Analyticity.
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Suppose f is a function and suppose that I
can compute the derivative of f of 0 the
second
derivative of f of 0, the third derivative
of
f of 0, the fourth derivative of f of 0.
And so on.
Suppose that I can compute the nth
derivative
of f at 0 regardless of what n is.
So in short, suppose that f is infinitely
differentiable at 0, meaning that although
might be really hard, atleast in principle
I could compute the millionth derivative
at 0.
The billionth derivative at 0, I can
compute any
higher derivative that I want, at the
point 0.
Well then the power series for f around 0
is the sum n
goes from 0 to infinity of the nth
derivative of f at 0.
This is a thing that makes sense, because
f is infinitely differentiable.
Divided by n factorial times x to the n.
But what I don't know, is whether that
power series converges to f, when x is
near but not equal to zero.
What I'm asking is whether this Taylor
series is actually equal to the function,
right?
What's the relationship between this
Taylor series
and the original function that I'm
studying?
Sometimes it happens that the function is
equal to its Taylor series.
So let's give a name to
that phenomenon.
Here's how we're going to talk about this.
The function f, which I'm assuming to
be infinitely differentiable, is said to
be real
analytic at the point zero if there's some
number big R positive, so that this
happens.
So that the function is equal to it's
Taylor series.
Maybe not everywhere.
But at least when x is within big R of
zero.
I can make this a bit more general.
Let's think for what this is really
talking about, Right.
What this definition is trying to get at
is the idea
that the function is equal to its tailor
series around zero.
That's what real analytic at zero means.
It means the function has a power series
representation around zero.
When you generalize this and you talk
about power series around other points.
Alright.
So, here's the definition of what it means
for a function to
be real analytic, not just at zero, but
real analytic at some arbitrary
point A.
Well, it should still mean that there's
some big R which will
measure how for x is from a now, instead
of from zero.
And here I've written down the Taylor
series around zero,
but I should write down the Taylor series
around a.
So, instead of differentiating at zero,
I'll differentiate f n times and
evaluate at a, and then I'll multiply that
not just by x.
But by x minus a to the nth power.
And then I don't want to say that x is
close to zero.
I want to say that x is within big R of a.
So this is saying that at least near a,
when you're within big R of a.
F can be written as a power series.
And that's what it means to say that f is
real analytic at a point a.
Let me draw a diagram to try to convey,
you know, what is going on here.
Let's suppose that this piece of paper
represents
all of the functions from the real numbers
to the real numbers.
And plenty of those functions are just
miserable functions.
Here's a graph of a terrible looking
function, that's got a lot of
discontinuities.
Some of these functions though are
continuous.
So within the collection of all functions,
I've got the continuous functions.
Here's a graph of what looks like to be a
continuous function.
Still not great, because it's still got
these spikes but at least it's continuous.
Now some of these continuous functions are
differentiable.
A function as differentiable is
necessarily continuous.
So within the collection of all continuous
functions, is a smaller
collection just a differentiable
functions,
they don't have these terrible spikes.
But just because you're differentiable,
doesn't mean that
you're say, twice differentiable or three
times differentiable.
So within the collection of all the
differentiable functions,
there's an even collection of functions
which are infinitely differentiable.
These are functions that I can
differentiate once,
twice, three times.
These are functions I can differentiate as
many times as I like.
Sometimes people call these functions
smooth functions or C infinity functions.
This is a very restrictive class of
functions.
But there's an even more restrictive class
of functions.
Within the smooth functions, there's a
smaller collection of functions.
The real analytic functions.
These functions aren't just smooth.
Right?
It's not just that I can differentiate
these functions.
These functions also
have the property that if I write down
their tailor series around
some point, that tailor series converges
near that point to the function.
So these real analytic functions are
really quite special.
I mean, not every function is real
analytic.
And yet the surprise is that so many of
the functions
that we care the most about turn out to be
real analytic.
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