Complex Numbers

Complex NumbersPreben AlsholmSeptember 4, 20081 Complex Numbers1.1 Sets of numbersSets of numbers N is the set of natural numbers, 1, 2, 3, 4, 5, . . . Z is the set of integers . . . 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, . . .. Q is the set of rational numbers, i.e. fractions and integers. R is the set of real numbers. Can be identified with the set of points ona straight line, the real number line. Irrational numbers are real numbersthat are not rational. C is the set of complex numbers. Can be identified with the set of pointsin the plane, the complex plane. Complex numbers that are not real arecalled imaginary. We have N Z Q R C.1.2 Rules for addition and multiplicationRules for addition and multiplication1. a + b = b + a (the commutative law of addition)2. (a + b) + c = a + (b + c) (the associative law of addition)3. ab = ba (the commutative law of multiplication)4. (ab) c = a (bc) (the associative law of multiplication)5. a (b + c) = ab + ac (the distributive law)6. a + 0 = a7. 1a = a8. a + x = 0 has precisely one solution for x9. ax = 1 has precisely one solution for x, provided a 6= 010. Every Cauchy sequence has a limit1

1.3 What is a Cauchy sequence?What is a Cauchy sequence? A Cauchy sequence is a sequence of numbers (x n ) ∞ n=1 = x 1, x 2 , x 3 , . . . , x n , . . .satisfying x n x m ! 0 for n, m ! ∞. That every Cauchy sequence has a limit means that x n x m ! 0 forn, m ! ∞ implies that there exists a number x, such that x n ! x forn ! ∞. The claim: Every Cauchy sequence has a limit is valid for R and C, not forQ. C has the very important property that every polynomial of degree 1 hasat least one root. (The Fundamental Theorem of Algebra).2 Description of the complex numbersDescription of the complex numbers As a set C equals the set of points in the plane. The plane is identifiedwith R 2 , thus C = R 2 . The point (a 1 , a 2 ) is written a 1 + a 2 i. Thus i is the point (0, 1) and 1 is thepoint (1, 0). Definition of addition. If a = a 1 + a 2 i and b = b 1 + b 2 i then a + b =(a 1 + b 1 ) + (a 2 + b 2 ) i. Definition of multiplication. If a = a 1 + a 2 i and b = b 1 + b 2 i thenab = (a 1 + a 2 i) (b 1 + b 2 i) = (a 1 b 1a 2 b 2 ) + (a 1 b 2 + a 2 b 1 ) i It follows that i 2 = 1.2.1 Division?Division? The solution to the equation az = 1 exists if a 6= 0 and is uniquely determined.It is denoted a 1 or 1 a . By b a we mean ba 1 . It is the solution to the equation az = b The usual method of calculating b a :2 + 3i4 + 7i==(2 + 3i) ( 4 7i) (2 + 3i) ( 4 7i)=( 4 + 7i) ( 4 7i) ( 4) 2 (7i) 2(2 + 3i) ( 4 7i)16 + 49=13 26i65= 1 525 i2

2.2 Real and imaginary parts etc.Real and imaginary parts etc. Real part: Re (a 1 + ia 2 ) = a 1 . Imaginary part: Im (a 1 + ia 2 ) = a 2 Complex conjugate: a = a 1 + ia 2 = a 1 ia 2 a + b = a + b and (ab) = ab Modulus, absolute value: jaj = ja 1 + ia 2 j = jabj = jaj jbj The triangle inequality: ja + bj jaj + jbj2.3 Polar form IPolar form Iqa 2 1 + a2 2 Let r = jaj and v be an angle measured from the positive real axis tothe line connecting 0 and a (measured positive in the counterclockwisedirection). v is an argument for a. Notation: arg (a). The set of arguments for a isfv + p2π jp 2 Z g. Any complex number can be written in polar form: a = r (cos v + i sin v),where r is the modulus and v is an argument of a.2.4 Polar form IIPolar form II The principal value: Arg (a) is the uniquely given argument in the interval] π, π]. By arg τ(a) we mean the unique argument in the interval ]τ, τ + 2π], thusArg (a) = arg π(a). arg (ab) = arg a + arg b3

arg (a n ) = n arg a arg a b = arg a arg b These must be properly understood: Thus arg (ab) = arg a + arg b meansthat one of the arguments for ab is obtained by adding an argument for aand an argument for b.2.5 The Complex ExponentialThe Complex Exponential The real exponential function exp has the fundamental propertyi.e. e x+y = e x e y for all x, y 2 R.exp (x + y) = exp (x) exp (y) Definition. If z = x + iy (x, y 2 R) thenexp (z) = exp x (cos y + i sin y) ex+iy = e x and arg e x+iy = y when x, y 2 R. exp (z 1 + z 2 ) = exp z 1 exp z 2 for all z 1 , z 2 2 C, i.e. e z 1+z 2 = e z 1e z 2. Proof: Let z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 , thene x 1+iy 1 e x 2 +iy 2 = ex 1 e x 2je z 1 e z 2j = je z 1j je z 2j == e x 1+x 2= e x 1+x 2 +i(y 1 +y 2 ) = e z 1 +z 2 arg (e z 1 e z 2) =arg (e z 1) + arg (e z 2) = arge x 1+iy 1= y 1 + y 2 = arge x 1+x 2 +i(y 1 +y 2 )= arg e z 1+z 2+ arge x 2+iy 22.6 The polar form once moreThe polar form once more The polar form for the number a having modulus r and argument v waswrittena = r (cos v + i sin v)In the future we shall write:a = r exp (iv) = re ivrp p 2 Example. The polar form for 3 i. Modulus: 3 + ( 1) 2 =5π2. An argument is 6. Thusp3 i = 2 exp i 5π = 2e i 5π 664

2.7 De Moivre’s formulaDe Moivre’s formula For n 2 N og θ 2 R gælder(cos θ + i sin θ) n = cos nθ + i sin nθ Proof: (cos θ + i sin θ) n =e iθ n= e inθ = cos nθ + i sin nθ Example.cos 3x = Re (cos 3x + i sin 3x) = Re(cos x + i sin x) 3= Re cos 3 x + 3i cos 2 x sin x 3 cos x sin 2 x i sin 3 x = cos 3 x 3 cos x sin 2 x = cos 3 x 3 cos x 1 cos 2 x= 4 cos 3 x 3 cos x By replacing Re with Im above we get the formulasin 3x = 3 cos 2 x sin x sin 3 x = 3 1 sin 2 x sin xsin 3 x= 4 sin 3 x + 3 sin x5