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HERBERT GROSS: Hi,
our lesson today
is going to be
concerned with looking
at some of the
ramifications of what
it means for a complex valued
function to be analytic.
And one of the side results,
I hope, of today's lesson
will be to show why, if the
complex numbers had never
been invented up
until this point,
they would have
been invented, most
likely to discuss mappings of
the xy plane into the uv plane.
While I hope to make that
clearer as we go along,
for the time being, let's
simply entitle today's lesson
Conformal Mappings.
And by way of review--
and what I want to
do here, you see,
is emphasize the real part
of complex variable theory.
Suppose I have the usual
mapping from the xy plane
into the uv plane given by u as
some function of x and y and v
as some function of x and y.
And you may recall this
is back much earlier
in our course in
block 4, block 3,
where we were labeling this
as f bar of x, y equals u,
v. No necessity here of talking
about complex variables.
We're mapping two space
into two space, the xy plane
into the uv plane,
by the mapping f
bar defined by u and v being
these functions of x and y.
And what we saw was that
this particular mapping was,
at least locally, meaning in a
neighborhood of a given point,
in vertical, provided that the
determinant of the Jacobian
matrix was not 0.
In other words, that u sub
x, v sub y minus u sub y,
v sub x was not equal to zero.
And again, what
I want you to see
is that, up until
this point, there
is absolutely no knowledge that
we need of complex variables.
Now, what is
interesting is this.
Remember that, when we
introduced the complex numbers,
we said that, graphically, their
domain would be the xy plane,
and that would be called
the Argand diagram.
Now, you see, coming
back to this little
aside that we were
making over here,
if I now view the xy plane
as being the Argand diagram,
then what we're saying is that
x, y names the complex number
z.
u, v names the complex
number w, namely u plus iv.
And since we're talking, now,
about mapping complex numbers
into complex numbers, f bar
simply becomes some function f.
Now, the question
that's more important
is, why would one want this
particular interpretation?
What particular interpretation?
The particular interpretation
that I have in mind is
to simply observe that, whenever
I have a change of variables--
say, two equations
and two unknowns--
u equals some
function of x and y,
v equals some
function of x and y--
what I'm saying is I can always
view this pair of equations
as the single complex
variable equation that f of z
equals u plus iv, that,
mechanically, by letting
u be the real part and
v be the imaginary part,
I can invent the complex
valued function u plus iv.
By the way, this is in
much the same spirit
that, when one talks
about curves in the plane,
one does not have to know
anything about vectors.
I can talk about the curve x
equals x of t, y equals y of t
and then say, OK, let's
introduce the radius vector R
and then write that R
is equal to xi plus yj,
and this allows me to
take two scalar equations
and write them as
one vector equation.
So I can certainly take two
scalar, two real functions
of two real variables
and write them
in terms of the language
of complex functions.
The question, of course, is
why would I want to do this?
We have already seen this
in our very first lecture
on complex variables.
For example, we knew how to
handle things like cosine n
theta without ever
having to have
heard of complex variables.
Yet we found that, by such
things as [INAUDIBLE] theorem,
that there was some
very nice enlightenment
that the structure
of complex numbers
brought to real numbers.
In other words, then, perhaps
by taking this pair of equations
and writing them in the
language of complex variables,
I might, from the vantage point
of the complex number system,
be able to look down at
the real number system
and see some things
very elegantly
from a computational
point of view,
from a philosophical
point of view,
that I might not have been
able to notice otherwise.
Now, because this is
becoming a harangue,
let's get on with
the specific details
and see if this doesn't
become clear as we go along.
What we've already
said is we can
view this change of
variables as being
the real and imaginary parts
of the complex function f of z.
In other words, let's
invent the function f of z
to be u plus iv.
Now, the point is--
let's assume, for the sake
of argument, that f of z
happens to be an
analytic function.
What does it mean to say
that f of z is analytic?
It means that f prime exists.
In particular, from our
lecture of last time,
not only does f prime
exist, but if we
want to write this
quite simply, it
turns out to be the
partial of u with respect
to x plus i times the partial
of v with respect to x.
In particular, the
Cauchy-Riemann conditions hold,
which means that u
sub x equals v sub y,
and u sub y equals
minus v sub x.
And by the way, to
reinforce what I was just
saying before,
please observe that
this particular
relationship does not
necessitate my knowing
anything about complex numbers.
Given two functions u and
v, functions of x and y,
I can compute the partial
of u with respect to x.
I can compute the
partial of v with respect
to y, the partial of
u with respect to y,
the partial of v
with respect to x
and see if these
conditions hold.
At any rate, if these
conditions do hold--
and notice how nicely the
language of complex variables
allows me to say that
these conditions hold,
namely all I have to say is
that f of z is analytic--
then coming back to what
the Jacobian meant--
remember what the
Jacobian meant?
It was u sub x v sub y
minus u sub y v sub x.
Coming back to this, I can
now compute this very simply,
simply by observing that
another way of saying v sub y
is to replace it by u sub x.
And if I do that, this term
becomes u sub x squared.
If I now replace u sub
y by minus v sub x,
this becomes v sub x squared.
And so this expression
becomes u sub
x squared plus v sub x squared.
And if we now look here, notice
that, since the magnitude
of a complex number is just
the positive square root
of the sum of the squares of the
real and the imaginary parts,
this expression
here is precisely
the square of the
magnitude of f prime of z.
Now, look, the only way
that this can be 0, then,
is if this is 0.
The only way this can
be 0 is if f prime of z
is itself 0 because the
only complex number whose
magnitude is 0 is 0 itself.
Now, what does this
tell me, therefore?
This tells me that the
system of real equations, u
equals u of xy, v equals
v of xy, is invertible if,
when we write the complex
function, u plus iv
and call that f--
if f is analytic and
f prime of z is not 0.
Again, notice, I
could have stated that
without the language of
complex variables at all.
I could have said, look, suppose
the partial of u with respect
to x equals the partial
of v with respect to y,
and the partial of
u with respect to y
is minus the partial
of v respect to x.
And suppose that u sub x squared
plus v sub x squared is not 0.
Then this will be invertible.
Now, notice one
of the properties
is that there are a lot
of invertible functions
which do not obey these
stringent conditions.
Consequently, one would
like to believe that,
if we're going to introduce the
language of complex variables,
that we would like to get much
more out of this than just
the fact that f of z maps the
xy plane into the uv plane
in a one-to-one, onto manner
as long as f prime is not 0.
Let me point out that some
invertible mappings are
nicer than others.
Now, what do I mean by nicer?
Well, let's make up
a definition here.
An invertible mapping
is called conformal--
and maybe you can guess
what this is going to mean--
if it preserves angles.
Now, what do I mean
by preserving angles?
What I'm saying is--
let's suppose two curves meet
at a certain angle in the xy
plane.
When I map the xy
plane into the uv
plane in a one-to-one fashion,
the two curves in the xy plane
have images in the uv plane.
The question that comes
up is, will the two curves
intersect at the same
angle in the xy plane
that their images
intersect in the uv plane?
Well, heck, maybe it's easier
to do by means of an example.
Let's look at the
usual linear mappings
that we were talking about
when we introduced the Jacobian
in double integration.
Remember, one of the
problems that we tackled
was looking to see how we
map, say, a parallelogram
R to the unit square S by a
linear mapping of the form u
equals ax plus by,
v equals cx plus dy.
Notice that, even though
this map was linear,
even though it was invertible,
it obviously is not conformal.
Why isn't it conformal?
Well, the image of the line of
the vector OA in the xy plane
is O prime A prime
in the uv plane.
The image of OB in the xy
plane is given by O prime B
prime in the uv plane.
Notice that in the xy
plane, the vectors OA and OB
meet at an angle, theta,
which clearly is not
90 degrees, whereas their images
intersect at a right angle.
In other words, this mapping
did not preserve angles.
The image of an
angle in the xy plane
did not have to be the angle
of the image in the uv plane.
Now, why, for
example, would it be
important to want
to preserve angles?
Well, among other things,
when we change variables,
we sometimes don't
want to change
the physical significance
of a problem.
In other words, we
may be trying to solve
a problem in the xy plane.
For convenience, we map the
problem into the uv plane.
Well, it may happen that
certain physical properties
are present in the xy plane.
We may be talking about
potential and force.
And maybe one family of lines
intersects another family
of lines at right angles.
We'd like to believe
that we could
have a mapping into the uv plane
where, if the two lines met
at right angles in the xy
plane, their images would
meet at right angles
in the uv plane.
Well, let's not worry
about that right now.
Let's simply emphasize the
mathematics of a situation.
A conformal mapping is one
which preserves angles.
And I now claim the
very interesting thing,
namely, if the mapping u as
some function of x and y,
v is some function of x
and y-- given that mapping,
suppose I form the complex
function u plus iv equals f,
and suppose that that function
turns out to be analytic,
and the derivative is not 0.
We just saw that that guaranteed
that the mapping would
be invertible.
I now claim that, in
this special case,
this mapping is also conformal.
That's a very beautiful
result. In other words,
the mapping u equals u
of xy, v equals v is xy,
is conformable as
soon as we can be sure
that the function f given
by u plus iv is analytic
and that f prime is not 0.
Now, why is this
mapping conformal?
And again, the arithmetic
of complex numbers
comes in in a very
handy way over here.
Namely, let's look to see what
happens under the mapping.
All we're going to do now
is change the language,
not from the xy plane
and the uv plane,
but we are now going
to look at the xy plane
as the Argand diagram,
where the point x0, y0
is represented by the
complex number z0.
What I'm going to
assume now is this.
Let's take the point
z0, and let's take
two nearby points, z1 and z2.
Now, z0 has some image under f.
Let's call it w0.
z1 has an image.
We'll call it w1
in the uv plane.
And z2 has some image,
which we'll call w2.
Now, what we would like
to do is the following.
Let's take the straight lines
that join z0 to z1, z0 to z2.
Let's take the straight lines
that join w0 to w2 and w0
to w2.
This is a vector.
We'll call that delta w1.
This is a vector.
In other words, we'll leave
you as a complex number.
It's w2 minus w0, is delta W2.
This we'll call delta z1, for
reference, and this vector
we'll call delta
z2, for reference.
Now, remember how one
divides two complex numbers.
We divide the magnitudes, and
we subtract the arguments.
Consequently, if I let theta
be the angle in reference here,
notice that theta is the
angle obtained by dividing
delta z2 by delta z1.
Namely, if I divide
delta z2 by delta z1,
I subtract the angles.
If I take the angle
that delta z2 makes
with the positive
x-axis, subtract
from that the angle the delta z1
makes with the positive x-axis,
what's left is the angle theta.
Similarly, notice that
phi, this angle over here,
is the argument of delta
w2 divided by delta w1.
Namely, I simply
subtract the angle
that delta w2 makes with
the positive u axis.
I subtract from
that angle the angle
that delta w1 makes with the
axis, and that result is phi.
Because to divide
two complex numbers,
we divide the magnitudes,
subtract the arguments.
What I want to show is that,
if the complex value function
f is analytic and
f prime is not 0,
I want to show that
theta equals phi.
Now, the way I'm
going to do this
is I will assume that we're in
a very small neighborhood here
so that what I'm
saying is, what?
What does it mean to
say that f prime exists?
Remember, f prime was
the quotient delta w
divided by delta z.
So for small values of delta
z, delta w divided by delta z
must be approximately
f prime of z0.
We talked about that last time.
In particular then, delta
w2 divided by delta z2
is one such ratio.
Delta w1 divided by delta
z1 is another such ratio.
So for small changes
in z, these two ratios
should be approximately
equal, meaning
that the error is negligible
when we go to the limit
and that that approximate
ratio is f prime of z0.
Now, looking at
this ratio, the fact
that these are
approximately equal
says that delta z2
divided by delta z1
is approximately equal
to delta w2 divided
by delta w1, noticing, by the
way, that if f prime were 0,
delta w1 and delta
w2 would both be 0.
And that would give me a 0 over
0 form, which is indeterminant.
From a geometric point of
view, notice that f prime of 0
may be viewed as a
vector in the uv plane.
It has a magnitude,
and it has a direction.
If f prime is 0, that
vector is just a point,
and a point does not determine
a magnitude or a direction
when you're talking about
this type of ratios.
At least, it doesn't
determine the direction.
It may determine a magnitude 0.
But at any rate,
all we're saying
is that, by the
analyticness of f,
we now know this property here.
But since these two numbers
are approximately equal,
their arguments must
be approximately equal,
and that says that theta is
approximately equal to phi,
where, again, I want
to emphasize-- when
I say approximately
equal, I mean
they are equal up to errors of
second order infinitesimals,
and those are the
terms that go to 0 so
fast that, in the limit,
they don't appear.
This does not mean that
these are almost equal.
It means, in the
limit, they are equal,
the usual way that I'm using
approximations whenever
I've talked about linearity.
Now, one application
of conformal mappings
comes up when we
discuss the problem
that we introduced in our
discussion of Green's theorem
about boundary value,
steady state temperature.
Remember, we were talking
about a region, R,
enclosed by some curve,
C. And I had a temperature
distribution on
C and inside R. I
knew that the temperature
satisfied Laplace's
equation in R. That was called
the steady state condition.
namely, the second
partial of T with respect
to x plus the second partial
of T with respect to y
was zero in R. I
knew what he looked
like on the boundary
of R, namely on C.
And now what we said was,
back in our study of Green's
theorem, that this determined
a unique function, T is xy,
defined in this entire
region R. And the problem
is how do you determine what T
looks like in the entire region
R, just from this
information alone?
Notice, again, at this
point, I would never
have had to have heard
of complex variables
to understand this problem.
How can I prove that?
Hopefully, in our discussion
of Green's theorem
and the exercises, you
understood this problem.
Otherwise, you couldn't
have done the exercise.
Well, at that stage,
we hadn't talked
about analytic functions.
Consequently,
that's all the proof
that we need that this problem
makes perfectly good sense
without complex variables.
This is a real problem defined
in terms of real variables
in a real world situation.
The key point--
and that's what's
going to be the rest
of this lecture-- is
to prove that key point.
See, what happens is I don't
like the problem the way
it's stated over here.
So I say, OK, let me make
a change of variables,
the same way as I make
a change of variables
in solving definite integrals
in ordinary calculus
of a single real variable.
I make the change of
variables, hopefully,
to arrive at an integrand that
is easier for me to handle.
There's no guarantee
that the new integrand
grand will be any more
palatable than the old.
But the idea is I say OK let
me map R from the xy plane
into the uv plane
by some mapping f
bar that maps e2, into e2,
two space into two space.
In terms of the language
of complex variables,
all I'm saying is I
can view that mapping
as a complex valued function
of a complex variable.
I replace f bar by f,
as I mentioned earlier
in the lecture.
Well, the idea is, regardless
of how I want to do that,
suppose I now map
R into some region,
s, in the uv plane with a
new boundary-- say, C prime.
That translates my problem from
the xy plane into the uv plane.
If it happens I can solve
that problem in the uv plane,
the inverse mapping
then comes back
to give me the solution
in the xy plane.
Now, the big problem
that comes up
is that, in general,
invertible transformations
do not preserve
statements made in terms
of a coordinate system.
In other words,
for example, if u
is some function of x and y and
v is some function of x and y
and that gives me an invertible
mapping that maps the region
R into a region S,
there is no reason
to assume that, just because
this equation is obeyed in R,
that t sub uu plus T sub
vv will equal 0 in s.
In other words,
this is a statement
that depends on the
coordinate system.
It's like saying
that, when you wanted
to compute the
magnitude of a vector,
you just took the square root
of the sum of the squares
of the components.
That was only true if you were
using Cartesian coordinates.
If you use polar
coordinates, you
had to use a more
elaborate computational
recipe for a distance function.
See, the trouble is I can
map this into the uv plane,
but it may happen that
Laplace's equation is not
obeyed on the new region, S.
But the key amazing
point is this.
If I now take the mapping
induced by u and v and call
that, again, u plus iv,
call that function f--
in other words, if f maps the
region R in the Argand diagram
interpretation of the xy plane
into the region S in the uv
plane, where f is you plus iv,
and f is analytic and f prime
is not 0, then the amazing thing
is that this is equal to 0 in R
if and only if--
I might as well put this
in here because it does
go both ways by invertibility--
this is obeyed in s.
In other words, a
conformal mapping
preserves Laplace's equation.
What does that mean?
It means this.
Let's suppose that f is a
conformal mapping-- namely,
it's analytic, and its
derivative is never 0.
I'm given that T equals
T sub 0 of xy on C
and that it satisfies
Laplace's equation in R. I now
make the mapping f.
Since f is invertible, it
carries the closed curve C
into a closed curve C prime.
And the interior of c is carried
into the interior of C prime,
which is S. And that
gives me a new problem,
namely T is some
function of u and v
in the uv plane on C
prime, and it still
satisfies Laplace's equation.
Suppose it happens that,
because of the geometry
here, I can solve this
problem in the uv plane.
If I can do that, I
simply find what t of uv
looks like in S. Remembering
that u is u of xy,
v is v of xy, I plug this in.
This gives me t in
terms of x and y,
and that would be a solution in
the region R because, you see,
Laplace's equation is obeyed.
You see the key point is what?
That conformal mappings
preserve the solution
of Laplace's
equation, and that is
one of the very
important applications
of conformal mappings in
the study of the real world.
And again notice I
could have defined
conformal without any
reference to complex numbers,
just in terms of
preserving angles,
and then have invented what I
mean by an analytic function
by studying the geometry,
inventing the Argand diagram,
et cetera.
But why not take advantage
of the structure which
already exists?
Well, at any rate,
what I would like
to do for the finale
for today's lesson
is to prove this
particular result.
And the reason I would like
to prove that is that, once
and for all, this should
review the chain rule
for real variables.
It should show you how
the chain rule is used
and what happens with
ordinary transformations
from an algebraic point of view.
And finally, because
the proof never makes
use of complex numbers directly
but only properties of u and v,
where u and v are the
real imaginary parts
of a complex function,
I think that this
should psychologically eliminate
the traumatic experience
that complex valued functions
have no real application.
You see, the thing I
want to do is this.
I want to compute
T sub xx plus T sub
yy, given the
transformation that u
is some function of x and y and
v is some function of x and y.
No assumptions about the
mapping being invertible yet.
All I'm going to assume is
that u and v are continuously
differentiable
functions of x and y
so I can make
whatever manipulations
I want with these.
And what I would like to do is
now compute the Laplacian T sub
xx plus T sub yy as it would
look in terms of u and v.
And the first thing I
point out, quite simply,
by the chain rule, is
to take the partial of T
with respect to x.
I simply do not what?
Take the partial
of T with respect
to u times the partial
of u with respect
to x, plus the partial
of T with respect
to v times the partial
of u with respect x.
It's as if the us
and the vs cancel.
This is the contribution
of T sub x due to u,
contribution of T sub x
due to v. I add them up
because you and v
are independent.
I hope by now you remember
this almost automatically.
Then, I want the partial
of this with respect to x.
That means I want the partial
of this expression with respect
to x.
The partial of a sum is
the sum of the partials.
That brings me
from here to here.
Each of these is a product.
The partial of a product, I have
to use the product rule for.
That's what?
The first times the partial
of the second with respect
to x plus the
partial of the first,
T sub u, with respect
to x times the second.
Similarly, this term
here becomes this?
The first times the
partial of the second plus
the partial of the
first times the second.
And the key point is that
I put these in parentheses
here to emphasize the
fact that these are still
single functions.
Consequently, what I can now
do is apply the chain rule
to each of these
expressions again.
Don't be thrown off by the
subscripts u and v. Think
of the whole thing
in parentheses
as being some
function of u and v.
To take the partial of what's
in parentheses with respect
to x, I take the partial
first with respect to u times
the partial of u with
respect to x, then
the partial with respect
to v times the partial of v
respect to x--
in other words,
leaving the details,
again, for you to review.
This expression
here becomes this.
This expression is what?
The partial of this
with respect to u
times the partial
of u respect to x,
the partial of this with
respect to v times the partial
of v respect to x.
And now taking these
expressions and replacing these
by this in our
previous expression.
And, again, leaving
the details to you,
I wind up with the
fairly complicated result
that, in terms of the partials
with respect to u and v,
t sub xx is this fairly
messy expression.
By the way, I do not have to
do this whole thing over again
to find the second
partial of T with respect
to y because this derivation
is symmetric in x and y.
And if you don't
believe this, you
can do the thing
over as an exercise
and see what does happen.
I claim all I have
to do now to get
what the second partial
of T with respect to y
looks like is to go
through this result.
And every place I see
an x I replace it by a y
because I'm just taking
partials with respect
to y rather than
with respect to x.
Everything else stays the same.
So I now wind up with
this expression here.
And now the interesting
thing happens.
I just add these
two expressions.
And I get-- well, I
guess the technical word
for it is a "mess."
I get T sub xx plus T sub yy--
actually involves
five different terms.
There's a second partial
of T with respect to u.
And the coefficient of that,
you see, would be what?
It's u sub of x squared here.
It's u sub of y squared here.
So the coefficient
of that is u sub
x squared plus u sub y squared.
Again, without going
through the details,
I get a second partial of
T with respect to v term
and that, multiplied by v sub
x squared plus v sub y squared.
Observing that
the mixed partials
are equal by continuity, I
can combine the T sub uv terms
and get that the
coefficient is twice u sub
x, v sub x plus
u sub y, v sub y.
There's a term involving T sub
u, whose coefficient u sub xx
plus u sub yy and
a term involving
T sub v, the partial
of T with with respect
to v, whose coefficient
is v sub xx plus v sub yy.
And so far, I've imposed
no conditions on u and v
other than that u and v were
continuously differentiable
functions of x and y.
And I hope that what this
proves to you conclusively
is that, when you
translate Laplace's
equation into an arbitrary
uv coordinate system,
you do not get just a T
sub uu and T sub vv term.
There are five terms.
And if you're lucky, some
of them happen to drop out.
And by the way,
one hint here is,
as you may remember from
our lecture of last time,
it happened that if
u and v were the real
and the imaginary parts
of an analytic function
or, without using the language
of analytic functions,
if u sub x equaled v sub y
and u sub y was minus v sub x,
it turned out that u sub
xx plus u sub yy was 0
and v sub xx plus v sub yy is 0.
So in that special
case, notice that we
have the good luck that both of
these two terms would vanish.
But in general, it's not true
that for arbitrary functions, u
and v, that they satisfy the
Cauchy-Riemann conditions.
So now we're going
to invoke what we
know about conformal mappings.
Now we say, look.
Let's form the complex valued
function, u plus iv, where
u and v are as given over here.
If it turns out that
the function, f, defined
by u plus iv is
analytic, then what
do we already know from before?
We know from before that f
prime is u sub x plus iv sub x,
that u sub x is v sub y, that
u sub y is minus v sub x.
We know that the square of
the magnitude of f prime
is u sub x squared
plus v sub x squared.
By the way, since v sub x is
just the negative of u sub y,
v sub x squared is
u sub of y squared.
So this can be written
in this equivalent form.
Similarly, since u sub
x is equal to v sub y,
u sub x squared is equal
to v sub y squared,
so these are three
different forms
for expressing the square
of the magnitude of f prime.
We also knew that,
if this was analytic,
that u sub xx plus u sub yy and
v sub xx plus v sub yy is 0.
By the way, notice
that what this does
is that this is giving us a
hold on all of the coefficients
that we had over here.
In fact, it seems that the
only thing left to worry about
is what is u sub x, v sub
x plus u sub y, v sub y.
Well, look at.
u sub x, v sub x plus u sub
y, v sub y is simply this.
Notice that another name
for u sub of x is v sub y.
And another name for u
sub of y is minus v sub x.
Consequently, this
expression is just v sub y v
sub x minus v sub s v sub y.
These are numbers, so
it's communitative here.
In other words, this is just
this, so when you subtract them
the result is 0.
That means, by the way,
that, under the assumption
that u plus iv is
analytic, then we
can say that u sub x v sub
x plus u sub y v sub y is 0,
so the T sub uv term drops out.
To make a long story
short, all of our terms
drop out except for the terms
that involve t sub uu and t sub
vv, where what we showed was
that their coefficients were
simply the square of the
magnitude of F prime of z.
So in the special case that the
mapping u plus iv is conformal,
we have the remarkable result
that the Laplacian is changed
only by a non-negative factor
as we go from the xy plane
to the uv plane.
In particular, assuming
that f prime of z is not 0--
and notice that that's the
condition for the mapping
to be conformal.
If f prime of z is not 0,
notice that if this is not 0,
this expression can
be zero if and only
if this expression is 0.
And that's precisely the result
that we needed when we talked
about the fact that Laplace's
equation was preserved
by a common formal
mapping, and that
is one of the most
important reasons as to why
many branches of
physics, for example, use
the theory of
complex variables--
is that the derivative in
terms of complex variables
gives us a very nice
language and some very
nice computational holds
that, technically speaking,
we could have done without.
We could technically
have speaking
have done this all in terms
of the language of real
valued functions and constructed
a very artificial language
a very unnatural language.
But notice that, because the
calculus of complex variables
mimics that of
real variables, we
wind up with a very natural
language from which we
can operate, and this gives
us a tremendous vantage
point over the real world.
In other words, the complex
numbers are not only real,
but they are an advantage point
over the so-called real number
world.
At any rate, hopefully the
exercises in today's assignment
will help clarify this further.
Next time, we will
look at, still,
another aspect of the
value of complex variables
in the study of real
variable theory.
But at any rate, until
next time, goodbye.
Funding for the
publication of this video
was provided by the Gabriella
and Paul Rosenbaum Foundation.
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