Introduction to Rouché's Theorem
Rouché's theorem is a fundamental result in complex analysis relating the zeros of two analytic functions within a simple closed contour.
Statement of Rouché's Theorem
- Let C be a simple closed contour in the complex plane.
- Suppose F and G are analytic inside and on C with no poles.
- If |F(z)| > |G(z)| holds for every point z on the contour C, then F and F + G have the same number of zeros inside C, counting multiplicities.
Proof Outline of Rouché's Theorem
- Since |F(z)| > |G(z)| on C, F(z) has no zeros on the contour (its modulus can't be zero).
- Similarly, F + G has no zeros on C by applying the triangle inequality to show |F + G| > 0.
- Using the argument principle, the number of zeros minus poles inside C equals the normalized change in argument over C.
- Both F and F+G are analytic and pole-free inside C, making pole counts zero.
- Factor the argument of F + G as the sum of arguments of F and 1 + (G / F).
- Since |G(z)/F(z)| < 1 on C, the function H(z) = 1 + (G(z)/F(z)) is a closed contour around 1 with radius less than 1, thus it doesn't wind around the origin, and its winding number is zero.
- This implies the change in argument of H(z) is zero.
- Therefore, the zeros of F and F + G inside C match in count and multiplicity.
Application: Proving the Fundamental Theorem of Algebra
The fundamental theorem states that any polynomial P(z) of degree n with complex coefficients has exactly n roots (counting multiplicities) in the complex plane.
Proof Steps Using Rouché's Theorem
- Decompose the polynomial: let F(z) = a_n z^n (leading term), and G(z) = remaining lower degree terms.
- Define C as a circle of very large radius R centered at the origin.
- On C, |F(z)| = |a_n| R^n dominates |G(z)|, which is bounded by the sum of magnitudes of coefficients times powers of R less than n.
- For sufficiently large R, |F(z)| > |G(z)| on C.
- By Rouché's theorem, P(z) = F(z) + G(z) and F(z) have the same number of zeros inside C.
- F(z) = a_n z^n has n zeros at the origin (with multiplicity n).
- Hence, P(z) has n zeros inside the contour, proving the fundamental theorem of algebra.
Key Takeaways
- Rouché's theorem provides a powerful technique to compare zeros of complex functions inside contours.
- The argument principle underpins the proof by linking zeros and poles to the change in argument.
- Choosing a sufficiently large contour is crucial for applying Rouché's theorem to polynomials.
- The fundamental theorem of algebra follows naturally from these complex analysis tools.
Additional Notes
- Understanding the modulus and argument properties of complex numbers aids in grasping contour behavior.
- The winding number concept ensures functions do not wrap around points, impacting zeros count.
- This proof strategy is widely applicable in complex variable problems and analytic function theory.
For a deeper foundation, consider reviewing Introduction to Functions of Complex Variables and Holomorphicity to solidify your grasp of analytic functions and their properties.
greeting students and welcome back to another lesson on complex variables in this video we're gonna prove Boucher's
theorem and then use rachets theorem to prove the fundamental theorem of algebra rachets theorem follows quite neatly
from the argument principle which we discussed in the previous video the statement of rachets theorem goes
something like this suppose C is a simple closed contour on the complex plane suppose also that I
have two complex functions F and G that are both analytic inside C and on C so f and G don't have any poles in addition
suppose that the magnitude or modulus of F is greater than the magnitude or modulus of G at each point on the curve
C if these conditions hold then F of C and F of Z plus G of Z have the same number of zeros counting multiplicities
inside the closed contour C will begin the proof by considering the modulus of G of Z on the closed contour C because
of the definition of the modulus of the complex number the modulus of G of Z must be greater than or equal to 0 it's
a magnitude it obviously can't be negative now because the modulus of F of Z is greater than the modulus of G of Z
on the curve C according to this statement in the theorem we can add this additional inequality to the front of
this expression the implication of this inequality is that the modulus of f of z is greater than 0 on the curve C which
means that F of Z is nonzero on the curve C it has no zeros if it did have zeros then its modulus would not be
positive everywhere on the curve C because the modulus of a 0 would be 0 in addition since the modulus of f of z is
positive and greater than the modulus of G of Z we can move this modulus of G of Z to the left hand side of this
inequality and say that the modulus of f of z minus the modulus of G of Z is greater than 0 now there's a theorem in
complex analysis that the modulus of the sum of two complex numbers is greater than or equal to the absolute value of
the difference of their moduli if we apply this theorem to F and G here's what we'll get
since the modulus of the sum of F and G is now greater than zero according to this inequality we can also say that the
function f plus G also has no zeros on C using the same logic we use for the function f alone now that we've
established that f of Z and F of Z plus G of Z have no zeros on the contour C then combined with the fact that F and G
are analytic inside and on C as well as the fact that C is a simple closed contour we've satisfied all the
prerequisites for the argument principle this means that we can apply the argument principle to F and F plus G if
we apply it to F then we'll get Z F minus P F equals 1 over 2 pi times the change in the argument of F of Z as we
traverse the curve C if we apply it to f of Z plus G of Z we'll get Z FG minus p FG equals 1 over 2 pi times the change
in the argument of F plus G over the curve C I mentioned earlier in the statement of rachets theorem that F and
G are both analytic inside C which means that they have no poles inside C as a result these Peas are zeros and I can
cancel them from these two equations f has no poles G has no poles so F plus G obviously will not have any poles let's
take the change in the argument of F plus G and analyze it over on the side we can start by factorizing the F within
the argument expression now the argument of the product of f of Z and 1 plus G of Z over F of Z is just the sum of their
individual arguments now let me illustrate why this is the case on the side let's say I had a complex number Z
1 with an argument theta 1 and another complex number Z 2 with an argument Theda 2 now if I multiply Z 1 and Z 2
their argument will consist of theta 1 plus theta 2 because of the property of powers that powers with the same base
that multiplied are added together in the exponent this property that the argument of Z 1 times Z 2 is the
argument of Z 1 plus the argument of Z 2 is what allows us to split up the functions in the argument up here let's
take this equation and multiply it by 100 two pi we do that here's what we'll get
now the expression on the left is just equal to Z F G and the first expression on the right is just equal to Z F
according to the argument principle we applied above all that's left is dealing with the winding number of 1 plus G of Z
over F of Z and I'm going to denote this function by H of Z we already know that on the contour C the modulus of G of Z
is less than the modulus of F of Z from the original assumptions of the theorem this means that if I take the modulus of
G of Z over F of Z I'll end up with an answer less than 1 and since the modulus of G of Z over f of Z which is the same
as the modulus of the ratio of G of Z and F of Z since this modulus is less than 1 it follows from the definition of
H of Z that the modulus of H of Z minus 1 is also less than 1 so if I were to draw the contour representing W equals H
of Z in the complex plane then that contour would be centered at W equals 1 and because the modulus of all the
complex numbers on that contour is less than 1 the contour representing H of Z will never encircle the origin because
it will never deviate a distance greater than 1 from the center at W equals 1 and since this contour will never encircle
the origin its winding number up here must be 0 remember if a contour does not encircle the origin its winding number
is 0 that's the property of winding numbers and if we apply this we'll leave with the following expression relating
the zeroes of F inside C to the zeroes of F plus G inside see that the zeroes of F plus G counting multiplicities
inside C equals the zeroes of F inside C and this is what we needed to prove with rachets theorem that F and F plus G have
the same number of zeroes inside the contour C counting multiplicities so this final statement should complete our
proof let's solve an example problem involving rucious theorem and this is a pretty involved example because it
involves using rachets theorem to prove the fundamental theorem of algebra we'll start off this example by first
stating the fundamental theorem of algebra which says that if I have a polynomial P of Z of degree
with complex coefficients then that polynomial has n complex roots or n complex zeros counting the
multiplicities note that n here is a positive integer to prove this theorem we'll start by setting F of C to the
last term of this polynomial and G of Z to every other term before it will let our simple closed contour C be a circle
with a very large radius R on this circle the modulus of the function f of Z is just the modulus of a n times
capital R to the power n now the number of zeroes of F of Z inside this contour C is n counting multiplicity of course
because since f of Z is a n times Z to the N f of C has one 0 at Z equals 0 but this 0 it has a multiplicity of n
because Z is raised to the power n therefore the number of zeros of F of Z inside the contour C is just n taking
multiplicity into account let's now look at the modulus of G of Z on the contour C the magnitude of G of Z is the modulus
of the sum of all these terms one of the properties of the modulus is that the modulus of the sum of a bunch of complex
numbers must be less than or equal to the sum of the individual moduli we can split up these individual moduli into
the separate moduli products then we can plug in the fact that on the contour c the modulus of Z is r when we do that
here's what we'll get for the modulus of G of Z let's perform another special operation let's multiply each of these
individual terms on the right by R raised to some power such that each term now multiplies R to the N minus 1 so for
instance this a naught term I'm going to multiply by R to the N minus 1 the a 1 term I'm going to multiply by R to the N
minus 2 and so on now if we do that so that each term now multiplies R to the N minus 1 we'll find
that our new right-hand side is now even greater than the older right hand side because R again is much much greater
than 1 so as a result this inequality will still hold what we'll do now is divide this inequality by the modulus of
F of Z on both the modulus of f of z is positive so that won't change the sign of our
inequality at all so we can just divide by the modulus of f of z without changing anything but we already know
that the modulus of f of z is the modulus of a n times R to the N so we can make the substitution but we'll only
do it on the right-hand side of this inequality we can then cancel the R's in the numerator and denominator on the
right and here's what we'll end up with now if R were large enough that it were greater than this expression the sum of
the magnitudes of the lower order coefficients divided by a n then this term on the right hand side of the
inequality would obviously be less than 1 so as a result by the transitive property the braciole the moduli of G of
Z and F of Z would be less than 1 which therefore means that the modulus of G of Z on the contour C would be less than
the modulus of F of Z on the contour C at this point we're ready to apply rachets theorem the first condition is
obviously satisfied C is just a very large circle so it's a simple closed contour the second condition of rachets
theorem is also satisfied because f and g are just simple polynomials so therefore they must be analytic inside
and on C they don't have any poles and the third condition is what we just proved that the modulus of F is greater
than the modulus of G on the contour C because all these conditions are satisfied we can apply rachets theorem
and say that the function f of Z has the same number of zeros as the function f of Z plus G of Z which is just the whole
polynomial P of C but how many zeros does f of z have well we showed that a couple of minutes ago f of Z has n zeros
inside C counting multiplicity as a result the polynomial P of Z also has n zeros inside C counting multiplicity and
this proves the fundamental theorem of algebra that a polynomial of degree n with complex coefficients has n roots in
the complex plane counting multiplicities of course our proof is contingent on the fact that we choose a
large enough contour C with a large enough radius R that this condition is satisfied however this shouldn't be a
problem the fundamental theorem of algebra doesn't restrict just to a particular region in the
complex flame when it comes to the roots pretty much the entirety of the complex plane is fair game which is why we can
be very liberal in how large we want our contour C to be anyway that should do it for the video 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
Rouché's theorem states that if two analytic functions F and G satisfy |F(z)| > |G(z)| on a simple closed contour C, then F and F + G have the same number of zeros inside C, counting multiplicities. It is applied in complex analysis to compare zeros of functions within contours, especially when one function dominates another in modulus on the boundary.
The argument principle links the number of zeros and poles of an analytic function inside a contour to the change in the argument of the function along that contour. In proving Rouché's theorem, it helps show that the winding number of the function H(z) = 1 + (G(z)/F(z)) is zero, leading to the conclusion that F and F + G have the same number of zeros inside the contour.
For a polynomial P(z), choosing a sufficiently large radius R ensures the leading term F(z) = a_n z^n dominates the lower degree terms G(z) on the contour |z|=R. This dominance (|F(z)| > |G(z)|) is key to applying Rouché's theorem, confirming that P(z) has as many zeros inside the contour as its degree n.
The proof decomposes a degree n polynomial P(z) into F(z) = a_n z^n and G(z) as the remaining terms. By picking a large circle C where |F(z)| > |G(z)|, Rouché's theorem states P and F have the same number of zeros inside C. Since F(z) has exactly n zeros (counting multiplicities) at the origin, P(z) must also have n zeros, proving the theorem.
H(z) is used to compare F and F + G by normalizing the perturbation G relative to F. Since |G(z)/F(z)| < 1 on the contour, H(z) forms a loop around 1 with radius less than 1, which does not enclose the origin. This means H(z) has zero winding number around zero, so the argument change contributed by H(z) is zero, preserving the count of zeros.
The modulus helps determine dominance of functions on contours, crucial for applying Rouché's theorem, while the winding number measures how many times a function's image wraps around a point, indicating zeros inside the contour. Together, they help analyze and count zeros of analytic functions within specified regions in the complex plane.
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