Understanding Jordan's Lemma in Complex Variables: Statement and Proof

greeting students and welcome back to my series on complex variables in this lesson I'm gonna talk about a pretty

famous concept in complex variables known as jordan's lemma i used to believe that it was pronounced jordan

but it's actually a french name so it's pronounced jordan anyway let's start by drawing what we need to properly state

jordan's lemma we have our complex plane with the real axis and the imaginary axis we've also got a smaller circle

with a radius R naught that's centered at the origin and we've got a larger semi circle with radius R that's also

centered at the origin where capital R is greater than capital R naught I'm going to label the semicircle as Z sub

capital R so now that we have everything we're ready to state jordan's lemma suppose we had a function f of z that

was analytic or holomorphic so differentiable everywhere in the upper half plane that was outside the circle

given by the magnitude of Z equals capital R naught so outside the smaller circle in the upper half plane suppose

also that as we just mentioned C sub R is a semicircle of radius capital R whose equation is given by Z equals

capital R times the exponential of I theta where theta lies between 0 and pi and capital R is greater than capital R

naught finally we'll suppose that for all points on the semicircle C R the function f of Z had an upper limit of M

sub capital R that approached 0 as the radius of the semicircle approached infinity if all of these statements

apply then jordan's lemma says that the contour integral of f of z times the exponential of i a z over the semicircle

in the anti-clockwise direction approaches 0 as capital R approaches infinity

note that a here is a positive constant now for this video we're gonna go ahead and prove this lemma the proof starts

out a bit more awkwardly because it doesn't seem very related to jordan's lemma but it will come together at the

end we're gonna start by graphing two functions on the Cartesian plane y equals sine theta and y equals 2 theta

over pi to theta over pi is just a straight line going through the origin that becomes

one at theta equals PI by two just like how sine theta becomes one at theta equals PI by two now you can see

from this graph that sine theta is actually greater than or equal to two theta over PI for theta between 0 and PI

by 2 this is actually important because it means that the negative exponential of capital R times sine theta is less

than or equal to the negative exponential of capital R times 2 theta over pi where capital R is obviously a

positive constant which it is because it's the radius of the semicircle and since the exponential of negative

capital R sine theta is less than the exponential of negative to capital R times theta over PI that means the

integrals of these quantities from 0 to PI by 2 also follow the same inequality order now you can actually compute this

ladder integral and when you do that you'll get PI over 2 times R times 1 minus the exponential of negative R

where R is greater than 0 and since R is greater than 0 you can safely say that this quantity is also less than PI over

2 R because the exponential of a negative number is a positive number that's less than 1 and if you take one

minus of that and multiply by PI over 2 R you end up with something that is in fact less than PI over 2 R so therefore

the integral from 0 to PI by 2 of the exponential of negative capital R times sine theta is less than or equal to PI

over 2 R where R is greater than 0 let's go back and look at the graph of sine beta we can see that sine theta is

symmetric when you reflect it about theta equals PI over 2 so what we can say is that the integral from 0 to PI of

the exponential of negative capital R times sine theta is twice the integral from 0 to PI by 2 therefore the integral

from 0 to PI would then be less than or equal to PI divided by 1 capital are using the same inequality that we wrote

above now this inequality actually has a name it's called jordan's inequality and jordan's

inequality is what we will use to prove jordan's lemma so let's continue the proof of Jordans

lemma we'll start by taking the integral from the limit that we want to ultimately prove which is the integral

over C R of F of C times the exponential of I times a times Z where C R is a semicircle on the upper half of the

complex plane with radius capital R and since we're integrating f of Z over a semicircular arc what we can do is

convert this contour integral into an integral over an interval by using the polar representation of complex numbers

and since the radius of the semicircle is a constant we can write the complex variable Z as capital R times the

exponential of I theta and then replace all the Z's in this contour integral by capital R times e to the I theta and if

we do that replacement we'll get this integral on the right the only remaining step is to replace

the DZ by D theta and adjust the limits on the integral accordingly now from the equation for Z above we know that DZ by

D theta is just I times capital R times the exponential of I theta this means that DZ is just I R times the

exponential of I theta D theta so we can replace the DZ in the integral to get the following

and since we're integrating over theta now we just have to change the region of integration from the contour CR to the

interval from 0 to PI let's now take the magnitude of this entire integral if you recall a theorem that I proved way back

in my ml inequality video links in the description you will remember that the magnitude of the integral is less than

or equal to the integral of the magnitude this theorem applies to a complex function that's piecewise

continuous over the interval where we're integrating and if you go back up to the assumptions we made before proving this

theorem we can see that f of Z is analytic everywhere outside the smaller circle of radius R naught therefore this

integral magnitude theorem can be safely applied to our situation let's recall another fact from complex analysis if I

have two numbers two complex numbers z1 and z2 then the magnitude of the product of Z 1 and Z 2 equals the product of

their magnitudes I can then apply this exact same logic to the expression inside the integral to split up all the

magnitudes for this last magnitude we can further split it up to get the magnitude of I which is just 1 the

magnitude of the positive real constant capital R which is just capital R and the magnitude of e to the I theta which

is also 1 now if you don't believe me on the last one you can apply the Euler formula to the exponential of I theta

you'll get cosine theta plus I sine theta and the magnitude of that is just the square root of cosine squared plus

sine squared which is just 1 now what about the exponential term how do we deal with this let's take it aside

and make some more room now the exponential of I theta inside the exponential can be expanded to cosine

and sine using the Euler formula we can now split up the exponential because there's a plus in the middle I

is the imaginary number so I squared is negative one which means that this second exponential becomes the

exponential of negative a RS times sine theta again the magnitude of the product of

two complex numbers is the product of their magnitude so we can now split up these two terms now the exponential of

negative AR sine theta is a real number and the magnitude of a simple real numbers just that real number which

means we can erase the magnitude bars from it that just leaves us with the first term which is the magnitude of a

pure imaginary exponential which is just 1 again if you end up using Euler's formula because you don't believe me

you'll get a cosine term plus I times a sine term and the magnitude of that will just be 1 therefore we can effectively

eliminate this first term because its magnitude is 1 and when we do that our final answer is that the exponential of

I a times R times the exponential of I theta is just the exponential of negative a R sine theta so now we can go

back cross off this exponential magnitude and perform the replacement on the integral that we were previously

working on now what about this first magnitude the magnitude of F well it's actually pretty easy we know from the

assumptions of our proof if we scroll all the way up that M sub capital R is the upper limit of F of R times the

exponential of I theta which is the same as f of Z because we just applied the polar representation of complex numbers

so what we can do is that we can say that the function f is less than or equal to M sub R but we can substitute

this directly into the integral because it's an inequality however what we can say is that because the function f is

less than or equal to M sub R and integral involving the function f multiplied by a couple of positive

numbers will therefore be less than or equal to the integral of its upper limit M sub R multiplied by those same

positive numbers again M sub R and R are just constants so we can take them outside this integral so now you have M

sub capital R and capital R multiplying the integral of this exponential now comes the fun part because at this point

we can apply sure Don's inequality the integral of the exponential that we have very strongly resembles the integral in

Jordans and equality the only difference is the extra a factor but that doesn't really matter because you can easily

treat a and capital R together so that if you apply jour don's inequality this integral becomes less than or equal to

the following now the capital R's will cancel and the expression just becomes M sub R times PI over 8 we're almost done

throughout this proof you'll recall that we've used a fairly long chain of equalities and inequalities and if you

go back up to the start of the chain you'll have the integral over the contour C sub r of f of z times the

exponential of i a z what we're gonna do is we're going to copy/paste this term from the beginning of our chain down

below from all the computations and simplifications that we've done in this proof which have either involved the

Equality or the less than or equal to sign we can therefore see that from the transitive property of inequality the

magnitude of the integral over the contour C R of F of Z times the exponential of I a Z is less than or

equal to M R times pi over a now I'm going to label this inequality as 1 so now that we have this final inequality

we're almost done with the proof if we go back up to the original assumptions we made in the statement of the theorem

we can see that the limit of the upper bound mr as capital R approaches infinity is 0 therefore if we take the

limit of inequality one as our approaches infinity will find that the limit as our approaches infinity of the

integral over C R of F of Z times the exponential of I a Z is less than or equal to 0 and since the magnitude of a

complex number is never negative we can get rid of the less than 0 bit and change this to the Equality because

obviously having the limit of this magnitude be negative wouldn't make any sense zero is the only possibility it

shouldn't be difficult to extend this equality further and say that since the limit of the magnitude of the contour

integral equals 0 the limit of the contour integral itself must be zero this is because as the magnitude of the

complex number approaches zero the complex number itself approaches zero and gets closer

to the origin so finally after all of this work we've proven jordan's lemma in the next video on complex variables i'm

going to show you how to apply your don's lemma to calculate some improper integrals via the residue theorem I'd

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