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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 rect
angles 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 cotan
gent 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
0 0
ieiz cos z i sin 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
iw
eiw e
z cos w
2
or
e2iw 2zeiw 1 0
Solving the quadratic over C
p
p
2z 4z2 4
eiw z z2 1
2
Hence
p
w i log z z2 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.
6
This 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
V
C
C
L
VR R
VL
E(t)
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 Fara
day 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
Z
t
1
VC I d 2.4
C
If we have a periodic driving EMF as would arise naturally from any gener
ator 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
1
VR I R VL i LI VC I
i C
Adding up the voltages of equation 2.1 we get
1
E R i L I
C
and the quantity
1
R i L
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 al
lows us to carry through phase information since the phase of the voltage is
changed by inductances or capacitances which is of very considerable prac
tical 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
Chapter 3
C Di erentiable Functions
3.1 Two sorts of Di erentiability
Suppose f C C is a function taking x iy to u iv. We know that if
it is di erentiable regarded as a map from R2 to R2 then the derivative is a
matrix of partial derivatives
u u
x y
v v
x y
If you learnt nothing else from second year Mathematics you may still be
able to hold your head up high if you grasped the idea that the above matrix [ Pobierz całość w formacie PDF ]