2.9 Trigonometric Functions

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The argument through in_nite series that showed that

ei_ = cos _ + i sin _

and the argument that we can replace the usual series for the real exponential

by simply putting in complex values, is asking to be carried the extra mile.

Suppose we put a complex value in place of a real value for the functions sin

and cos? Would we get respectable complex functions out? Yes indeed we

do.

I de_ne the complex trig functions as follows:

De_nition 2.9.1

eiz = cos z + i sin z

for any complex number z.

It follows immediately that

cos z =

1

2

(eiz + e􀀀iz)

and putting z = x + iy and hence iz = 􀀀y + ix, 􀀀iz = y 􀀀 ix we get

cos z =

1

2 􀀀e􀀀y(cos x + i sin x) + ey(cos x 􀀀 i sin x)_

or

cos(x + iy) = cos(x) cosh(y) 􀀀 i sin(x) sinh(y)

Similarly we obtain:

sin(x + iy) = sin(x) cosh(y) + i cos(x) sinh(y)

Example 2.9.1 Solve: sin z = i.

Solution We have sin x cosh y = 0, and since cosh y _ 1 it follows that

x = n_ for some integer n.

We also have cos x sinh y = 1, hence sinh y = _1 follows.

So x = n_; y = sinh􀀀1

_1 with the positive value when n is even and the

negative when it is odd. x = 0; y = ln(1 + p2) is a solution.

2.9. TRIGONOMETRIC FUNCTIONS 83

Figure 2.26: The image by the sine function of the unit square

Exercise 2.9.1 Figure 2.26 shows the image of the unit square by the sin

function. Show the top curved edge is a part of an ellipse, and the right

curved edge is part of a hyperbola.

It would appear that the edges of the image meet at right angles. Can you

explain this?

Going back to the images we have for complex functions of squares and rectangles,

you might notice that the images of square corners almost always

come out as curves meting at right angles. There is one exception to this.

Can you (a) give an explanation of the phenomenon and (b) account for the

exception?

It follows from my de_nition that there are power series expansions of the

usual sort for the trig functions sin z and cos z. The tangent, secant, cotangent

and cosecant functions are de_ned in the obvious ways. Inverse functions

are de_ned in the obvious way also. The rest is algebra, but there's a lot of

it.

Di_erentiating the trig functions proceeds from the de_nition:

eiz = cos z + i sin z

) ieiz = cos0 z + i sin0 z

84 CHAPTER 2. EXAMPLES OF COMPLEX FUNCTIONS

= 􀀀sin z + i cos z

where the second line is obtained by di_erentiating the top line, and the

last line is obtained by multiplying the top line by i. This tells us that the

derivative of cos is 􀀀sin and the derivative of sin is cos, as in the real case.

The de_nitions also imply that cos z is just the usual function when z is real,

and likewise for sin.

The inverse trig functions can be obtained from the de_nitions:

Example 2.9.2 If w = arccos z, obtain an expression for w in terms of the

functions de_ned earlier.

Solution

We have

z = cos w =

eiw + e􀀀iw

2

or

e2iw

􀀀 2zeiw + 1 = 0

Solving the quadratic (over C !)

eiw =

2z + p4z2 􀀀 4

2

= z + pz2 􀀀 1

Hence

w = 􀀀i log(z + pz2 􀀀 1)

We have all the problems of multiple values in both the square root and the

log functions.

Exercise 2.9.2 Find arcsin 3.

It is worth exploring the derivatives of these functions, if only so as to be able

to do some nasty integrals later by knowing they have easy antiderivatives6.

6This sort of thing used to be a cottage industry in the seventeenth and eighteenth

centuries: mathematicians would issue public challenges to solve horrible integration prob-

lems which they made up by doing a lot of di_erentiations. This is cheating, something

Mathematicians are good at.

2.9. TRIGONOMETRIC FUNCTIONS 85

E(t)

L

C

R

C

L V

V

R V

Figure 2.27: A simple LCR circuit

Exercise 2.9.3 Compute the derivatives of as many of the trig functions

and their inverses as you can.

There is a standard application of the use of complex functions to LCR

circuits which it would be a pity to pass up:

Example 2.9.3 (LCR circuits)

The _gure shows a series LCR circuit with applied EMF E(t); the voltage

drop across each component is shown by VR; VC; VL respectively. We have

E(t) = VR +VC +VL (2.1)

at every time t.

It is well known that the current I in a resistance satis_es Ohms Law, so we

have immediately

VR = IR (2.2)

and since what goes in must come out, the current I through each component

is the same.

86 CHAPTER 2. EXAMPLES OF COMPLEX FUNCTIONS

The current and voltage drop across an inductance or choke is given by

VL = LI_ (2.3)

since the impedance is due to the self induced magnetic _eld which by Faraday's

Laws is proportional to the rate of change of current.

Finally, the voltage drop across a capacitor or condenser is proportional to

the charge on the plates, so we have

VC =

1

C Z t

0

I(_ )d_ (2.4)

If we have a periodic driving EMF as would arise naturally from any generator,

we can write

I(t) = I0 cos(!t) (2.5)

where ! is the frequency.

I now assume that the current is the real part of a complex current I_, which

will make keeping track of things simpler.

Then

I_(t) = I0ei(!t) (2.6)

and similarly for complex voltages:

V _ R = I_R; V _ L = i!LI_; V _ C =

1

i!C

I_

Adding up the voltages of equation 2.1 we get:

E_ = _R + i_!L 􀀀

1

!C__I_

and the quantity

R + i_!L 􀀀

1

!C_

is called the complex impedance usually denoted by Z.

Then Ohm's Law holds for complex voltages and currents.

2.9. TRIGONOMETRIC FUNCTIONS 87

This notation may seem puzzling; it is little more than a notation, but it allows

us to carry through phase information (since the phase of the voltage is

changed by inductances or capacitances) which is of very considerable practical

signi_cance in Power distribution, for example. But I shall leave this to

your Engineering lecturers to develop.

Since you ought to be getting the idea by now as to what to look for, I shall

_nish the chapter in a spirit of optimism, believing that you have sorted

out at least a few functions from C to C and that you have some ideas

of how to go about investigating others if they are sprung on you in an

examination. I leave you to think about some possibilities by working out

which real functions have not yet been extended to complex functions. There

is a lot of room for some experimenting here to investigate the behaviour of

lots of functions I haven't mentioned as well as lots that I have. Life being

short, I have to leave it to you to do some investigation. You will _nd it

more fun than most of what's on television.

In the next chapter we continue to work out parallels between R and C and

the functions between them, but we take a big jump in generality. We ask

what it would mean to di_erentiate a complex function.

88 CHAPTER 2. EXAMPLES OF COMPLEX FUNCTIONS

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