A complex function is merely a rule for assigning certain
complex numbers to other complex numbers. The simplest
(nonconstant) assignment is the identity function
fz≡z
f
z
z
. Perhaps the next simplest function assigns to each
number its square, i.e.,
fz≡z2
f
z
z
2
.
As we decomposed the argument of
ff, namely
zz, into its real and imaginary
parts, we shall also find it convenient to partition the
value of ff,
z2
z
2
in this case, into its real and imaginary parts. In general, we write
fx+iy=uxy+ivxy
f
x
y
u
x
y
v
x
y
(1)
where
uu and
vv are both realvalued functions
of two real variables. In the case that
fz≡z2
f
z
z
2
we find
uxy=x2−y2
u
x
y
x
2
y
2
and
vxy=2xy
v
x
y
2
x
y
With the tools of complex
numbers, we may produce complex polynomials
fz=zm+
c
m

1
zm−1+…+c1z+c0
f
z
z
m
c
m

1
z
m
1
…
c1
z
c0
(2)
We say that such an
ff is order
mm. We shall often find it
convenient to represent polynomials as the product of their
factors, namely
fz=z−λ1d1z−λ2d2…z−λhdh
f
z
z
λ1
d1
z
λ2
d2
…
z
λh
dh
(3)
Each
λjλj
is a
root of
degree
djdj. Here
hh is the number of
distinct roots of
ff. We call
λjλj
a
simple root when
dj=1
dj
1
.
We can observe the appearance of ratios of polynomials or so
called
rational functions. Suppose
qz=fzgz
q
z
f
z
g
z
in rational, that
ff is of order at most
m−1
m
1
while
gg is of order
mm with the simple roots
λ1…λm
λ1
…
λm
.
It should come as no surprise that such
qq should admit a
Partial
Fraction Expansion
qz=∑j=1mqjz−λj
q
z
j
1
m
qj
z
λj
(4)
One uncovers the
qjqj
by first multiplying each side by
z−λj
z
λj
and then letting
zz tend to
λjλj.
For example, if
1z2+1=q1z+i+q2z−i
1
z
2
1
q1
z
q2
z
(5)
then multiplying each side by
z+i
z
produces
1z−i=q1+q2(z+i)z−i
1
z
q1
q2
z
z
(6)
Now, in order to isolate
q1q1 it is clear that we should set
z=−i
z
.
So doing we find that
q1=i2
q1
2
.
In order to find
q2q2 we multiply
Equation 5 by
z−i
z
.
and then set
z=i
z
.
So doing we find
q2=−i2
q2
2
,
and so
1z2+i=i2z+i+−i2z−i
1
z
2
2
z
2
z
(7).
Returning to the general case, we encode the above in the
simple formula
qj=limit zz→λj(z−λj)qz
qj
z
z
λj
z
λj
q
z
(8)
You should be able to use this to confirm that
zz2+1=1/2z+i+1/2z−i
z
z
2
1
12
z
12
z
(9)
Recall that the transfer function we met in The Laplace Transform
module was in fact a matrix of rational functions. Now, the
partial fraction expansion of a matrix of rational functions
is simply the matrix of partial fraction expansions of each of
its elements. This is easier done than said. For example,
the transfer function of
B=(
01
10
)
B
0
1
1
0
is
zI−B1=1z2+1(
z1
1z
)=1z+i(
1/2i2
−i21/2
)+1z−i(
1/2−i2
i21/2
)
z
I
B
1
1
z
2
1
z
1
1
z
1
z
12
2
2
12
1
z
12
2
2
12
(10)
The first line comes form either GaussJordan by hand or via
the symbolic toolbox in Matlab. More importantly, the second
line is simply an amalgamation of
Equation 5 and
Equation 7.
Complex matrices have finally entered the picture. We shall
devote all of Chapter 10 to uncovering the remarkable
properties enjoyed by the matrices that appear in the partial
fraction expansion of
zI−B1
z
I
B
1
Have you noticed that, in our example, the two matrices are
each projections, and they sum to
II, and that their product is
00? Could this be an accident?
In The Laplace
Transform module we were confronted with the complex
exponential. By analogy to the real exponential we define
ez≡∑n=0∞znn!
z
n
0
z
n
n
(11)
and find that
ee=1+iθ+iθ22+iθ33!+iθ44!+…=1−θ22+θ44!−…+i(θ−θ33!+θ55!−…)=cosθ+isinθ
e
θ
1
θ
θ
2
2
θ
3
3
θ
4
4
…
1
θ
2
2
θ
4
4
…
θ
θ
3
3
θ
5
5
…
θ
θ
(12)
With this observation, the
polar form is now
simply
z=zeiθ
z
z
θ
.
One may just as easily verify that
cosθ=eiθ+e(−i)θ2
θ
θ
θ
2
and
sinθ=eiθ−e(−i)θ2i
θ
θ
θ
2
These suggest the definitions, for complex
zz, of
cosz≡eiz+e(i−z)2
z
z
z
2
(13)
and
sinz≡eiz−e(−i)z2i
z
z
z
2
(14)
As in the real case the exponential enjoys the property that
ez1+z2=ez1ez2
z1
z2
z1
z2
and in particular
ex+iy=exeiy=excosy+iexsiny
x
y
x
y
x
y
x
y
(15)
Finally, the inverse of the complex exponential is the complex
logarithm,
for
z=zeiθ
z
z
θ
.
One finds that
ln1+i=ln2+i3π4
1
2
3
4
.