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Analyticity.

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[MUSIC]

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Suppose f is a function and suppose that I

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can compute the derivative of f of 0 the
second

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derivative of f of 0, the third derivative
of

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f of 0, the fourth derivative of f of 0.

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And so on.

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Suppose that I can compute the nth
derivative

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of f at 0 regardless of what n is.

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So in short, suppose that f is infinitely
differentiable at 0, meaning that although

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might be really hard, atleast in principle

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I could compute the millionth derivative
at 0.

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The billionth derivative at 0, I can
compute any

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higher derivative that I want, at the
point 0.

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Well then the power series for f around 0
is the sum n

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goes from 0 to infinity of the nth
derivative of f at 0.

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This is a thing that makes sense, because
f is infinitely differentiable.

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Divided by n factorial times x to the n.
But what I don't know, is whether that

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power series converges to f, when x is
near but not equal to zero.

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What I'm asking is whether this Taylor

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series is actually equal to the function,
right?

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What's the relationship between this
Taylor series

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and the original function that I'm
studying?

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Sometimes it happens that the function is
equal to its Taylor series.

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So let's give a name to

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that phenomenon.
Here's how we're going to talk about this.

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The function f, which I'm assuming to

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be infinitely differentiable, is said to
be real

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analytic at the point zero if there's some

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number big R positive, so that this
happens.

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So that the function is equal to it's
Taylor series.

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Maybe not everywhere.

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But at least when x is within big R of
zero.

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I can make this a bit more general.

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Let's think for what this is really
talking about, Right.

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What this definition is trying to get at
is the idea

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that the function is equal to its tailor
series around zero.

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That's what real analytic at zero means.

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It means the function has a power series
representation around zero.

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When you generalize this and you talk
about power series around other points.

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Alright.

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So, here's the definition of what it means
for a function to

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be real analytic, not just at zero, but
real analytic at some arbitrary

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point A.

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Well, it should still mean that there's
some big R which will

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measure how for x is from a now, instead
of from zero.

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And here I've written down the Taylor
series around zero,

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but I should write down the Taylor series
around a.

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So, instead of differentiating at zero,
I'll differentiate f n times and

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evaluate at a, and then I'll multiply that
not just by x.

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But by x minus a to the nth power.

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And then I don't want to say that x is
close to zero.

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I want to say that x is within big R of a.

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So this is saying that at least near a,
when you're within big R of a.

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F can be written as a power series.

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And that's what it means to say that f is
real analytic at a point a.

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Let me draw a diagram to try to convey,
you know, what is going on here.

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Let's suppose that this piece of paper
represents

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all of the functions from the real numbers
to the real numbers.

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And plenty of those functions are just
miserable functions.

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Here's a graph of a terrible looking

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function, that's got a lot of
discontinuities.

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Some of these functions though are
continuous.

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So within the collection of all functions,
I've got the continuous functions.

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Here's a graph of what looks like to be a
continuous function.

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Still not great, because it's still got
these spikes but at least it's continuous.

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Now some of these continuous functions are
differentiable.

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A function as differentiable is
necessarily continuous.

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So within the collection of all continuous
functions, is a smaller

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collection just a differentiable
functions,

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they don't have these terrible spikes.

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But just because you're differentiable,
doesn't mean that

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you're say, twice differentiable or three
times differentiable.

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So within the collection of all the
differentiable functions,

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there's an even collection of functions
which are infinitely differentiable.

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These are functions that I can
differentiate once,

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twice, three times.

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These are functions I can differentiate as
many times as I like.

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Sometimes people call these functions
smooth functions or C infinity functions.

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This is a very restrictive class of
functions.

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But there's an even more restrictive class
of functions.

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Within the smooth functions, there's a
smaller collection of functions.

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The real analytic functions.
These functions aren't just smooth.

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Right?

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It's not just that I can differentiate
these functions.

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These functions also

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have the property that if I write down
their tailor series around

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some point, that tailor series converges
near that point to the function.

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So these real analytic functions are
really quite special.

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I mean, not every function is real
analytic.

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And yet the surprise is that so many of
the functions

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that we care the most about turn out to be
real analytic.

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[NOISE]

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[SOUND]