1 Complex numbers and the complex plane

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1 L1: Complex numbers and complex-valued functions. Contents: The field of complex numbers. Real and imaginary part. Conjugation and modulus or absolute valued. Inequalities: The triangular and the Cauchy. Polar representation: the argument. The exponential notation and the De Moivre theorem. Circles and lines by using complex numbers. Complex valued functions and its differential. 1 Complex numbers and the complex plane It is completely obvious that the equation x = 0 cannot be solved using the real numbers. Mathematicians have therefore invented the imaginary number i 1. In other words one declares i 2 = 1, and naturally also ( i) 2 = 1. Once i is defined as a number, we must make sure we can compute 2 i, 4 i, 1, i i2 5, 4 i +2 i 3 5 and the like. So mathematicians have invented the number i, but this in turn immediately generates many other numbers, namely those that can be written as a 0 +a 1 i +a 2 i 2 + +a n i n b 0 +b 1 i +b 2 i 2 + +b m i m with a 1,, a n, b 1,, b m real. Once i is born, a whole set C, called set of complex numbers a 0+a 1 i +a 2 i 2 + +a n i n b 0 +b 1 i +b 2 i 2 + +b m i m with a 1,, a n, b 1,, b m comes along. Notice R is a subset of C, for if a R then a = a i 2 C, and R C. An important theorem is the following. Theorem 1.1. If z C, i.e. z = a 0+a 1 i +a 2 i 2 + +a n i n b 0 +b 1 i +b 2 i 2 + +b m i m, a 1,, a n, b 1,, b m, then z can be written uniquely as: z = x + i y with x, y R. The real number x is called the real part of z, Re(z) := x, while the real number y is the imaginary part and we write Im(z), so that z = Re(z) + i Im(z). 1 Sometimes the symbol 1 is used to indicate i, but be careful because ( 1)( 1) is not the same as 1 1. Mathematical Methods, L1 1 Mathematical Methods

2 proof. From i 2 = 1 it follows that z = a 0+a 1 i +a 2 i 2 + +a n i n b 0 +b 1 i +b 2 i 2 + +b m i m R. Besides, z = ( A+B i C+D i x = AC+BD and y = BC AD C 2 +D 2 C 2 +D 2 z = x + y i = x + y i and y y, we would have = A+B i, where A, B, C, D C+D i C 2 +D 2. Therefore z = x + y i, with C D i )( ) = (AC+BD)+(BC AD) i C D i. Uniqueness is easy to prove. If there were z such that i = x x y y which implies i is real! Contradiction. Hence y = y and necessarily x = x. The inverse 1 z and the conjugate z Let z denote a complex number z = x + i y. The previous proof provides us with the idea of how to find the inverse 1 of a complex number. It is convenient to define first the z conjugate z = x y i of z as the complex number whose imaginary part is opposite to the imaginary part of z. Geometrically the conjugate is the symmetric point to z with respect to the x-axis. Two important properties are: Proposition 1.2. Let z, w be complex numbers. Then: zw = zw z + w = z + w Using the first property repeatedly we have z n = z n. Observe z = z iff Im(z) = 0, i.e. iff z is real. Now the important remark: z.z = x 2 + y 2 The product of a number by its conjugate equals the distance squared of the point from the origin, i.e.the square of the modulus of the vector z. Since the modulus is z, z.z = z 2. From this the inverse is easy. Mathematical Methods, L1 2 Mathematical Methods

3 Proposition 1.3. The inverse 1 z of z 0 is 1 z = z z 2 = x x 2 + y 2 y x 2 + y 2 i Proof. From the product z z z 2 = zz z 2 = z 2 z 2 = 1 we see that 1 z = z z 2, by definition. Example 1.4. The inverse of i is 1 i = i The inverse 1, together with the sum and the product allows to see C as a number z field 2. Two complex numbers z = x + y i, w = a + b i are multiplied as follows: z.w = (x + y i)(a + b i) = xa + xb i +ya i +yb i 2 = xa + (xb + ya) i +yb( 1) so z.w = (xa yb) + (xb + ya) i. A famous formula: z 2 w 2 = (x 2 + y 2 )(a 2 + b 2 ) = (xa yb) 2 + (xb + ya) 2 = zw 2. This was Euler s starting point for proving Fermat s theorem 3, which states that the natural numbers of the form 4k + 1 are the sum of two squares. Notice at last that the real and imaginary parts can be found using conjugates: Re(z) = z + z 2 Im(z) = z z 2 i The number z is said purely imaginary if Re(z) = 0. Thus z is purely imaginary iff z = y i, y R. The condition z = z is necessary and sufficient for z to be purely imaginary. 2 First noticed by the mathematician Raffaele Bombelli from Bologna, in di Fermat sulle somme di due quadrati Mathematical Methods, L1 3 Mathematical Methods

4 2 Inequalities Here are two important inequalities: (triangle) z + w z + w (1) and (Cauchy s) a 1 b a n b n 2 ( a a n 2 )( b b n 2 ) (2) Mathematical Methods, L1 4 Mathematical Methods

5 3 The exponential, polar coordinates z = ρe i θ := ρ (cos(θ) + i sin(θ)) where ρ = z is called modulus or norm. The angle θ, i.e. a real number determined up to integer multiples of 2π, is called argument or amplitude and denoted by arg(z) by using the trigonometric identities we have e i θ e i ψ = e i θ+ψ and so arg(zw) = arg(z) + arg(w) arg( z ) = arg(z) arg(w) w we have also the so called de Moivre s formula (cos(θ) + i sin(θ)) n = (cos(nθ) + i sin(nθ)) which is useful in order to obtain the n solutions of z n = a namely z = n r (cos( ψn + k 2πn ) + i sin(ψn + k 2πn ) ) k = 0, 1,, n 1 where a = re i ψ. Mathematical Methods, L1 5 Mathematical Methods

6 4 Circles and lines by using complex numbers Here is the equation of the circle of radius r centered at z 0 : an straight line can be written as where r is a real number. Here is an example: can be written as: with a = 1 2 i. z z 0 2 = r 2 az + az = r x + 2y = 1 az + az = 1 Mathematical Methods, L1 6 Mathematical Methods

7 5 Complex valued functions and its differential A complex valued function is a function from a certain set S to the complex numbers. Namely, f : S C. So if p S then f(p) = u(p) + i v(p) where u(p), v(p) R. A complex valued function can be also regarded as function to R 2. If S R n is an open subset then the function f : S C is called smooth if all partial derivatives of u and v do exists. We are mainly interested in complex functions. Namely, when S C. In this case we will regard f as a function of (x, y): f(x, y) = (u(x, y), v(x, y)) Recall that if f has partial derivatives we can form the so called Jacobian matrix J f : J f = When the Jacobian matrix J f ( u x v x differential df of the function f. Namely, u y v y ) ( ) ux u = y v x v y is multiplied by the column ( ) dx df = J f = (u dy x dx + u y dy, v x dx + v y dy) By using complex numbers we write: df = (u x dx + u y dy) + i (v x dx + v y dy) ( ) dx dy we get the the Mathematical Methods, L1 7 Mathematical Methods