MTH 362: Advanced Engineering Mathematics Lecture 1 Jonathan A. Chávez Casillas 1 1 University of Rhode Island Department of Mathematics September 7, 2017
Course Name and number: MTH 362: Advanced Engineering Mathematics Instructor: Jonathan } {{ } First Name Allan } {{ } Middle Name Chávez } {{ Casillas } Last Name Office Hours: Wednesday 11 am - 2pm (or by appointment) Office Room: Lippitt Hall 200A E-mail: jchavezc@uri.edu
Complex Numbers Jonathan Cha vez
Complex Numbers Jonathan Cha vez
1 Complex Numbers and the Complex Plane 2 of Complex Numbers. Powers and Roots
The Complex Numbers Since high school we have been taught to solve equations such as: Exercise: Solve: x 2 + 2x = 1 x 3 + 6x 2 40x = 0 The solution to such equations belong to the real numbers.
The Complex Numbers However, if we modify slightly the equations, the solutions are not real anymore: Exercise: Try to solve: x 2 = 1 x 3 10x 2 40x = 0 The solutions of the previous equations belong to a set of numbers called the Complex Numbers
The Complex Numbers By definition, a complex number has two parts: A real part and an imaginary part. Because of this duo, there are two basic ways of writing complex numbers: Complex numbers notation: A complex number z consists of an ORDERED pair of real numbers x and y. That is, z = (x, y) is thought as a complex number and x is called the real part and y is called the imaginary part. In notation, x = Re z and y = Im z. The imaginary unit, (0, 1), is denoted by i. Thus, a more common notation is: z=(x,y)=x+iy It is important to notice that two complex numbers z 1 = (x 1, y 1) = x 1 + iy 1 and z 2 = (x 2, y 2) = x 2 + iy 2 are equal ONLY if x 1 = x 2 AND y 1 = y 2.
Arithmetic of Complex Numbers Addition: The addition of two complex numbers z 1 = (x 1, y 1) = x 1 + iy 1 and z 2 = (x 2, y 2) = x 2 + iy 2 is defined as: z 1 + z 2 := (x 1 + x 2, y 1 + y 2) = x 1 + x 2 + i(y 1 + y 2) Multiplication: The product of two complex numbers z 1 = (x 1, y 1) = x 1 + iy 1 and z 2 = (x 2, y 2) = x 2 + iy 2 is defined as: z 1z 2 := (x 1x 2 y 1y 2, x 1y 2 + x 2y 1) = x 1x 2 y 1y 2 + i(x 1y 2 + x 2y 1) Particular Example: A very important example and a shortcut to multiply complex numbers is that if z 1 = (0, 1) = i and z 2 = (0, 1) = i, then z 1z 2 = i i = i 2 = ( 1, 0) = 1. That is i 2 = 1.
Arithmetic of Complex Numbers Subtraction: The subtraction of two complex numbers z 1 = (x 1, y 1) = x 1 + iy 1 and z 2 = (x 2, y 2) = x 2 + iy 2 is defined as: z 1 z 2 := (x 1 x 2, y 1 y 2) = x 1 x 2 + i(y 1 y 2) Quotient: The quotient of two complex numbers z 1 = (x 1, y 1) = x 1 + iy 1 and z 2 = (x 2, y 2) = x 2 + iy 2 is defined as: ( ) z 1 x1x 2 + y 1y 2 x2y1 x1y2 x1x2 + y1y2 x2y1 x1y2 := z 2 x2 2 + y 2 2, x2 2 + y 2 2 = x2 2 + y 2 2 + i x2 2 + y 2 2 An easy way to remember the quotient is to multiply and divide by the conjugate (see below) of the divisor.
The Complex Plane As the notation z = (x, y) suggests, we can identify the complex number z with the pair (x, y) and thus, we can plot it in the plane as we would for a point in the two dimensional Cartesian plane. This plane composed of two coordinates axis (the horizontal, or x axis, represents the real part and the vertical or y axis represents the imaginary part) is called the complex plane. Exercise: For the complex numbers z 1 = (3, 1) = 3 + i and z 2 = (2, 4) = 2 + 4i, plot z 1, z 2, z 1 + z 2, z 1 z 2. How does the addition and substraction relates to the one of vectors in R 2?
The Complex Conjugate The complex conjugate of a complex number z = (x, y) = x + iy is defined as z = (x, y) = x iy. It is obtained geometrically by just reflecting about the real (x-)axis. Exercise: Show that for any two complex numbers z 1 = (x 1, y 1) and z 2 = (x 2, y 2): z 1 z 1 = x 2 1 + y 2 1. Re z 1 = x 1 = 1 2 (z1 + z1). Im z 1 = y 1 = 1 2 (z z1). (z1 + z2) = z1 + z2 and z1z2 = z1 z2.
Exercise: Let z 1 = 4 + 3i and z 2 = 1 i. Find: (2z 1 + 3z 2) 2. Re(1/ z 1). Im (1 + i)z 2. z 2/(z 1 3).
Table of Contents 1 Complex Numbers and the Complex Plane 2 of Complex Numbers. Powers and Roots
of Complex Numbers As in two dimensions, when plotting a complex number, we may look at its polar form. That is, the angle it makes with the real axis and the distance from the origin the point has. Remember that for a point (x, y) in the cartesian plane, its polar coordinates are: x = r cos θ y = r sin θ Thus, it is natural to assume that the polar representation of the complex number z = x + iy is given by z = r(cos θ + i sin θ). r = z = x 2 + y 2 is called the modulus of the complex number and θ = arg(z) = arctan y x is the argument of the complex number z. As in calculus, all angles are measured in radians and a positive angle traverse in counterclockwise sense.
of Complex Numbers Remember that you may write the same angle in a lot of different ways. For example, π/3 is the same angle as π/3 + 6π, but all angles are unique up to a multiple of 2π, which is the periodicity of the trigonometric functions sine and cosine. Thus, a Principal Argument, denoted as Argz, is defined for complex numbers. The principal argument satisfy the inequality π < Argz π Please notice that the equal sign is on the right side of the inequality! Then, the way to think of the Principal argument is as follows:
of Complex Numbers Exercise: Obtain the polar form of z 1 = 2 2i and z 2 = 1 3i. Find their modulus and all possible arguments. Find also their principal argument.
Multiplication and Division in So far, we understand the geometric meaning of adding and subtracting two complex numbers. But, what does it mean geometrically to multiply and divide two complex numbers? Home Exercise: Show that if z 1 = r 1(cos θ 1 + i sin θ 1) and z 2 = r 2(cos θ 2 + i sin θ 2), then z 1 z 1z 2 = r 1r 2 [cos(θ 1 + θ 2) + i sin(θ 1 + θ 2)] = r1 [cos(θ 1 θ 2) + i sin(θ 1 θ 2)] z 2 r 2 Exercise: Using the formulas above show that z 1z 2 = z 1 z 2 and z 1 z 2 = z 1 z 2 arg(z 1z 2) = arg(z 1) + arg(z 2) and arg ( z 1 z 2 ) = arg(z1) arg(z 2) (up to multiples of 2π)
Integer Powers of z and De Moivre s Formula Using the product formula with z 1 = z 2 = z = r(cos θ + i sin θ) we obtain that z 2 = r 2 [cos(2θ) + i sin(2θ)]. We can continue by induction to show that for n a positive integer, Home Exercise: Show that 1 Equation (1) holds also for negative integers n. 2 (De Moivre s Formula) (cos θ + i sin θ) n = cos(nθ) + i sin(nθ). z n = r n (cos(nθ) + i sin(nθ)) (1) 3 Using De Moivre s formula with n=2, expand the binomial in the left and equate the real an imaginary parts from both sides of the formula. Do you recognize those formulas? Hint: For (a) use Equation (1) and the quotient formula with one complex number being 1. For (b) use again Equation (1) for a complex number with modulus 1.
Roots of Complex numbers If P(z) is a polynomial of degree n (i.e., P(z) = a nz n + a n 1z n 1 +... + a 1z + a 0), then the equation P(z) = 0 has ALWAYS n complex solutions (Is this true if we restrict z to be just real?) Then, it should follow that the equation z n = w, where z and w are complex numbers, should have also n roots. Each of those roots is represented as z = n w Thus, the expression above has not just value but n. The natural question is to find out which are those n values.
Roots of Complex numbers Exercise: Use the polar form of a complex number and De Moivre formula to find the n roots of the equation z n = w. In particular, find the n roots of the unity, i.e., what are the roots of the equation z n = 1.
Roots of Complex numbers Exercise: As seen above, n 1 = cos ( 2kπ n ) + i sin ( ) 2kπ, k = 0, 1, 2,..., n 1. n These n values are called the nth roots of unity. They lie on the circle of radius 1 and center 0, called the unit circle, and constitute the vertices of a regular polygon of n sides (inscribed in the circle of radius 1). The next figure depicts the roots of unity when n = 3, 4, 5.