Module 5 - Complex Analysis (self Study)

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self Study Course

MODULE 5

COMPLEX NUMBERS I

Module Topics 1. Graphical representation of complex numbers 2. Algebra of complex numbers 3. Polar form of a complex number 4. Euler’s formula and exponential form of a complex number

A:

Work Scheme based on JAMES (THIRD EDITION)

1. Read section 3.1, the Introduction, and in particular the section on quadratic equations on p.159. 2. Study the beginning of section 3.2. Although an unknown complex number z is usually written z = x+jy, most people would write a particular complex number as 3 − 2j, for instance, and not the form 3 − j2 as adopted in J. However, with the normal rules for multiplication of numbers (2)(j) = (j)(2) and so the representations 3 − 2j and 3 − j2 are identical. p √ √ √ −16 = 16(−1) = ±4 −1 √ = ±4j = ±j4. Using a similar Having defined j = −1 it follows that √ method it is easy to show that for any positive number b then −b = ±j b. 3. Study section 3.2.1 on the Argand diagram and Example 3.1. ***Do Exercise 1 on p.166*** 4. Study cases (i) and (ii) in section 3.2.2 and Examples 3.2 and 3.3. Addition and subtraction of complex numbers is straightforward. There is no need to remember the general formulae since it is easy to perform the calculations as necessary. 5. It is easily seen from case (iii) in section 3.2.2 √ that to multiply complex numbers the normal rules for multiplying out brackets are used. Since j = −1 it follows that j 2 = −1 and this must be used when simplifying expressions. Study Example 3.4. Note that

j 2 = −1,

j 3 = j(j 2 ) = j(−1) = −j,

j 4 = j 2 j 2 = (−1)(−1) = 1,

and using these results all higher powers of j must equal ±1 or ±j. 6. Division of complex numbers is more complicated. Study case (iv) on p.163 and Examples 3.5 and 3.6. Note that from the definition of the complex conjugate it can be shown that (z1 + z2 )∗ = z1∗ + z2∗ ,

(z1 z2 )∗ = z1∗ z2∗ .

***Do Exercises 2(a),(b),(d),(f ),(g), 3(a),(b), 5(a),(b),(c), 7, 12 on p.166*** –1–

7. Study section 3.2.3 on modulus and argument. For later work it is important to emphasise that adding any multiple of 2π to arg z leaves the complex number z unaltered. However, the principal value of arg z lies in the range −π < arg z ≤ +π. Work through Example 3.7 and then study section 3.2.5 up to and including Example 3.8. ***Do Exercises 16(a),(c),(e),(g),(i) on p.177*** 8. Study the short section on multiplication in polar form at the bottom of p.167 and the top of p.168. Study Example 3.9. 9. Study the notes on multiplying by j beginning at the bottom of p.168 and on division in polar form on p.169. Study Example 3.10. 10. Before working through Example 3.11 it is important to emphasise that by taking z1 = z2 = z in equations 3.4 and 3.5 it follows that |z 2 | = |z|2

and

arg(z 2 ) = 2 arg z.

and

arg(z n ) = n arg z,

Extending the argument it follows that |z n | = |z|n

although some adjustment (adding or subtracting integral multiples of 2 π ) may be necessary to bring the answer for the argument into the range between −π and +π. Study Example 3.11. 11. Study section 3.2.6 on Euler’s formula. The formula 3.9 is very important and J. justifies the latter by multiplying out and using trigonometric identities. Other authors justify equation 3.9 by considering the series expansions for ex , cos x and sin x. Those of you who know these expansions might like to use them to derive equation 3.9. Work through Examples 3.12 and 3.13. 12. Note that Euler’s formula and the properties of exponentials can also be used to derive results 3.6 and the one above 3.4 as follows. If z1 = r1 ejθ1 and z2 = r2 ejθ2 then z1 z2 = r1 ejθ1 r2 ejθ2 = r1 r2 ej(θ1 +θ2 ) = r1 r2 {cos(θ1 + θ2 ) + j sin(θ1 + θ2 )} , and

r1 ejθ1 r1 r1 z1 = = ej(θ1 −θ2 ) = {cos(θ1 − θ2 ) + j sin(θ1 − θ2 )} . z2 r2 ejθ2 r2 r2

***Do Exercises 14 and 15 on p.177*** 13. To complete the module turn briefly to p.214 and study section 4.2.5 on representing complex numbers as vectors. Vectors may be new to some of you but they will be studied in detail later in the year. Work through Example 4.17.

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B:

Work Scheme based on STROUD (FIFTH EDITION)

Work through Programme 1, Complex Numbers, Part 1, leaving out only section 58. Note the following notations. Given the complex number z = x + jy, then its conjugate z = x − jy is usually written z ∗ (or z ). In writing a complex number in polar form S. chooses arg z to lie in the range 0 < arg z ≤ 2π (in radians). J. and most other authors use the range −π < arg z ≤ π, and in this unit you should use the latter range if asked for the principal value of arg z. (Clearly adding any integral multiple of 2π on to arg z leaves z unaltered). Study Programme 2, Complex Numbers, Part 2, sections 1-26 omitting sections 11 and 12. Note that the comment above on the range of arg z applies when finding the angle in the proper form (i.e. the principal value of arg z).

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Specimen Test 5 1.

If z = 2 + j and w = −3 + j, evaluate the following: (i) Re (z),

2.

(ii) Im (w),

(iii) w∗ ,

(iv) jz − 2w,

(v) z/w.

Illustrate on the diagram the sum z1 + z2 and the difference z1 − z2 of two complex numbers z1 and z2 . y •

z2

z1• x

√ 3. (i) If z = − 3 − j (a) find |z| and the principal argument of z, (b) write down all possible values of arg z, (c) plot z on an Argand diagram indicating its modulus and the principal value of its argument. √ (ii) Write z = − 3 − j in exponential form and hence write the following in exponential form, giving the principal value of the argument in each case: (a) z 6 ,

(b) z −2 .

4. (i) Express the following quantities in terms of |z1 | and |z2 |: z1 (c) |z1n |. (b) , (a) |z1 z2 | , z2 (ii) Use the results in (i) to find the modulus of

(2 + j)11 . (−1 − 2j)9

5. (i) Complete the following statement of Euler’s formula

ejα = cos α+

(ii) Write down the corresponding expression for e−jα , and hence express cos α and sin α in terms of complex exponentials.

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