Holograms.
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They tell me that if you take a total little bit of a
hologram and you look through it, you can still see the whole picture.
The same is sort of true of analytic functions.
All right? Let's think about sine.
So here, I've got the graph of a function, y equals f of x.
And in this case, the function that I'm graphing is just sine.
I'm going to think about this graph a little bit
and just focus on this region right around the origin.
All right, imagine, if you will, that I've taken.
Away all of the graph and I'm just looking
at just this little tiny piece right around the origin.
What can I figure out about sine just by looking right around the origin.
Well, if I look around the origin, I can
figure out the value of sine at the origin.
If I'm just looking the origin I can
figure out the derivative of sine at the origin.
I can figure out the second derivative sine at
the origin.
I can figure out the third derivative of sine.
Just by looking at this little tiny piece of the graph.
I can figure out all of the derivatives of sine, just
by looking at this little tiny piece of the graph of sine.
And if I can calculate all the derivatives of sine, that means that just by looking
at that piece, I can write down the Taylor series for sine at 0.
And we've already seen that the Taylor
series for sine converges to sine everywhere.
That's exactly right.
We've seen that sine of x is equal to its Taylor series for all values of x.
And what that means is that just this little tiny piece of
sine, just around the origin, manages to encode the entire story about sine.
All of the values of sine can be recovered just
by knowing a little tiny piece of sine around the origin.
Because that's enough to calculate all the derivatives at 0 which
is enough to recover the Taylor series, which recovers the entire function.
Contrast that with something involving absolute value.
Well, here's the graph. Of a function, so y=f(x).
But in this case, the function is f(x)=|x-1|-1.
Now, what happens if I zoom in on the origin?
Well, this little piece of the graph just around the origin is enough to
calculate some facts about f, right?
What can I calculate just by looking right around the origin?
Well I can calculate the value of this function is 0 when I plug in
0, I can calculate that the derivative of this function is negative 1 at 0.
And I can calculate that all of the higher derivatives, all of the
second, third, fourth and so on derivatives at 0 are equal to 0.
So yeah, this thing looks like a straight line around here and
that infinitesimal information at the point 0.
Just the derivatives at the point 0 are enough to
recover the function around 0 on a whole interval around 0.
Now of course, I might have been tricked into
thinking that the entire function just looked like this, right.
I can't tell that there's this point where the function
fails to be differentiable, just when I'm looking over here.
But nevertheless, just this infinitesimal information
at a point, is still enough to recover some information
about the function in a whole interval, around that point.
Real analytic functions are really quite surprising.
Infinitesimal information, just at a point, is
giving you information at a whole interval.
Sine, cosine, and e to the x are even better.
In that case, the Taylor series converges to those functions everywhere.
Which means infinitesimal information at a single
point, on say the graph of sine, alright, a single point on the graph
of cosine, or at a single point on the graph of e to the x.
That information is telling you everything about the entire function.
Yeah en, en, entire is actually the word that, that's used.
Entire describes functions like sin where just the infinitesimal information at a
point is enough to recover a power series with an infinite radius of convergence.
Converges everywhere to the entire function.
So, it's really like a hologram.
Alright?
Just a little tiny bit of a hologram records everything
in the scene and so too would say sine of x.
Alright?
Just a little tiny piece of sine reveals everything about the function.
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