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MM Vercelli. 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 x2 + 1 = 0 cannot be solved using the real numbers. Mathematicians have therefore invented the imaginary number ”i” 1 . In other words one declares i2 = −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, 1i , i2 5, 4 i +2 i3 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 a0 +a1 i +a2 i2 +···+an in with a1 , · · · , an , b1 , · · · , bm real. b0 +b1 i +b2 i2 +···+bm im 2 n +a1 i +a2 i +···+an i Once ”i” is born, a whole set C, called set of complex numbers ba00+b 2 m 1 i +b2 i +···+bm i with a1 , · · · , an , b1 , · · · , bm comes along. Notice R is a subset of C, for if a ∈ R then a = −a i2 ∈ C, and R ⊂ C. An important theorem is the following. Theorem 1.1. If z ∈ C, i.e. z = be written uniquely as: a0 +a1 i +a2 i2 +···+an in b0 +b1 i +b2 i2 +···+bm im , a1 , · · · , an , b1 , · · · , bm , then z can z = x + iy 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 same as −1 −1 . Mathematical Methods, L1 √ −1 is used to indicate i, but be careful because 1 p (−1)(−1) is not the Mathematical Methods MM Vercelli. a0 +a1 i +a2 i2 +···+an in A+B i , = C+D i b0 +b1 i +b2 i2 +···+bm im (AC+BD)+(BC−AD) i . Therefore z C 2 +D2 proof. From i2 = −1 it follows that z = where A, B, C, D ∈ A+B i C−D i Besides, z = ( C+D )( ) = i C−D i AC+BD BC−AD and y = C 2 +D2 . Uniqueness C 2 +D2 0 0 0 R. = x + y i, with x= is easy to prove. If there were z such that z = x + y i = x + y i and y 6= y , we would have i= x − x0 y0 − y which implies i is real! Contradiction. Hence y = y 0 and necessarily x = x0 . 2 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 z1 of a complex number. It is convenient to define first the 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 = x2 + 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 MM Vercelli. Proposition 1.3. The inverse 1 z of z 6= 0 is 1 z x y = 2 = 2 − 2 i 2 z |z| x +y x + y2 Proof . From the product z |z|z 2 = 2 zz |z|2 Example 1.4. The inverse of i is = |z|2 |z|2 = 1 we see that 1 z = z |z|2 , by definition. 1 = −i i The inverse z1 , together with the sum and the product allows to see C as a number 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 i2 = xa + (xb + ya) i +yb(−1) so z.w = (xa − yb) + (xb + ya) i. A famous formula: |z|2 |w|2 = (x2 + y 2 )(a2 + b2 ) = (xa − yb)2 + (xb + ya)2 = |zw|2 . This was Euler’s starting point for proving Fermat’s theorem3 , 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 2i 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 3 First noticed by the mathematician Raffaele Bombelli from Bologna, in 1572. http://it.wikipedia.org/wiki/Teorema di Fermat sulle somme di due quadrati Mathematical Methods, L1 3 Mathematical Methods MM Vercelli. 2 Inequalities Here are two important inequalities: (triangle) |z + w| ≤ |z| + |w| (1) and (Cauchy 0 s) Mathematical Methods, L1 |a1 b1 + · · · + an bn |2 ≤ (|a1 |2 + · · · + |an |2 )(|b1 |2 + · · · + |bn |2 ) 4 Mathematical Methods (2) MM Vercelli. 3 The exponential, polar coordinates z = ρei 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 ei θ ei ψ = ei θ+ψ and so arg(zw) = arg(z) + arg(w) z arg( ) = 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 zn = a namely z= √ n µ ¶ ψ 2π ψ 2π r cos( + k ) + i sin( + k ) n n n n k = 0, 1, · · · , n − 1 where a = rei ψ . Mathematical Methods, L1 5 Mathematical Methods MM Vercelli. 4 Circles and lines by using complex numbers Here is the equation of the circle of radius r centered at z0 : |z − z0 |2 = r2 an straight line can be written as az + az = r where r is a real number. Here is an example: x + 2y = 1 can be written as: az + az = 1 with a = 1 2 − i. Mathematical Methods, L1 6 Mathematical Methods MM Vercelli. 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 R2 . If S ⊂ Rn 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 Jf : µ ∂u Jf = ∂x ∂v ∂x ∂u ¶ ∂y ∂v ∂y µ = ux uy vx vy ¶ µ When the Jacobian matrix Jf is multiplied by the column dx dy ¶ we get the the differential df of the function f . Namely, µ df = Jf dx dy ¶ = (ux dx + uy dy, vx dx + vy dy) By using complex numbers we write: df = (ux dx + uy dy) + i (vx dx + vy dy) Mathematical Methods, L1 7 Mathematical Methods