SolitaryRoad.com

Website owner: James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

Zeros, poles, Argument principle, Rouche’s theorem

Zeros and poles

Let f(z) be analytic inside and on a simple closed curve C, except for possibly a finite number of poles in the region R inside C. Consider the curve that is traced out in the w plane by f(z) as the point z goes around C once in the positive direction. As z goes around C, f(z) describes some closed curve. See Fig. 1. How f(z) behaves depends on the number of zeros and poles of f(z) in R.

Case 1. There are no poles in R. Suppose f(z) has one simple zero in R. Then as z goes around C once in the positive direction, f(z) will wind around the origin once in the positive (counterclockwise) direction. Suppose there are two simple zeros in R. Then as z goes around C once in the positive direction, f(z) will wind around the origin twice in the positive direction. Suppose there is one zero of order three in R. Then as z goes around C once in the positive direction, f(z) will wind around the origin three times in the positive direction.

Case 2. There are no zeros in R. Let f(z) have one simple pole in R. Then as z goes around C once in the positive direction, f(z) will wind around the origin once in the negative (clockwise) direction. If there is one pole of order 3 in R, then as z goes around C, f(z) will wind around the origin three times in the negative direction. If there are 3 simple poles in R, then as z goes around C, f(z) will wind around the origin three times in the negative direction.

Case 3. There are both zeros and poles in R. Suppose f(z) has one simple zero and two simple poles in R. Then as z goes around C once in the positive direction, f(z) will wind around the origin once in the negative direction. Suppose f(z) has three simple zeros and one simple pole in R. Then as z goes around C once in the positive direction, f(z) will wind around the origin twice in the positive direction.

Thus we see that as point z makes one trip around C in the positive direction, f(z) makes n trips around the origin where n is given by the total number of zeros within C minus the total number of poles within C. If n is negative it indicates trips in the negative direction. A zero (or pole) of order m is counted as m zeros (or poles).

How f(z) behaves as z goes around C can also be expressed in terms of the change in the
argument of f(z), Δ arg f(z), when z makes one trip about C. If there is a single zero and no
poles, Δ arg f(z) = 2π. If there are three zeros and no poles, Δ_{C} arg f(z) = 3 ·2π = 6π, etc.

Let us now consider an example that will give insight into what is happening. Let f(z) be the polynomial

1) f(z) = a_{0}z^{n} + a_{1}z^{n-1} + ... + a_{n} = a_{0}(z - z_{1})(z - z_{2}) ... (z - z_{n})

which we have decomposed into linear factors. The numbers z_{1}, z_{2}, ... , z_{n } correspond to the
zeros of f(z).

From the formula

we know the product of several complex numbers is equal to the sum of the arguments of the factors. Thus

2) arg f(z) = arg a_{0} + arg (z - z_{1}) + arg (z - z_{2}) + ........ + arg (z - z_{n}) .

Let Δ arg f(z) be the change in the argument of f(z) when z makes one trip about C. Then Δ arg f(z) will be equal to 2π times the number of times the point f(z) winds around the origin. Now it is clear that when z makes one trip about C

3) Δ arg f(z) = Δ arg a_{0} + Δ arg (z - z_{1}) + Δ arg (z
- z_{2}) + ........ + Δ arg (z - z_{n}) .

Now Δ arg a_{0} = 0 since a_{0} is a constant. We note that z
- z_{1} corresponds to a vector extending from point z_{1} to
point z. Let us assume that z_{1} lies in the interior of the
region. See Fig. 2. From the figure it is clear that as
the point z goes around the curve C the vector z - z_{1}
makes a complete revolution about its initial point.
Consequently Δ arg (z - z_{1}) = 2π. Let us now assume
that point z_{2} lies outside the region as shown in the figure. In this case the vector z - z_{2} moves
from side to side and returns to its original position without making a revolution about its initial
point. Thus Δ arg (z - z_{1}) = 0. We can reason the same way about all the roots. Thus Δ arg f(z)
is equal to 2π times the number of roots of f(z) lying in the interior of the region. And the
number of roots inside the region is equal to the number of times the point f(z) winds around the
origin.

We just considered a function that had zeros and no poles. Let us now consider one with poles. Consider the function

Here q_{1}, q_{2}, ... , q_{m} are the zeros and p_{1}, p_{2}, ... , p_{n } the poles of f(z). Then

5) arg f(z) = arg a_{0} + arg (z - q_{1}) + arg (z - q_{2}) + ........ + arg (z - q_{n})

- arg b_{0} - arg (z - p_{1}) - arg (z - p_{2}) - ........ - arg (z - p_{n})

When z makes one trip about C, Δ arg f(z) is given by

6) Δ arg f(z) = Δ arg a_{0} + Δ arg (z - q_{1}) + Δ arg (z - q_{2}) + ........ + Δ arg (z - q_{n})

- Δ arg b_{0} - Δ arg (z - p_{1}) - Δ arg (z - p_{2}) - ........ - Δ arg (z - p_{n})

We now reason exactly as we did in the previous example. If the pole p_{1} is inside C, then as the
point z goes around the curve C the vector z - p_{1} makes a complete revolution about its initial
point. Consequently Δ arg (z - p_{1}) = 2π. In 6) this increment has a negative sign attached to it,
so it is subtracted from the total for Δ arg f(z). All zeros and poles lying inside the region
contribute to the total for Δ arg f(z). Contributions from the zeros are added and contributions
from the poles are subtracted. Zeros and poles lying outside the region don’t contribute. Thus Δ
arg f(z) is equal to 2π(N - P) where N is the number of zeros and P is the number of poles inside
C. The number of times the point f(z) winds around the origin is given by N - P.

Theorem 1. If f(z) is analytic inside and on a simple, closed curve C , except for a pole of order p at z = a inside C, then

If f(z) is analytic inside and on a simple, closed curve C , except for a zero of order n at z = a inside C, then

Theorem 2. Let f(z) be analytic inside and on a simple, closed curve C , except for j poles of
orders m_{1}, m_{2}, ... m_{j} at points α_{1}, α_{2}, ..., α_{j} and k zeros of orders n_{1}, n_{2}, ... n_{k} at points β_{1}, β_{2}, ..., β_{k},
all located in the region inside C. Suppose that f(z)
0 on C. Then

If, in the above theorem, we let P = m_{1} + m_{2} + ... + m_{j} and Z = n_{1} + n_{2} + ... + n_{k},

then

Argument principle. Let f(z) be analytic inside and on a simple closed curve C, except for a finite number of poles inside C. Let Z and P be the number of zeros and poles (multiplicities counted, each zero or pole counted a number of times equal to its order) inside C. Let Δ arg f(z) be the change in arg f(z) as z makes one trip about C in the positive (counterclockwise) direction. Then

Δ arg f(z) = 2π (Z - P)

Theorem 3. Let C be a simple, closed curve and R be the open region interior to C. Let f and g be functions analytic on C and its interior R, except for a finite number of zeros and poles in its interior, and satisfying | f | > | g | on C. Then the difference between the number of zeros and poles of f in R (multiplicities counted, each zero or pole counted a number of times equal to its order) is equal to the difference between the number of zeros and poles of f + g in R.

Rouche’s theorem. If f(z) and g(z) are analytic inside and on a simple closed curve C and if |g(z)| < |f(z)| on C, then f(z) + g(z) and f(z) have the same number of zeros inside C.

References

Mathematics, Its Content, Method and Meaning. Vol. I

Wylie. Advanced Engineering Mathematics

Hauser. Complex Variables with Physical Applications

More from SolitaryRoad.com:

Jesus Christ and His Teachings

Way of enlightenment, wisdom, and understanding

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

On Self-sufficient Country Living, Homesteading

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

Theory on the Formation of Character

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-control, self-restraint, self-discipline basic to so much in life

[ Home ] [ Up ] [ Info ] [ Mail ]