Introduction to the Argument Principle
The argument principle is a fundamental theorem in complex analysis that connects the behavior of a meromorphic function inside a closed contour to the change in the function’s argument on that contour. It should not be confused with informal debate tactics; instead, it deals with the argument (angle) of complex numbers.
Key Definitions and Assumptions
- Contour C: A positively oriented, simple closed curve that encloses an area without crossing itself.
- Function f(z): Meromorphic inside C, analytic and nonzero on C itself.
- Zeros and Poles: Let Z be the count of zeros and P the count of poles of f(z) inside C, both counted with multiplicities and orders.
Statement of the Argument Principle
The winding number of the image of contour C under f, which is the total change in the argument of f(z) as z moves along C, equals the number of zeros minus the number of poles of f inside C:
[ \text{Winding Number} = Z - P ]
For a deeper understanding of the winding number concept and its relation with meromorphic functions, see Understanding Winding Numbers and Meromorphic Functions in Complex Analysis.
Proof Overview
Integral Setup
Consider the contour integral:
[ \oint_C \frac{f'(z)}{f(z)} dz ]
This integral will be evaluated in two ways.
Parametrization and Argument Change
- Parametrize C by a real parameter t from a to b with z(t).
- Express f(z) in polar form as ( \rho(t) e^{i \phi(t)} ).
- Using the chain rule, rewrite the integral in terms of t, separating real and imaginary parts.
- The real part vanishes since the start and end points match, leaving:
[ \oint_C \frac{f'(z)}{f(z)} dz = i( \phi(b) - \phi(a) ) ]
where ( \phi(b) - \phi(a) ) is the total change in the argument around C.
Residue Theorem Application
- Express f(z) near zeros ( z_i ) as ( (z - z_i)^{m_i} g_i(z) ), where ( m_i ) is the multiplicity.
- The residue of ( \frac{f'(z)}{f(z)} ) at zero ( z_i ) equals its multiplicity ( m_i ).
- Similarly, express f(z) near poles ( z_j ) as ( \frac{h_j(z)}{(z - z_j)^{k_j}} ), ( k_j ) being the order.
- The residue at pole ( z_j ) is ( -k_j ).
- Summing residues for all zeros and poles inside C:
[ \sum_{i=1}^N m_i - \sum_{j=1}^P k_j = Z - P ]
- By the residue theorem:
[ \oint_C \frac{f'(z)}{f(z)} dz = 2 \pi i (Z - P) ]
For comprehensive details on applying the residue theorem in such contexts, consider reading Understanding the Residue Theorem in Complex Variables.
Conclusion
Equating the two expressions for the integral yields:
[ i \Delta \arg f(z) = 2 \pi i (Z - P) \implies Z - P = \frac{1}{2 \pi} \Delta \arg f(z) ]
Thus, the winding number (change in argument divided by ( 2\pi )) equals the difference between zeros and poles inside the contour, which is the argument principle.
This principle is essential in complex analysis for locating zeros and poles and understanding function behavior inside contours, with practical implications in engineering, physics, and applied mathematics.
To further expand on integral theorems and contours related to this topic, you might find Understanding Cauchy’s Theorem and Complex Integrals Explained useful.
greeting students and welcome back to another lesson on complex variables in this video we're gonna discuss and prove
the argument principle this should be confused with the argument principle of internet forums which states that the
person who comes up with the most ad hominem attacks wins the argument instead the argument principle and
complex variables is related to the argument of a complex number which I defined in a previous video let's say
that we have a positively oriented simple closed contour C by closed I mean that the contour completely encloses an
area and by simple I mean that the contour doesn't cross itself let's also say that f of Z is a meromorphic
function inside C and that it's analytic and nonzero on C so it's meromorphic inside C but it has no zeroes and no
poles on the curve C finally suppose that capital Z and P represent the number of zeros and poles of F of Z
inside C respectively I'll explain later what it means when it says counting multiplicities and orders
if all these assumptions are true then the winding number of f of Z as we traverse the image of the curve C is
equal to the number of zeros of F of Z inside C minus the number of poles of F of Z inside C counting multiplicities
and orders of course we'll start the proof by looking at the integral over C of F prime Z over f of Z to perform this
integral we'll use a parametric representation of C with the parameter T such that the curve C is represented by
Z of T where T is a real parameter that runs from A to B the limits on tv' a and the B are selected such that as T goes
from A to B the complex number Z encircles the closed contour C exactly once now using this parametric
representation our integral in Z becomes an integral in T from A to B note that the Z prime T is there because in order
to convert DZ to DT we have to use the chain rule where DZ is DZ over DT or Z prime T times DT as we've discussed
before when we take the complex number Z that's parameterised by T and run it through a function f we
yet another complex number W which is also dependent on T and just as the complex number Z has a polar
representation so too does the complex number W I'll write the polar representation of W as Rho times the
exponential of I times Phi in general because Z depends on the parameter T W also depends on the parameter T such
that in the polar representation W can be written as Rho of T times the exponential of I times Phi of T and
because W is just a function of Z we can write f of Z as this polar representation of W I'll call this
equation a let's go back up to our integral and put this numerator in terms of Rho and Phi we can write the F prime
as DF by DZ and we can write the Z prime as DZ by DT once we do that we can use the chain rule to write the whole
numerator as DF by DT using equation a I can differentiate F with respect to T quite easily
I'll get Rho prime of T times the exponential of I times Phi plus I times Rho of T times Phi prime of T times the
exponential of I times Phi if we plug this back into our integral here's what we'll get note that I've replaced the F
of Z in the denominator by an expression involving Rho and Phi that I got from Equation a we can then split up the
numerator and get two integrals as follows after we cancel the common terms we finally end up with these two
relatively simple integrals the first integral is just the natural log of Rho of T and the second integral is just I
times Phi of T of course the limits of a and B still apply now recall that the limits a and B were chosen such that as
T went from A to B we went around the closed contour C exactly once because we go around exactly once we end up at the
same place on C that we started at which means that if we go through the function W equals f of Z we will end up at the
same point on the W plane that we started at so if we started at W naught equals F of Z naught then as we go
around the curve C we'll end up back at Z naught and so therefore we will end up back at W naught in general
this new w-not point will have a different argument it's going to be shifted by some multiple of 2pi however
its distance from the origin when T equals B will be the same as its distance from the origin when T equals a
obviously because it's the same point that we end back up at therefore row if a equals row of B and we can cancel the
contribution of the natural log leaving us with Phi of B minus Phi of a as the answer to our integral but Phi of B
minus Phi of a is actually related to the winding number it's the change in the argument of F of Z as we make a full
traversal of the closed contour C we discuss this in a previous video where I define the winding number so in
conclusion the integral over the closed curve C of F prime of Z over F of Z is I times the change with respect to the
contour C of the argument of F of Z I'm going to call this equation 1 now the integral on the left hand side of
equation 1 can be evaluated using a second method let me show you how if we go back to the assumptions of this
theorem then F of Z is analytic and nonzero on the contour C but it's meromorphic inside the contour C well
suppose that f of z has n zeros inside the contour C that I'll denote by Zi where I is an index from 1 to n note
that I over here is an index and not the imaginary number I couldn't really use other letters because I was running out
so unfortunately I had to go with I even though it's a bit confusing with the notation we'll also suppose that F has P
poles inside the contour C that I'll denote by Z J where J varies from 1 to P remember we're allowed to have poles
inside the contour because f of Z is meromorphic inside the contour so it has a finite number of discontinuities
inside but we can't have poles on the contour let's start by considering the I at 0 of f of z zi if f has a 0 at z i
then we can take out the contribution of z i as a factor of z minus CI and write f as the following now in general the
factor z minus z i that's making f 0 at z i appears i times and f it doesn't necessarily appear just once if we
completely take it out of F will be left with a function G of Z that no longer has zeros at Z I because we fully taken
away the contribution of Z I to making F 0 by the way this mi is called the multiplicity of the 0 Z I now if we
differentiate this F of Z that's written in terms of Zi here's what we'll get using the
product rule and if we take the ratio F prime of Z over F of Z when they're expressed in terms of Z I will get em I
over z minus di plus G prime Z over G of Z after simplification I'll call this equation B since G prime of Z and G of Z
contain no zeros at Z I this ratio doesn't matter when it comes to determining the residue of F prime Z
over F of Z at Z I therefore it should be easy to conclude that F prime Z over F of Z has a residue at zi equal to the
multiplicity of zi in the function f of z which is just meye meye is the coefficient of z minus zi to the
negative 1 in f prime Z over F of Z therefore it's the residue of F prime Z over f of z ad zi by definition in the
same vein let's consider the jf pole of f of z Z J if F has a pole at Z J then we can take out the contribution of Z J
as a factor of Z minus Z J from the denominator and write f as the following now in general the factor Z minus Z J
that's acting as the pole of F at ZJ appears KJ x and f it doesn't necessarily appear just once if we
completely take it out of F we'll be left with the function H of Z that no longer has poles at cj because we've
fully taken away the contribution of Z J as a pole by the way this KJ is called the order of the pole Z J next if we
differentiate this F of Z that's written in terms of Z J here's what we'll get using the product rule and if we take
the ratio F prime Z over F of Z when they're expressed in terms of Z J we'll get negative K J over Z minus Z J plus h
prime Z over H of Z after simplification I'll call this equation C since H prime of Z and H of Z contain no zeroes or
poles at Z J this ratio doesn't matter when it comes to determining the residue of F prime Z over F of Z at Z J
therefore it should be easy to conclude that F prime Z over F of Z has a residue zj equal to the negative of the order of
the pole ZJ in the function f of Z which is just negative k j- k j is the coefficient of z minus z j to the
negative 1 therefore it's the residue of F prime Z over F of Z at Z J by definition now when we perform the
integration of F prime Z over F of Z over the contour C we can use the residues theorem to say that this
integral is equal to 2 pi I times the sum of the residues of F prime Z over F of Z and in order to evaluate the sum of
the residues we'll need to add all the residues due to the contributions of the zeroes of F inside the contour C this
means we'll need to add all the residues corresponding to the zeroes of F of Z which means that we'll need to add all
the multiplicities mi of the end zeroes in addition we'll need to add all the residues due to the contribution of the
poles of F of Z inside the contour C which means we'll need to add all the negative K J's of the people´s therefore
the total sum of the residues of F prime Z over F of Z inside the contour C can be written as the sum from I equals 1 to
N of mi minus the sum from J equals 1 to P of KJ I'll denote the first summation by capital Z which is the number of
zeros of F inside the contour C counting multiplicities so I'm adding the multiplicities of all the zeros together
in addition I'll denote the second summation by P which is the number of poles of F inside the contour C counting
the orders so I'm adding the orders of all the poles together so the integral of F prime Z over F of Z over the closed
contour C is 2 pi I times Capital Z minus P I'll call this equation 2 now if I go up to equation 1 then you can see
that the left-hand sides of both equations 1 & 2 are equal this means that the right-hand sides must be equal
by the transitive property which means that I times the change in the argument of F of Z with respect to the traversal
of the curve C equals 2 pi I times Capital Z minus P if we simplify this we'll finally end up with capital Z
minus P equaling the wine of F of Z as we traverse the curve C where capital Z and P are the zeros and
poles of F of Z inside the curve C counting multiplicities and orders and this is what we needed to prove the
argument principle so that's it I'd like to thank the following patrons for supporting me at the five-dollar level
or higher and if you enjoyed the video feel free to like and subscribe this is the Faculty of Khan signing out
To apply the argument principle, first ensure the function is meromorphic inside and analytic on the contour. Then, compute the total change in the argument of the function values along the contour. Dividing this change by 2π gives the difference between the number of zeros and poles inside. This method helps locate zeros and poles without explicitly solving the function.
The argument principle states that the total change in the argument (angle) of a meromorphic function as you traverse a closed contour equals 2π times the difference between the number of zeros and poles inside that contour, counted with multiplicities. In other words, the winding number of the image of the contour under the function equals the number of zeros minus the number of poles enclosed.
The function f'(z)/f(z) has simple poles at the zeros and poles of f(z) with residues equal to their multiplicities (positive for zeros, negative for poles). By applying the residue theorem, the contour integral of f'(z)/f(z) sums these residues, thus equating to 2πi times the difference between the number of zeros and poles inside the contour.
The winding number measures how many times the image of the contour under the function wraps around the origin in the complex plane. In the argument principle, it equals the change in the function's argument divided by 2π, which directly corresponds to the difference between the zeros and poles inside the contour, linking geometric behavior to analytic properties.
Multiplicity of a zero refers to how many times the function equals zero at that point, while the order of a pole indicates the degree of the function's singularity. Both affect the residues of f'(z)/f(z) and thus contribute proportionally to the integral; the argument principle counts zeros and poles with their multiplicities and orders, providing an accurate net count inside the contour.
Parametrizing the contour with a real parameter allows expressing the function in polar form and applying the chain rule to rewrite the integral of f'(z)/f(z). This splits the integral into real and imaginary parts, showing that the integral corresponds to the total change in the argument of the function around the contour, which is crucial for establishing the argument principle.
The argument principle is key in determining the location and count of zeros and poles of complex functions, which is essential in engineering for system stability analysis, in physics for wave phenomena, and in applied mathematics for solving complex equations and integral evaluations. It provides a powerful tool without needing explicit solutions.
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