In previous chapters we have been discussing real numbers and their algebraic representation. Real numbers are part of a larger set called complex numbers. In this chapter we start by showing how the latter arise and then discuss their properties and how they are represented. Complex numbers and complex variables are of great practical importance in a wide range of topics, including vibrations and waves, and quantum theory.

6.1 Complex numbers

Given a positive real number q (not necessarily an integer) we know that its square roots are also real numbers. But situations also arise where we meet the square root of a negative number. In Section 2.1.1, for example, we saw that the solution of a general quadratic equation ax² + bx + c = 0 is of the form

(6.1)

and there is no restriction on the sign of (b² − 4ac). Thus we have to face the question: can we find an interpretation of the quantity , where q > 0? It cannot be the same as because squaring would produce a contradiction. A new definition is required. Since

it follows that the only new definition needed is for . This is denoted by the letter i, with i² = −1, and is called an imaginary number.¹ Thus, and is also an imaginary number. If x and y are two real numbers, then the number z = x + iy is called a complex number and x and y are called its real and imaginary parts, denoted Re z = x and Im z = y. Formally, the quantities Re and Im are functions, whose argument is a complex number z and whose results are the real and imaginary parts of z, respectively. Note that both functions produce real outputs, and in particular the imaginary part y is a real number; it is always understood that it is multiplied by i to give an imaginary number.

A first sight it may appear that imaginary numbers have no applications in physical science because all physical measurements yield a real number. In fact the converse is true: complex numbers play a vital role in the mathematical analysis of numerous physical phenomena. We will see that apparently making a problem more complicated by introducing complex variables, can in practice actually simplify analyses by allowing the use of powerful techniques available in the theory of complex quantities.

We can now interpret solutions of equations such as

Using the standard formula (6.1) gives

(6.2)

that is, the roots z_{1, 2} are complex numbers, with , and . In this case, the two roots only differ by the sign of the imaginary parts. Pairs of complex numbers related in this way are said to be complex conjugates of each other. Thus, if a complex number z = x + iy, then its complex conjugate, written z*, is z* = x − iy.² It is straightforward to show that complex conjugation has the properties:

and so on for several complex numbers.

Two complex numbers are defined to be equal only if the real parts of both numbers are equal and the imaginary parts of both numbers are equal. Complex numbers obey the usual rules of addition, subtraction and multiplication, including the commutative, associative and distributive laws obeyed by real numbers, as discussed in Section 1.1.1. For example,

(6.3a)

and

(6.3b)

where i² = −1 has been used in (6.3b). Division of a complex number by a real number is straightforward; the real number divides the real and imaginary parts of the complex number separately. Division by a complex number is a little more complicated. If we have two complex numbers p and q, their quotient is in general also a complex number, whose real and imaginary parts are found by rationalisation. In the case of complex numbers, this means multiplying the numerator and denominator by the complex conjugate of the latter to give a new real denominator, which then divides the real and imaginary parts of the numerator. Explicitly, if p and q are two complex numbers,

(6.4a)

The quantity (Re q)² + (Im q)² that appears in the denominator is the square of the modulus, or absolute value, of q, written , or |q|. Thus,

(6.4b)

It follows from (6.4b), that for a general complex number z,

(6.5)

6.2 Complex plane: Argand diagrams

The complex number z = x + iy is an ordered pair of real numbers that can be written (x, y) and these can be viewed as the Cartesian co-ordinates of a point P(x, y) in a plane, called in this context the complex plane. The diagram in which complex numbers are represented in this way is called an Argand diagram. This is shown in Figure 6.1, with the general point P(x, y) plotted.

An alternative way of representing a complex number is to use two-dimensional polar co-ordinates (r, θ), where r is the positive distance to P from the origin and θ is measured in the counter-clockwise sense from the x-axis. The quantities r and θ are also shown in Figure 6.1, from which we see that

(6.6a)

so that

(6.6b)

This is called the polar form of z. The quantity r, given by , is the modulus of z, that is, . The angle θ is called the argument of z and is written θ = arg z.³ It may be found using (6.6b) to be θ = arctan (y/x), but care must be taken when using this result to take account of the signs of both x and y separately, otherwise not all the values of θ found from their ratio will satisfy (6.6a). As the latter equations only define θ up to an additive integral multiple of 2π, it is usual to quote the so-called principal value of θ, that is, the value for which − π < θ ≤ π. For example, if z = x + iy = 2 − 2i, then and , where the latter has solutions

with n any integer. However, only the first of these has x = rcos θ > 0 and y = rsin θ < 0, as required, and choosing n = 0, we obtain the principal value θ = −π/4.

The Argand diagram provides a geometrical interpretation of arithmetical operations involving complex numbers. For example, Figure 6.2 shows two complex numbers z₁ and z₂. It is easy to see from the construction shown that their sum

is the point plotted at z₃. An analogous diagram describes subtraction.

Multiplication and division are particularly simple in polar form. In the case of multiplication of two complex numbers z₁ and z₂, we have from (6.6a)

which, using the trigonometric identities (2.36a), is

Thus the product z₁z₂ is a complex number with modulus r₁r₂ and argument (θ₁ + θ₂).

For the of division of two complex numbers z₁/z₂ we can write z₁/z₂ = (z₁z*₂)/|z₂|² and then again use (6.6a), to give

which, using the trigonometric identities (2.36a), is

Thus the quantity z₁/z₂ is a complex number with modulus r₁/r₂ and argument (θ₁ − θ₂). This relation can also be demonstrated on an Argand diagram.

Finally, simple equations define curves in the complex plane. For example, consider the equation

Using z = x + iy gives z² = x² − y² + 2ixy and hence

This is a hyperbola, the two branches of which pass through , respectively. Another example is the equation |z + 3| = 5, which may be written

This is the equation of a circle of radius 5 and centre at ( − 3, 0).

6.3 Complex variables and series

In Chapter 1, the discussion of real numbers was extended to their algebraic realisation, real variables. In the same way we may extend the current discussion to consider complex variables and their associated complex algebra. The rules discussed in Section 1.2 for the algebra of real variables hold, provided we remember that a complex variable is actually a pair of real variables. Thus, for example, the function

written in terms of its real and imaginary parts is

and may be rationalised by multiplying the numerator and denominator by the complex conjugate of the latter, that is, [(x + 3) − iy]. This gives

where

Similarly, an equation such as

(6.7)

may be written

and when expanded is

Because two complex quantities are only equal if both their real and imaginary parts are equal, (6.7) is equivalent to two equations,

We next consider series of the form

(6.8)

where the individual terms are now complex numbers or expressions. By writing

(6.9)

theses series may be expressed in terms of two real series, enabling the results established in Chapter 5 for real series to be easily extended to the complex case. In particular, one can show that d'Alembert's ratio test still holds,⁴ so that if

(6.10)

as n → ∞, the series converges if ρ < 1 and does not converge if ρ > 1, while the case ρ = 1 requires special treatment. Thus, for example, to test the convergence of the series

using the ratio test, we find the quantity

and hence the series is convergent.

We can also extend the previous discussion of power series. These now become series of the form

(6.11)

where the variable z = x + iy, with z₀ = x₀ + iy₀ and the coefficients a_n are complex numbers. Then by the ratio test, the series converges if

i.e. if

(6.12)

as in the case of a real series (5.19). In terms of the real and imaginary parts, this becomes

i.e.

corresponding to the interior of a circle in the complex plane, centred at z = z₀ with radius R. This circle is called the circle of convergence and R is called the radius of convergence. For example, consider the series

Then by (6.12), the series converges for

that is, inside a circle of radius unity centred at the origin z = 0 of the complex plane.

One very important power series is the complex exponential series

(6.13)

obtained by replacing the real variable x in (5.42) by the complex variable z = x + iy. D'Alembert's ratio test (6.10) shows that this series converges for all values of z, so that (6.13) can be used to define the exponential function over the whole complex plane. In the same way, the series for sin x and cos x in Table 5.1 can be generalised from real x to complex z, and used to define sin z and cos z in the whole complex plane, in which case sin z and cos z are no longer real or restricted to the range −1 to +1. Other functions can then be defined from these in analogy to the corresponding functions of a real variable. For example tan z ≡ sin z/cos z, while the hyperbolic functions are

(6.14)

in analogy to (2.57) for real z = x.

*6.3.1 Proof of the ratio test for complex series

For the series (6.8) to converge, it is necessary and sufficient for both the real series

(6.15)

that occur in (6.9) to converge. For ρ > 1, this is impossible because (6.10) then implies , so at least one of the quantities Re a_n or Im a_n does not tend to zero. Hence the corresponding real series, and by implication (6.8), cannot converge by (5.15).

It remains to prove that (6.8) does converge for ρ < 1. To do this, we first consider the series

This is a real series, so that d'Alembert's ratio test applies, and it converges if ρ < 1. But

so that the real positive series

(6.16)

converge by the comparison test (5.55) established in Section 5.5.1*. The series (6.15) are then said to be ‘absolutely convergent’ and, as shown in Section 5.5.2*, an absolutely convergent series is convergent, as the name implies. Hence both the series (6.16), and thus the complex series (6.8), converge for ρ < 1, as required.

6.4 Euler's formula

In this section we introduce an important formula due to Euler and illustrate some of its many applications. To derive this formula, we substitute z = iθ, where θ is real, in the exponential series (6.13). This gives:

(6.17)

Now from the results given in Table 5.1, the real part of (6.17) is seen to be the series for cos θ and the imaginary part is the series for sin θ. So we have deduced the important result

(6.18)

This is Euler's formula, and enables many useful relations to be derived. For example, from the definition of the hyperbolic functions (2.57) and Euler's formula, we have, for real angles θ,

and

Furthermore, using the polar forms (6.6b) together with (6.18), we can write any complex number z in the form

(6.19)

where r = |z| is the modulus and θ is the argument of z as usual. This exponential form is very useful in algebraic calculations involving complex variables, particularly multiplication and division. Using the law of exponents discussed in Section 1.1.2, and now extended to complex variables, we have for multiplication

(6.20a)

and for division,

(6.20b)

These are the same results that were obtained in Section 6.2, but derived here in a simpler way without using trigonometric identities.

6.4.1 Powers and roots

The exponential form provides a simple way of finding powers of a complex quantity, since if z = re^iθ, then

(6.21)

by repeated application of (6.20a). For example, to find the cube of z = (1 + i), we first convert it to exponential form

to give

The nth roots of a complex number z are the solutions w of the equation w = z^1/n, that is, the complex numbers whose nth power is z. There are always n such roots. To see this, we note that

for any integer k. Hence the roots are

(6.22)

However, it is easily to see that w_{k ± n} = w_k, so the only roots that are distinct are w₀, w₁, …, w_{n − 1}, with larger or smaller values of k merely reproducing the roots with k = 0, 1, …, n − 1. For example, to find the cube roots of z = (2 − 2i), we use the polar form with and θ = −π/4. Then using k = 0, 1, 2 gives the three solutions

Larger values of k just reproduce the solutions for k ≤ 2.

Of particular interest are the nth roots of unity. In this case zⁿ = 1 = e^2ikπ, where k is any integer, so z = e^2ikπ/n. Hence the solutions are

corresponding to k = 0, 1, 2, …, (n − 1). The solutions for n = 3, i.e.

are shown plotted on a circle of unit radius in Figure 6.3. Again, larger values of k just reproduce the solutions for k ≤ 2.

The polar representation of a complex number is also useful when finding the roots of a polynomial equation. To illustrate this, consider the polynomial equation

which factorises as

Hence solutions are given by z³ = 1, or z² = −2 or z = 3. In the first case we can use (6.21) to give the three solutions obtained given in (6.22)

The other solutions are , from z² = −2, and finally z₆ = 3. Thus we have six solutions in accord with the fundamental theorem of algebra. This example also illustrates the general result, that a polynomial equation with real coefficients has roots that occur in complex conjugate pairs. (See Problem 6.4)

6.4.2 Exponentials and logarithms

The exponential function was used in Chapter 2 to define natural logarithms, which we can also generalise for complex arguments. For a complex number z, the natural logarithm is defined by analogy with its definition for a real number. Thus e^ln z = z. Substituting ln z = α + iβ and z = re^iθ, where θ is the principal value, − π < θ ≤ π, we have

and since α and β are both real, e^α = r, i.e. α = ln r, and β = θ + 2πk(k = 0, 1, …), so

(6.23)

Thus the imaginary part of the logarithm is only defined up to additive multiples of 2π. The principal value of the logarithm is defined as the case when k = 0, for which⁵

(6.24)

It is straightforward to show that the results previously obtained in Chapter 2 for the logarithms of real variables also hold for complex variables. Thus, in general, using (6.19),

where k is an integer. If , this reduces to the result for principal values

Likewise, for division,

Extending the definition of logarithms to complex arguments enables us to generalise the discussion of Section 6.4.1 to complex powers and roots. For example, to evaluate (1 + i)^z where z = 1 − 2i, we have

Now using logarithms, 2^{− i} = e^{− iln 2} and so

So, finally, (1 + i)^z = 0.624 − 6.774i.

6.4.3 De Moivre's theorem

If we substitute Euler's formula into both sides of the simple identity

we immediately obtain De Moivre's theorem:

(6.25)

which is valid for all real values of n, whether integer or not.

De Moivre's theorem provides a very convenient way of obtaining expressions for powers of trigonometric functions, and expansions of these functions for multiple angles. Suppose we wish to express cos 4θ and sin 4θ in terms of powers of sin θ and cos θ. This could be done for each expansion separately by using the multiple-angle trigonometric formulas of Section 2.2.4, but by apparently making the problem more complicated by introducing complex variables, we can use the result (6.25). This gives

Then equating real and imaginary parts of both sides gives

and

This method may be applied in general to find the forms of cos nθ and sin nθ for any n > 0 by using the general results

(6.26)

that follow directly from De Moivre's theorem. In a similar way we can find expressions for cos ⁿθ and sin ⁿθ in terms of simple sines and cosines. For example, consider cos ⁴θ. From (6.25) this may be written

which using (6.25) again is

*6.4.4 Summation of series and evaluation of integrals

The Euler formula may also be used to sum many series involving sines and cosines. Consider the series

where a is a real constant. To find C we first form the analogous series for sines,

and then combine them to give the complex series

This is a geometric series with a common ratio R = ae^iθ and from Section 4.1 we know that it is

(6.27)

Finally, C is given by the real part of the right-hand side and, after some algebra, we find

As a bonus, S is given by the imaginary part of the right-hand side of (6.27).

A similar technique may be used for continuous variables, where the analogous quantities are integrals. Consider, for example, the integral

This could be evaluated directly by integration by parts, but as an illustration of the method we form the analogous integral involving sines,

and combine these to give

(6.28)

Finally, C is the real part of the right-hand side of (6.28), that is

and again, as a bonus, S is the imaginary part of the right-hand side.

Problems 6

1. If z₁ = 1 + 4i, z₂ = 2 − 3i and z₃ = 4 + 3i, find:

   (a) z₁ + z₂ − z₃, (b) (z₁ − z₂)* + z₃, (c) (z₁ + z₃)(z₂ + z₃), (d) (z₂ + z₁)(z₃ − z₂)*, (e) z₂/(z*₁z₃), (f) (z₁ + z₂)/(z₁ + z₃), (g) |z₁| and z^{− 1}₁.

2. If z₁ and z₂ are two complex numbers, verify that: (a) (z₁z₂)* = z*₁z*₂, (b) |z₁z₂| = |z₁||z₂|, (c) |z₁ + z₂| ≤ |z₁| + |z₂| and (d) |z₁ − z₂| ≥ |z₁| − |z₂|.

3. Simplify and rationalise the following expressions:

   • (a) , (b) , (c) .

4. Show that a polynomial equation of order n with real coefficients has roots that are either real, or in complex conjugate pairs.

5. Express the following numbers in polar form: (a) , (b) , (c) , using the principal value of the argument.

6. What are the plane curves represented by the equations (a) |z − 1| = 2, (b) |z + 1| = |z − i|?

7. Use the ratio test to find the circle of convergence for the following infinite series whose general terms R_n are (a) [(n!)²/(3n)!](z − 3i)ⁿ, (b) (n + 3)³(2iz)ⁿ, (c) ( − 1)ⁿz^{n + 3}/n!

8. What is the modulus and argument of (a) , (b) z = (1 + i)e^iπ/6, (c) z = [(2 + i)/(i − 3)]e^iπ/3.

9. Use Euler's formula to write the following complex numbers in the form x + iy:

   

10. Convert the following complex numbers to the form (x + iy):

    

11. Write the following complex numbers in the form (x + iy):

    

12. Express the following in the form x + iy:

    

13. Convert the following complex expressions to the form x + iy:

    

14. Use De Moivre's theorem to (a) write the expression

    

    in the form (x + iy); (b) show that

    

15. Use De Moivre's theorem to (a) simplify the expression

    

    and (b) show that

    

16. Evaluate the integral

    

17. Find the sum of the series

    

18. Use the binomial theorem for (1 + e^ix)ⁿ to show that

    

    and find the sum of the series

    

    where are the binomial coefficients.