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Holograms.

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[SOUND].

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They tell me that if you take a total
little bit of a

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hologram and you look through it, you can
still see the whole picture.

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The same is sort of true of analytic
functions.

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All right?
Let's think about sine.

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So here, I've got the graph of a function,
y equals f of x.

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And in this case, the function that I'm
graphing is just sine.

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I'm going to think about this graph a
little bit

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and just focus on this region right around
the origin.

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All right, imagine, if you will, that I've
taken.

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Away all of the graph and I'm just looking

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at just this little tiny piece right
around the origin.

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What can I figure out about sine just by
looking right around the origin.

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Well, if I look around the origin, I can

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figure out the value of sine at the
origin.

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If I'm just looking the origin I can

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figure out the derivative of sine at the
origin.

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I can figure out the second derivative
sine at

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the origin.

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I can figure out the third derivative of
sine.

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Just by looking at this little tiny piece
of the graph.

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I can figure out all of the derivatives of
sine, just

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by looking at this little tiny piece of
the graph of sine.

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And if I can calculate all the derivatives
of sine, that means that just by looking

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at that piece, I can write down the Taylor
series for sine at 0.

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And we've already seen that the Taylor

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series for sine converges to sine
everywhere.

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That's exactly right.

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We've seen that sine of x is equal to its
Taylor series for all values of x.

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And what that means is that just this
little tiny piece of

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sine, just around the origin, manages to
encode the entire story about sine.

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All of the values of sine can be recovered
just

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by knowing a little tiny piece of sine
around the origin.

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Because that's enough to calculate all the
derivatives at 0 which

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is enough to recover the Taylor series,
which recovers the entire function.

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Contrast that with something involving
absolute value.

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Well, here's the graph.
Of a function, so y=f(x).

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But in this case, the function is
f(x)=|x-1|-1.

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Now, what happens if I zoom in on the
origin?

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Well, this little piece of the graph just
around the origin is enough to

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calculate some facts about f, right?

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What can I calculate just by looking right
around the origin?

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Well I can calculate the value of this
function is 0 when I plug in

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0, I can calculate that the derivative of
this function is negative 1 at 0.

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And I can calculate that all of the higher
derivatives, all of the

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second, third, fourth and so on
derivatives at 0 are equal to 0.

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So yeah, this thing looks like a straight
line around here and

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that infinitesimal information at the
point 0.

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Just the derivatives at the point 0 are
enough to

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recover the function around 0 on a whole
interval around 0.

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Now of course, I might have been tricked
into

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thinking that the entire function just
looked like this, right.

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I can't tell that there's this point where
the function

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fails to be differentiable, just when I'm
looking over here.

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But nevertheless, just this infinitesimal
information

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at a point, is still enough to recover
some information

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about the function in a whole interval,
around that point.

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Real analytic functions are really quite
surprising.

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Infinitesimal information, just at a
point, is

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giving you information at a whole
interval.

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Sine, cosine, and e to the x are even
better.

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In that case, the Taylor series converges
to those functions everywhere.

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Which means infinitesimal information at a
single

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point, on say the graph of sine, alright,
a single point on the graph

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of cosine, or at a single point on the
graph of e to the x.

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That information is telling you everything
about the entire function.

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Yeah en, en, entire is actually the word
that, that's used.

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Entire describes functions like sin where
just the infinitesimal information at a

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point is enough to recover a power series
with an infinite radius of convergence.

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Converges everywhere to the entire
function.

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So, it's really like a hologram.

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Alright?

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Just a little tiny bit of a hologram
records everything

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in the scene and so too would say sine of
x.

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Alright?

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Just a little tiny piece of sine reveals
everything about the function.

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[SOUND]

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