z = a + ib (4.1) i 2 = 1. (4.2) <(z) = a, (4.3) =(z) = b. (4.4)

Transcription

1 Chapter 4 Complex Numbers 4.1 Definition of Complex Numbers A complex number is a number of the form where a and b are real numbers and i has the property that z a + ib (4.1) i 2 1. (4.2) a is called the real part of z and b is called the imaginary part of z. Weusethenotation <(z) a, (4.3) (z) b. (4.4) (We often use Re(z) and Im(z) instead because < and are hard to write!) Helpful note: Lots of people get stuck trying to understand what i 2 1 means. Don t. Sometimes mathematicians invent new rules, and then play with them to find out what the consequences are. Sometimes, but not always, this gives new mathematical structures that eventually turn out to be useful to science (for example, complex numbers turned out to be indispensable to quantum mechanics). But to begin with, it s best just to apply the rule without worrying about what, if anything, it means. 4.2 The Argand Diagram We often represent the complex number z by a line from the origin to the point (a, b). Such a diagram is called an Argand diagram, illustrated in Figure The Rules for Addition and Subtraction The rules for addition and subtraction of complex numbers are very simple. For complex numbers a + ib and c + id (a + ib)+(c + id) (a + c)+i(b + d) (4.5) and (a + ib) (c + id) (a c)+i(b d). (4.6) 31

2 Figure 4.1: The Argand diagram showing the complex number z a + ib. For example, (2 + 3i)+(4+5i) (2 + 4) + (3 + 5)i 6+8i and (6 i) (1 + 2i) (6 1) + ( 1 2)i 5 3i. Examples 4, Q1 4.4 Multiplication of Complex Numbers Multiplication proceeds as follows: (a + ib) (c + id) ac + iad + ibc + i 2 bd ac + iad + ibc bd because i 2 1 (ac bd)+i(ad + bc). (4.7) In practice, we simply multiply out each individual case rather than remembering this formula. For example (2 + 3i) (4 7i) i +3 4i 3 7i i +12i i. Examples 4, Q2 4.5 The Complex Conjugate The complex number z a ib is called the complex conjugate of z a + ib. So, for example, the complex conjugate of 4+2i is 4 2i. Notethat z z (a + ib) (a ib) a 2 iab + iab i 2 b 2 a 2 + b 2. (4.8) So the result of multiplying a complex number by its complex congjugate is always real. 32

3 Examples 4, Q3 4.6 The Modulus of a Complex Number The modulus of the complex number z a + ib is written as z and is defined by Note that (i) z is real; (ii) z 2 z z. z a 2 + b 2. (4.9) (iii) By Pythagoras, z is the length of the line on the Argand diagram, in Figure 4.1. Examples 4, Q4 4.7 Division of Complex Numbers To work out z 1 z 2, we multiply top and bottom by the complex conjugate of z 2, as follows: z 1 z 2 z 1 z 2 z 2 z 2 z 1 z 2 z 2 2. (4.10) By rearranging z 1 /z 2 in this way, it is easy to find the real and imaginary parts. For example 2+i 3 2i (2 + i)(3 + 2i) (3 2i)(3 + 2i) 6+4i +3i 2 9+6i 6i i i. Examples 4, Q5 4.8 Polar Form of Complex Numbers If we refer back to the Argand diagram Fig. 4.1, we can identify an alternative way of specifying the complex number z a + ib. Instead, we can specify the distance r of the point (a, b) from the origin, and the angle θ which z makes with the Re(z) (x) axis,as shown in Fig

4 Figure 4.2: The Argand diagram showing the complex number z a + ib in alternative polar (r, θ) form. It is clear from the figure that r 2 a 2 + b 2 and therefore r a 2 + b 2 (4.11) and tan(θ) b a and therefore θ tan 1 b a (4.12) In this notation, r is called the modulus of the complex number and θ is called the argument. Notethatifwereplacedθ by θ +2π, we would get exactly the same complex number. More generally, we could replace θ by θ ± 2nπ where n is a positive integer, and the complex number would still be unchanged. This means that the argument of a complex number is not unique. Again referring to Fig. 4.2, note that a r cos(θ) (4.13) and b r sin(θ) (4.14) It follows that an alternative way of writing the complex number z a + ib is z r(cos(θ)+i sin(θ)). (4.15) Examples 4, Q6 34

5 4.9 Multiplication and Division in Polar Form Consider z 1 a 1 + ib 1 r 1 (cos(θ 1 )+i sin(θ 1 )) and z 2 a 2 + ib 2 r 2 (cos(θ 2 )+i sin(θ 2 )). Then z 1 z 2 r 1 r 2 (cos(θ 1 )+isin(θ 1 ))(cos(θ 2 )+isin(θ 2 )) r 1 r 2 [(cos(θ 1 )cos(θ 2 ) sin(θ 1 )sin(θ 2 )) + i(cos(θ 1 )sin(θ 2 )+cos(θ 2 )sin(θ 1 ))] r 1 r 2 [cos(θ 1 + θ 2 )+isin(θ 1 + θ 2 )]. (4.16) using Eq. (1.25) and (1.27). So the modulus of z 1 z 2 is r 1 r 2 and the argument of z 1 z 2 is θ 1 + θ 2. So, when two complex numbers are multiplied together: 1. ModulusisMultiplied 2. Argument is Added It can similarly be shown that z 1 z 2 µ r1 r 2 [cos(θ 1 θ 2 )+i sin(θ 1 θ 2 )]. (4.17) 4.10 Exponential Form of Complex Numbers It can be shown that: e iθ cos(θ)+i sin(θ). (4.18) Without really proving this, here are a few good reasons why this works: 1. It works at θ 0: e i 0 e 0 1 cos(0) + i sin(0) 1 2. It obeys the same rules for multiplication (i.e. add the arguments) e iθ 1 e iθ 2 e iθ 1+iθ 2 e i(θ 1+θ 2 ) (cos(θ 1 )+i sin(θ 1 )) (cos(θ 2 )+i sin(θ 2 )) cos(θ 1 + θ 2 )+i sin(θ 1 + θ 2 ) 3. It works for derivatives. If z e iθ then but if z cos(θ)+i sin(θ) then dz dθ dz dθ ieiθ iz sin (θ)+i cos(θ) iz. 35

6 4. You can work it out from the Taylor series: e iθ 1+iθ + (iθ)2 2! 1+iθ θ2 1 θ2 2! + θ4 + (iθ)3 3! + (iθ)4 4! + (iθ)5 5! + (iθ)6 6! ! iθ3 3! + θ4 4! + iθ5 5! θ6 6! ! θ6 6! i θ θ3 3! + θ5 5!... [Taylor series for cos(θ)] + i [Taylor series for sin(θ)] So: e iθ cos(θ)+isin(θ). (4.19) Taking the complex conjugate of this, it follows that e iθ cos(θ) i sin(θ). (4.20) Adding Eq. (4.18) and (4.19) e iθ + e iθ 2cos(θ) and hence cos(θ) 1 e iθ + e iθ (4.21) 2 Similarly, subtracting Eq. (4.19) from (4.18) More generally, we know that and therefore where from Eq. (4.11) and (4.12) e iθ e iθ 2i sin(θ) and hence sin(θ) 1 e iθ e iθ. (4.22) 2i z a + ib r(cos(θ)+i sin(θ)) z a + ib re iθ (4.23) r a 2 + b 2 and θ tan 1 b a Examples 4, Q De Moivre s Theorem Consider the complex number z e iθ (cos(θ)+isin(θ)). Then z n e iθ n Hence e inθ cos(nθ)+i sin(nθ) (cos(θ)+i sin(θ)) n cos(nθ)+i sin(nθ). (4.24) Examples 4, Q8 36