7p/12) = 2*exp(i19p/12) = 2*exp[i(2p - 5p/12)] = 2*exp(i2p)*exp[i(-5p/12)] = 2*exp[i(-1.3090p), which represents the complex number with magnitude 2 and angle -1.3090 radians (note that we have used the fact that exp(i2p) = 1).

Roots

The nth root (n =1, 2, 3, ... ) of a complex number z^1/n = (reⁱ^q)^1/n has precisely n solutions. The formula for obtaining such roots is given by r^1/n {cos [(q + 2kp)/n] + i sin [(q + 2kp)/n]}, for k = 0, 1, 2, ..., n-1. Let us compute the 4th root of unity (should get 1, i, -1, -I):

r=1; % magnitude of z

a=0; % angle of z

n=4;

for k=0:n-1

r^(1/n)*(cos((a+2*k*pi)/n) + i*sin((a+2*k*pi)/n))

end

1

ans =

0.0000 + 1.0000i

ans =

-1.0000 + 0.0000i

ans =

0.0000 - 1.0000i

Roots of a complex coefficient polynomial

The solve command which solves for the roots of a symbolic polynomial also work for situations where some or all of the polynomial coefficients are complex valued. Here is an example:

z_roots = solve('z^2 - (5+i)*z + 8 + i = 0')

[5/2+1/2*i+1/2*2^(1/2)*(-4+3*i)^(1/2)]

[5/2+1/2*i-1/2*2^(1/2)*(-4+3*i)^(1/2)]

numeric(z_roots)

3.0000 + 2.0000i

2.0000 - 1.0000i

Answer verification:

expand('(x-(3+2*i))*(x-(2-i))')

x^2-5*x-i*x+8+i

V. Trigonometric Functions, Hyperbolic Functions

Euler's formula can be used to extend trigonometric functions for the case of complex arguments. By addition and subtraction of the two equations exp(ix) = cos x + i sin x and exp(-ix) = cos x - i sin x we obtain for the real cos and sin:

cos x = (1/2)[exp(ix) + exp(-ix)] and sin x = (1/2i)[exp(ix) - exp(-ix)]

This may be generalized to complex values z = x + iy:

cos z = (1/2)[exp(iz) + exp(-iz)] and sin z = (1/2i)[exp(iz) - exp(-iz)]

MatLab allows complex arguments for its sin & cos functions:

-4.1896 - 9.1092i

9.1545 - 4.1689i

0 + 1.1752i

MatLab also allows complex arguments for other built-in functions:

-3.7245 - 0.5118i

1.9596 + 3.1658i

log(1*exp(i*pi/2)) % ln(r*exp(iq)) = ln(r) + ln(exp(iq)) = ln(r) + iq

0 + 1.5708i

0 + 1.5708i

0.3466 + 0.7854i

0 + 3.1416i