Post on 29-Oct-2021
Singularities, Zeros Singularities, Zeros And PolesAnd Poles
Engineering Mathematics III
SingularitiesSingularities • We have seen that the function w = z3 is
analytic everywhere except at z = whilst the function w = z-1 is analytic everywhere except at z = 0.
• In fact, NO function except a constant is analytic throughout the complex plane, and every function except of a complex variable has one or more points in the z plane where it ceases to be analytic.
• These points are called “singularities”.
Engineering Mathematics III
Types of singularitiesTypes of singularities
• Three types of singularities exist: o Poles or unessential singularities
• “single-valued” functions
o Essential singularities
• “single-valued” functions
o Branch points
• “multivalued” functions
Engineering Mathematics III
Poles or unessential Poles or unessential singularitiessingularities
• A pole is a point in the complex plane at
which the value of a function becomes
infinite.
• For example, w = z-1 is infinite at z = 0, and
we say that the function w = z-1 has a pole
at the origin.
• A pole has an “order”:
o The pole in w = z-1 is first order.
o The pole in w = z-2 is second order.
Engineering Mathematics III
The order of a poleThe order of a pole
If w = f(z) becomes infinite at the point z = a, we define:
)()()( zfazzg n where n is an integer.
If it is possible to find a finite value of n which makes g(z) analytic at z = a, then, the pole of f(z) has been “removed” in forming g(z). The order of the pole is defined as the minimum integer value of n for whichg(z) is analytic at z = a.
pole, (a=0)
zw
1
)(1
)( zgz
z n Order = 1
Engineering Mathematics III
Essential singularitiesEssential singularities • Certain functions of complex variables have an infinite
number of terms which all approach infinity as the
complex variable approaches a specific value. These
could be thought of as poles of infinite order, but as the
singularity cannot be removed by multiplying the
function by a finite factor, they cannot be poles.
• This type of sigularity is called an essential singularity and
is portrayed by functions which can be expanded in a
descending power series of the variable.
• Example: e1/z has an essential sigularity at z = 0.
Engineering Mathematics III
Essential singularities can be distinguished from poles by the fact that they cannot be removed by multiplying by a factor of finite value.
Example:
..!
1...
!2
111
2
2/1 nznzz
ew infinite at the origin
We try to remove the singularity of the function at the origin by multiplying zp
..!
...!2
21
n
zzzzwz
nppppp
It consists of a finite number of positive powers of z, followed by an infinite number of negative powers of z.
All terms are positive
wzzAs p,0
It is impossible to find a finite value of p which will remove the singularity in e1/z at the origin. The singularity is “essential”. Engineering Mathematics III
Branch pointsBranch points • The singularities described above arise from
the non-analytic behaviour of single-valued
functions.
• However, multi-valued functions frequently
arise in the solution of engineering problems.
• For example:
2
1
zw irez i
erw 2
1
2
1
z w
For any value of z represented by a point on the circumference of the circle in the z plane, there will be two corresponding values of w represented by points in the w plane.
Engineering Mathematics III
ivuw
2
1
zw irez i
erw 2
1
2
1
2
1sin
2
1
2
1cos
2
1
ru
rr
u
sincos iei
2
1cos2
1
ru 2
1sin2
1
rv
2
1cos
2
1
2
1sin
2
1
rv
rr
v
and
u
rr
vand
v
rr
u 11
Cauchy-Riemann conditions in polar coordinates
when 0 2
A given range, where the function is single valued: the “branch”
The particular value of z at which the function becomes infinite or zero is called the “branch point”. The origin is the branch point here.
Engineering Mathematics III
Branch pointBranch point • A function is only multi-valued around
closed contours which enclose the branch point.
• It is only necessary to eliminate such contours and the function will become single valued. o The simplest way of doing this is to erect a barrier
from the branch point to infinity and not allow any curve to cross the barrier.
o The function becomes single valued and analytic for all permitted curves.
Engineering Mathematics III
Barrier Barrier -- branch cutbranch cut • The barrier must start from the branch point but
it can go to infinity in any direction in the z plane, and may be either curved or straight.
• In most normal applications, the barrier is drawn
along the negative real axis.
o The branch is termed the “principle branch”.
o The barrier is termed the “branch cut”.
o For the example given in the previous slide, the region,
the barrier confines the function to the region in
which the argument of z is within the range - < < .
Engineering Mathematics III
Engineering Mathematics III
Zeros and Poles of order m
Consider a function f that is analytic at a point z0.
(From Sec. 40). 0
( )( ) ( 1 2 ) exist atznf n , , .... z
0
0
0
0
)
)
( )
( )
( 0
'( 0
( 1) 0
( ) 0
If f z ,
f z :
m- f z
m f z
Then f is said to have a zero of order m at z0.
0
0
)
.
Lemma: ( ) ( ( )
analytic and non-zero at
m f z z z g z
z
Engineering Mathematics III
Example.
0
(
( 1) / when 0 is analytic at 0.
1 when 0
( ) ( 1)
22 1 )2! 3!
has a zero of order 2 at 0
( )ze z z
zz
zf z z e
z z z ......
m z
g z
Thm. Functions p and q are analytic at z0, and 0
( ) 0.p z
If q has a zero of order m at z0, then
( )( )
p zq z
has a pole of order m there.
0
0
)
)
( ) ( ( )
analytic and non zero
( ) ( ) ( )( ) (
mq z z z g z
p z p z /g zmq z z z
Engineering Mathematics III
Example. 0
1( ) has a pole of order 2 at 0( 1)z
f z zz e
Corollary: Let two functions p and q be analytic at a point z0.
0 0 0) ) ) If ( 0 ( 0 and ( 0 p z , q z , q' z
0
0
0 0
)Re
)
then is a simple pole of and
(( ) ( ) (z z
s
p(z)zq(z)
p zp z q z q' z
Pf: 0 0
0
( ) ( ) ( ), ( ) is analytic ard non zero at
( ) ( )/ ( )( )
q z z z g z g z z
p z p z g zz zq z
Form Theorem in sec 56, 0
0 0
0
0
)Res
)
)
)
(
(
(
(
z z
p zp(z) q(z) g z
p z
q' z
0 0) )But ( ( g z q' z
Engineering Mathematics III
Example. cos( ) cotsin
( ) cos sin both entire
zf z z z
p z z, q(z) z
The singularities of f(z) occur at zeros of q, or
( 0 1 2 )z n n , , , ...
Since ( ) ( 1) 0 ( ) 0 and ( ) 1 0n n p nπ , q nπ , q' nπ (- )
each singular point of is a simple pole,
( ) ( 1)with residue 1( ) ( 1)
z nπ fnp nπBn nq' nπ
try tan z