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Complex Functions

Module by: Steven J. Cox. E-mail the author

Summary: This module discusses complex functions and their relation to systems of linear equations.

Complex Functions

A complex function is merely a rule for assigning certain complex numbers to other complex numbers. The simplest (nonconstant) assignment is the identity function fzz f z z . Perhaps the next simplest function assigns to each number its square, i.e., fzz2 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 real-valued functions of two real variables. In the case that fzz2 f z z 2 we find uxy=x2y2 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 zm1++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=zλ1d1zλ2d2zλhdh 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 m1 m 1 while gg is of order mm with the simple roots λ1λm λ1 λm . It should come as no surprise that such qq should admit a Partial Fraction Expansion
qz=j=1mqjzλj q z j 1 m qj z λj
(4)
One uncovers the qjqj by first multiplying each side by zλj z λj and then letting zz tend to λjλj. For example, if
1z2+1=q1z+i+q2zi 1 z 2 1 q1 z q2 z
(5)
then multiplying each side by z+i z produces
1zi=q1+q2(z+i)zi 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 zi z . and then set z=i z . So doing we find q2=i2 q2 2 , and so
1z2+i=i2z+i+i2zi 1 z 2 2 z 2 z
(7)
. Returning to the general case, we encode the above in the simple formula
qj=limit  zzλj(zλj)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+1/2zi 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

zIB-1=1z2+1( z1 -1z )=1z+i( 1/2i2 i21/2 )+1zi( 1/2i2 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 Gauss-Jordan 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 zIB-1 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

ezn=0znn! z n 0 z n n
(11)
and find that
ee=1+iθ+iθ22+iθ33!+iθ44!+=1θ22+θ44!+i(θθ33!+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=|z|eiθ z z θ .

One may just as easily verify that cosθ=eiθ+e(i)θ2 θ θ θ 2 and sinθ=eiθe(i)θ2i θ θ θ 2 These suggest the definitions, for complex zz, of

coszeiz+e(iz)2 z z z 2
(13)
and
sinzeize(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,

lnzln|z|+iθ z z θ
(16)
for z=|z|eiθ z z θ . One finds that ln-1+i=ln2+i3π4 -1 2 3 4 .

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A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

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