Chapter 1. Complex Numbers. Dr. Pulak Sahoo
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1 Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistant Professor Department of Mathematics University Of Kalyani West Bengal, India sahoopulak1@gmail.com 1
2 Module-1: Basic Ideas 1 Introduction There is no real number x which satisfies the equation x 2 +1 = 0. To permit solutions of this and similar kind of equations, the set of complex numbers was introduced. The term complex number was introduced by C.F. Gauss, a German mathematician. Later on, the subject was enriched by the original work of A.L. Cauchy, B. Riemann, K. Weierstrass and many others. Let us denote by R the set of all real numbers. Consider R 2, the set of all ordered pairs of real numbers (x, y) defined by R 2 = {(x, y) : x R, y R}. Here ordered means (x, y) and (y, x) are distinct unless x = y. A complex number is defined as an ordered pair (x, y) of real numbers x and y satisfying the following rules : (i) (x, y) + (z, w) = (x + z, y + w), (ii) (x, y).(z, w) = (xz yw, xw + yz), (iii) (x, y) = (z, w) if and only if x = z and y = w, (iv) m(x, y) = (mx, my), where m is any scalar number. We denote by i the ordered pair (0, 1) and the real number x by the ordered pair (x, 0). From the above definition we see that (x, y) = (x, 0) + (0, y) = x(1, 0) + y(0, 1) = x + iy. 2
3 Also we see that i 2 = (0, 1).(0, 1) = ( 1, 0) = 1. Fundamental Operations of Complex Numbers For any two complex numbers z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 we define the operations of addition, subtraction, multiplication and division as follows: (i) (x 1 + iy 1 ) + (x 2 + iy 2 ) = (x 1 + x 2 ) + i(y 1 + y 2 ). (ii) (x 1 + iy 1 ) (x 2 + iy 2 ) = (x 1 x 2 ) + i(y 1 y 2 ). (iii) (x 1 + iy 1 ).(x 2 + iy 2 ) = (x 1 x 2 y 1 y 2 ) + i(x 1 y 2 + x 2 y 1 ). (iv) x 1+iy 1 x 2 +iy 2 = (x 1+iy 1 )(x 2 iy 2 ) = x 1x 2 +y 1 y 2 (x 2 +iy 2 )(x 2 iy 2 + i x 2y 1 x 1 y 2, provided z ) x 2 2 +y2 2 x y2 2 From the above definition, it is obvious that the set of all complex numbers forms a field with respect to addition and multiplication. 2 Complex Plane or Argand Plane Every complex number can be represented geometrically as a point in xy-plane. Now, we shall call this plane as complex plane or argand plane. Introducing a rectangular coordinate system in the plane, we can identify the complex number z = x + iy with the point P = (x, y). Obviously, the set of all real numbers (x, 0) corresponds to the x-axis, called the real axis, and the set of all purely imaginary numbers (0, y) corresponds to the y-axis, called the imaginary axis, while the set of all imaginary numbers corresponds to all points which do not lie on the real axis. The origin identifies the complex number 0 = 0 + i0. The complex plane is generally denoted by C and is often called the z-plane. The nonnegative number z, called modulus or absolute value of z = (x, y), is the distance of the complex number z from the origin, and hence z = x 2 + y 2 (see Fig. 1.1). The distance between two points z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 in the complex plane is given by z 1 z 2 = (x 1 x 2 ) 2 + (y 1 y 2 ) 2. We shall frequently use the following simple inequalities. x x x 2 + y 2 = Re(z) Re(z) z, y y x 2 + y 2 = Im(z) Im(z) z. 3
4 Fig. 1.1: Conjugate If z = x+iy is any complex number, then its complex conjugate denoted by z is z = x iy or z = (x, y). Clearly, z is the mirror image of the complex point z into real axis. It is obvious that z = z if and only if z is purely a real number. Also z = z. Following are the easy consequences of the above definition : (a) z + w = z + w; (b) z w = z w; (c) zw = z w; (d) z + z = 2Re(z); (e) z z = 2iIm(z); (f) zz is real and positive unless z = 0; (g) ( z ) = z ; w w (h) z = z. Polar Form of a Complex Number Let r and θ denote the polar coordinates of a point z = (x, y). For z 0, let x = r cos θ, y = r sin θ. Then z can be written as z = r(cos θ + i sin θ), which is called the polar form of a complex number. r = z = x 2 + y 2, a unique positive real number, is called the modulus or absolute value of z and θ = tan 1 (y/x), called the amplitude or argument of the complex number z and is denoted by arg z (see Fig. 1.2). Given a nonzero complex number z, its argument does not have a unique value; there are infinite values differing by multiples of 2π. For many purposes, it is useful to restrict the allowed range for the 4
5 Fig. 1.2: polar angle θ so as to get a unique value of θ for a given z. If the value of θ = arg z is restricted to lie in the range π < θ π, we obtain the principal value of arg z, which will be denoted as Arg z, i.e. Arg z = θ, π < θ π. It must be noted that arg z is not defined for z = 0. Now we summarize a few properties of the modulus and argument. Using r 1, r 2 and r 3 to denote the moduli and θ 1, θ 2, and θ 3 to denote the arguments of the complex numbers z 1, z 2 and z 3 respectively, we have : (i) If z 3 = z 1 z 2 then r 3 = r 1 r 2 and θ 3 = θ 1 + θ 2. (ii) If z 3 = z 1 z 2 then r 3 = r 1 r 2 and θ 3 = θ 1 θ 2, r 2 0. (iii) If z 2 = 1 z 1 then r 2 = 1 r 1 and θ 2 = θ 1, r 2 0. (iv) If z 2 = z n 1 then r 2 = r n 1 and θ 2 = nθ 1. Roots of a Complex Number A number w is called an n-th root of a complex number z if w n w = z 1/n. If n is a positive integer, then = z, and we write z 1/n = {r(cos θ + i sin θ)} 1/n ( ) ( )] 2kπ + θ 2kπ + θ = r [cos 1/n + i sin, n n where k = 0, 1, 2,..., n 1. From this it follows that there are n distinct values of z 1/n, provided z 0. 5
6 Example 1.1. For any two complex numbers z 1 and z 2 prove that z 1 + z 2 z 1 + z 2. Solution. We have z 1 + z 2 2 = (z 1 + z 2 )(z 1 + z 2 ) = (z 1 + z 2 )(z 1 + z 2 ) = z 1 z 1 + z 1 z 2 + z 2 z 1 + z 2 z 2 = z Re(z 1 z 2 )+ z 2 2 z z 1 z 2 + z 2 2 = z z 1 z 2 + z 2 2 = ( z 1 + z 2 ) 2. Taking the nonnegative square root of both sides we get z 1 + z 2 z 1 + z 2. Note 1.1. The above inequality is known as the triangle inequality. The geometrical meaning of the triangle inequality is that the length of one side of a triangle is less than or equal to the sum of the lengths of the other two sides. Note 1.2. The triangle inequality can be generalized immediately to the case of n complex numbers, using mathematical induction, which can be written as z 1 + z z n z 1 + z z n. Example 1.2. For any two complex numbers z 1 and z 2 prove that (a) z 1 z 2 z 1 z 2 z 1 + z 2 ; (b) z 1 z 2 z 1 + z 2. Solution. (a) For the second part of the inequality in (a), we have z 1 z 2 z 1 + z 2 = z 1 + z 2. For the first part of the inequality in (a), we have z 1 = z 1 z 2 + z 2 z 1 z 2 + z 2 6
7 This gives z 1 z 2 z 1 z 2. (1.1) Interchanging the role of z 1 and z 2 in (1.1), we obtain z 2 z 1 z 2 z 1 = z 1 z 2. (1.2) Combining (1.1) and (1.2), the result follows. (b) Replacing z 2 by z 2 in (a), we immediately obtain (b). Inequality (b) implies that the length of one side of a triangle is greater than or equal to the difference of the other two sides. Example 1.3. If the sum and product of two complex numbers are both real then show that two numbers must be either real or conjugate. Solution. Let z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2. Then z 1 + z 2 = (x 1 + x 2 ) + i(y 1 + y 2 ) and z 1 z 2 = (x 1 x 2 y 1 y 2 ) + i(x 1 y 2 + x 2 y 1 ). By the hypothesis we have y 1 + y 2 = 0 and x 1 y 2 + x 2 y 1 = 0. This gives y 2 = y 1 and x 1 = x 2 provided y 1 0. Hence the result follows. Example 1.4. If f(z) = a 0 z n + a 1 z n a n is a polynomial in a complex variable z with real coefficients a 0, a 1,..., a n then prove that f(z) = f(z). Solution. Since a 0, a 1,..., a n are all real, we have f(z) = a 0 z n + a 1 z n a n = a 0 z n + a 1 z n a n = a 0 z n + a 1 z n a n = a 0 (z) n + a 1 (z) n a n = f(z). 7
8 Example 1.5. Find the upper bound of the function z 2 +1 z 2 +z 6 Solution. We have for z = 1,, when z = 1. z z 2 + z 6 = z z 2 + z 6 z = (z + 3)(z 2) z 2 +1 ( z 3 )( z 2 ) = = 1. Therefore, the upper bound of the given function is 1. 8
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