Introduction to Complex Arguments
A complex number ( Z = x + yi ) can be expressed in polar form using Euler's formula as ( Z = r e^{i\theta} ), where:
- ( r = \sqrt{x^2 + y^2} ), the distance from the origin
- ( \theta ), the argument or angle relative to the positive real axis
For example, for ( a = 1 + i ), the argument ( \theta = \pi/4 ) radians because ( \theta = \arctan(1/1) ,). This ties closely with Introduction to Functions of Complex Variables and Holomorphicity, which discusses foundational complex function properties.
Meromorphic Functions Explained
- A function ( f(z) ) is meromorphic in a domain ( D ) if it is holomorphic everywhere except at a finite number of poles within ( D ).
- They occupy a middle ground between fully analytic functions (holomorphic everywhere) and those with discontinuities over an entire region.
- Meromorphic functions are differentiable everywhere in the domain except at isolated singularities (poles). Understanding these poles and their residues is elaborated in Mastering Residue Calculation Techniques in Complex Variables.
Concept of Winding Number
-
The winding number measures how many times a closed contour ( C ) wraps around a given point, typically the origin.
-
It is calculated by tracking how the argument of points on ( C ) changes as you traverse the contour once completely.
-
The winding number ( n ) is given by:
[ n = \frac{1}{2\pi} (\text{change in argument of } Z) = \frac{\theta_{final} - \theta_{initial}}{2\pi} ]
Examples:
- If traveling once around ( C ) results in a total argument change of ( 4\pi ) radians, the winding number is ( 2 ).
- For a contour enclosing the origin once with an argument change of ( 2\pi ), the winding number is ( 1 ).
- If the contour does not enclose the origin, the winding number is ( 0 ).
These computations often utilize Jordan curves, and their justification can be explored in Understanding Jordan's Lemma in Complex Variables: Statement and Proof.
Winding Number of Complex Functions
-
Given ( w = f(z) ), mapping a contour ( C ) in the ( z )-plane to a contour ( \Gamma ) in the ( w )-plane.
-
If ( f(z) ) has no zeros or poles on ( C ), then ( \Gamma ) is a closed contour that does not cross the origin in the ( w )-plane.
-
The winding number ( N ) of ( \Gamma ) around the origin is:
[ N = \frac{1}{2\pi} \Delta \arg(f(z)) ]
-
Here, ( \Delta \arg(f(z)) ) is the net change in the argument of ( f(z) ) as ( z ) traverses ( C ) once.
-
( N ) can be positive, negative, or zero based on the direction and whether ( \Gamma ) encircles the origin.
This concept plays a critical role when evaluating Computing Improper Fourier Integrals Using Complex Analysis Techniques which utilizes contour integrations and winding numbers.
Key Takeaways
- The argument is fundamental to understanding complex numbers in polar form.
- Meromorphic functions generalize holomorphic functions by allowing finite isolated poles.
- Winding number describes topological properties of contours and mappings in the complex plane, essential in complex analysis and related theorems such as the Argument Principle.
- Calculating winding numbers involves assessing the total change in argument around closed paths.
Understanding these concepts enriches the mathematical toolkit for analyzing complex functions and their behaviors on closed contours.
greeting students and welcome back to another lesson on complex variables in this video we're gonna discuss the ideas
of winding number and meromorphic functions before we do all that though let's set our foundation by discussing
the concept of argument say I have a complex number Z given by X plus y I where x and y are real numbers and I is
the imaginary number the square root of negative one as we've done in previous videos we can write this complex number
using polar coordinates with X equal to R cosine theta and y equal to R sine theta R by the way is the distance from
the origin and theta is the angle relative to the positive x-axis or the positive real axis on the complex plane
if we express x and y using polar coordinates we can use Euler's formula to write Z as R times e to the I theta
we've done this many times so far in our series so if you've been following along you should be no stranger to this the
argument of the complex number Z denoted by RZ is then equal to the theta in the polar representation of that complex
number it's the angle that a complex number makes relative to the positive real axis as a quick example consider
the complex number a equals 1 plus I we can convert a to polar form the distance from the origin is just the square root
of x squared plus y squared which is the square root of 1 squared plus 1 squared which is the square root of 2 the angle
of a relative to the positive real axis is just the inverse tangent of 1 over 1 which is 45 degrees or PI over 4
therefore the argument of a is just PI over 4 let's now move to the idea of a meromorphic function a complex function
f of Z is meromorphic in the domain D if it's holomorphic everywhere indeed that is if it's complex differentiable in D
except for a finite number of points which would be the poles of F of Z in D so a meromorphic function is allowed to
have a finite number of discontinuities it just can't be discontinuous over an entire region and D because that would
be infinite discontinuities intuitively you can think of meromorphic functions as lying on the middle of a spectrum of
continuity this left end of the spectrum is for the angry discontinuous functions that are discontinuous over entire
regions this right end is for the nice analytic functions that are continuous and differentiable everywhere and this
middle is occupied by meromorphic functions which aren't quite continuous everywhere in the domain D but only have
a finite number of discontinuities so they're not terribly discontinuous either the next concept I'd like to
discuss is the winding number say I have a closed contour C in the complex plane that looks something like this by closed
I mean that the contour completely encloses an area how many times does this contour go around the origin well
just from looking at it you can tell it goes around twice but let's explain why we'll start over here on this right end
of the contour which I'll call Z naught the angle Z naught makes with respect to the positive real axis is 0 so we
basically start with an argument of 0 now from Z naught we'll go around the curve until we end back up at Z naught
once we get to this left end for instance our argument will be PI once we get back to the right and our argument
will be 2 pi we keep going until we get back to Z naught in which case we'll be 3 PI over here and then 4 PI once we get
back to Z naught throughout this journey we've gone 720 degrees or 4 pi radians around the origin if we divide this
traveled angle by 2 pi we end up with 2 which is the number of times we've gone around the origin or in other words the
winding number of C with respect to the origin let me go over how we found this winding number we started at Z naught
whose argument was 0 we then traveled along the curve until we ended up in our original position
however our new argument theta naught prime was 4 pi because we had traversed 4 pi radians around the origin to end
back up in our original position if we divide this change in the argument by 2 pi we get the winding number of C with
respect to the origin let's find the winding number in a couple more cases say I have the same curve again but I
translate it over such that the origin is inside the curve here the winding number of C in this case around
the origin is just one here's why let's start at this point again which I'll call z1 now with the starting argument
of zero if I travel around this curve to end back up at z1 I'll first get to this left end which has an argument of pi
then I'll get to this right end which has an argument of 2 pi since we've gone around once then I travel around through
an angle slightly greater than 2 pi to end up at this point which still has an argument of 2 pi because we're still on
the right side of the origin and we haven't gone around it finally I end up back at z1 which again has an argument
of 2 pi because still I haven't gone around the origin so I go from an initial argument of 0 to a final
argument of 2 pi the winding number is therefore 1 another way to think about it is that I only circle the origin once
I only go around at 360 degrees or 2 pi radians as I traverse this curve I go around this point twice inside this
little loop over here but I only go around the origin once therefore the change in the argument of Z 1 when I end
up back at Z 1 is only 2 pi which makes the winding number equal to 1 finally say I have the same curve but now the
origin is completely outside it the winding number of C around the origin now is 0 to explain let me start at this
point which I'll call z2 which has a starting argument of 0 if I travel around this curve to end a back at z2
I'll first get to this left end by traveling through an angle slightly greater than 0 but once I get to the
left end I'll end up at a point which has an argument of 0 then I'll get to this right end by travelling through an
angle slightly less than 0 but again the argument at this point at this right end will be 0 I'll repeat the process to end
up at z2 and my argument there will still be 0 because I still haven't gone around the origin so I go from an
initial argument of 0 to a final argument of 0 the winding number is therefore 0 I don't encircle the origin
at all i encircle the points inside the curve but I do not encircle or why around the origin as a result the
winding number of this curve with respect to the origin is zero the take-home message of all this is that
the winding number with respect to a certain point of a curve is the number of times we go around that point as we
travel the curve once it should make sense to you that the winding number must be an integer you can only go
around something an integer number of times now there's an additional concept of the winding number of complex
functions instead of the more rudimentary winding number of a curve that we've discussed and I'll go over
that concept now suppose I have a function W equals f of Z that map's a complex number Z to another complex
number W suppose also that I make up a contour C and that f of z has no poles on this contour and it has no zeros on
this contour if we go around the contour C and evaluate F of Z at every point on that contour I can plot all the
resulting values of f of Z in the W plane the complex plane for the variable W these resulting values will form
another closed contour that we'll call capital gamma in the complex plane for W the reason it's a closed contour is that
when we start at some value Z naught and go around the closed curve C we will eventually return to Z naught because
we're on a closed curve now since W is a function of Z W cannot take on multiple values at Z naught so if Z starts at Z
naught and W starts at W naught then when Z returns to Z naught after traversing the closed curve C W will
also return to W naught this means that the curve gamma in the W plane that's basically the image of C under the
function f of Z this curve gamma must also be a closed curve in addition the curve gamma will not cross the origin in
the W plane the reason for this is that according to what we specified earlier F of Z has no zeros on the contour C
therefore gamma which is the image of C under the function f of Z does not cross the zero point it does not cross the
origin suppose that the polar representation of W is given by wrote times e to the I Phi where Rho is the
distance of the W point from the origin and the W plane and Phi is the angle relative to the positive real axis in
the W plane in that case the argument of W as we just defined would then just be Phi now in order to describe the idea of
a winding number of a function we're gonna start at Z naught on the curve C which corresponds to W naught equals f
of Z naught in the W plane the argument of this initial point w naught is Phi naught let's say now starting from Z
naught we're gonna traverse the entire contour C in the counterclockwise direction and in addition we're gonna
follow the corresponding points W on the image curve gamma at one stage while we're traversing the contour C we will
end up back at Z naught and at the same time we'll be back at W naught on the contour gamma once we've completed the
single traversal of the contour C we obviously end up back at W naught however the argument of W naught isn't
necessarily gonna be fine on its going to be a different angle Phi 1 which is Phi naught plus 2 pi times thumb integer
N and why do I say it's some integer N and not just fine not plus 2 pi it's because even though I go around the
contour C just once that single traversal of C might have correspondent to multiple traversals around the
contour gamma so when going around C once I could have actually gone around the image of seeing multiple times and
that's what the N represents it makes things more general and this integer multiple n is called the winding number
of gamma with respect to the origin in the W plane it tells us how many times gamma whines or circles around the
origin according to this equation above the winding number of a closed contour gamma is then 1 over 2 pi times pi 1
minus Phi naught now Phi 1 minus Phi naught can also be written as the change in the argument of F of Z or the
argument of W as we make a full traversal of the closed contour C therefore this is the equation which
defines our winding number n of a closed contour gamma which is the image of the closed contour see
just a couple things to note about the winding number one is that of gamma lines around the origin in a
counterclockwise direction as we wind around the contour C in a counterclockwise direction as well it's
winding number will be positive if gamma winds in a clockwise direction as we go around C in a counterclockwise manner
the winding number will be negative and if gamma does not encircle the origin at all as we go around the curve C in a
counterclockwise manner if add up if gamma does not surround the origin then the winding number will be zero as I
discussed earlier with my example anyway that should do it for this 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 cron signing out
The argument of a complex number is the angle ( \theta ) it makes with the positive real axis in the complex plane. To calculate it, convert the complex number ( Z = x + yi ) into polar form ( r e^{i\theta} ), where ( r = \sqrt{x^2 + y^2} ) and ( \theta = \arctan(y/x) ). For example, for ( 1 + i ), the argument is ( \pi/4 ) radians since ( \arctan(1/1) = \pi/4 ).
A meromorphic function is holomorphic (complex differentiable) throughout a domain except at a finite number of isolated poles, which are singularities where the function's value can approach infinity. Unlike fully analytic functions, meromorphic functions allow these poles but are still differentiable everywhere else. Understanding the poles and their residues is key to analyzing such functions.
The winding number measures how many times a closed contour wraps around a specific point, usually the origin. It is calculated by measuring the total change in the argument of points along the contour as you traverse it once. Mathematically, ( n = \frac{1}{2\pi}(\theta_{final} - \theta_{initial}) ). For instance, if the argument changes by ( 4\pi ), the winding number is 2. This number indicates the topological relationship between the contour and the point it encircles.
When a complex function ( w = f(z) ) maps a contour ( C ) in the ( z )-plane to a contour ( \Gamma ) in the ( w )-plane, the winding number ( N ) of ( \Gamma ) around the origin reflects how many times ( f(z) ) encircles zero as ( z ) completes one traversal of ( C ). Provided ( f(z) ) has no zeros or poles on ( C ), ( N = \frac{1}{2\pi} \Delta \arg(f(z)) ) helps analyze properties like zeros, poles, and behavior of integrals in complex analysis.
Winding numbers quantify how function values wind around points in the complex plane, a concept central to the Argument Principle. This principle relates the winding number of ( f(z) ) around zero to the difference between the number of zeros and poles inside the contour. Thus, calculating winding numbers enables the determination of zeros and poles without explicitly finding them, aiding in integral evaluations and complex function analysis.
Yes, for a contour ( C ) that goes once around the origin counterclockwise, the argument changes by ( 2\pi ) radians, so the winding number is 1. If the contour loops twice around the origin, the argument change is ( 4\pi ), and the winding number is 2. Conversely, if the contour does not enclose the origin, the argument change is zero, yielding a winding number of 0. Tracking the argument increment as you traverse ( C ) provides the winding number.
Meromorphic functions often appear in complex integrals used to evaluate improper Fourier integrals via contour integration techniques. Winding numbers help determine how many times contours in the complex plane encircle singularities or zeros, which is essential for applying residue theorems correctly. This understanding enables accurate calculation of integrals that are otherwise difficult to solve using real-variable methods alone.
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