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Comment by GlenTheMachine | Hacker News Reader

It is’t the exponential function that’s a miracle, it’s the Maclaurin series. Think about it. The Maclaurin series says that in a very real sense, the value of a function at zero, and the value of all of its derivatives at zero, contains all of the information there is to know about it. Intuitively, this should be completely unexpected. Why in the world would the derivatives of a function evaluated at a single point tell you anything whatsoever about the value of the function arbitrarily far away from that point?

EDIT: as mentioned below, this is not true of all functions, even all functions that are infinitely differentiable at zero. But it is true for very large classes of functions.

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> Intuitively, this should be completely unexpected.

It actually makes intuitive sense to me. I like to think of it this way:

1) If we have the value of a function at a given point and we want to extrapolate the function's values before and after that point, we need to know how the value is changing at that point: the derivative.

2) But if that derivative isn't constant, that won't get us very far. We also need to know how the derivative is changing at the given point: the 2nd derivative.

3) But if that 2nd derivative isn't constant, that won't get us very far. We also need to know how the 2nd derivative is changing at the given point: the 3rd derivative.

And so on, potentially up to infinity. But once we take into account all of the derivatives, we know how the value of the function is changing and how that change is itself changing, and as there is no additional change that comes "out of nowhere", so to speak, we have enough information to calculate the value at any other point.

I love math.

Thank you for the beautiful explanation.

But the problem is that the definitions of those derivatives are for infonitesimal change. What about far away points?

The definition works for any point. However often for higher functions you need to break the function into ranges to find closed form derivative series and get a useful result.

Far away points have properties resembling infinitesimally close points, as long as the function is continuous.

Now when there's a lack of continuity, then we run into a problem.

I'm not a mathematician but I assume there are functions with infinite number of derivatives. For such you cannot know all the values of the function by knowing the values of all derivatives because you would have to know infinitely many numbers.

Whereas if there are a fixed number of derivatives then at some level the derivative is constant and thus you can intuitively think that based on those derivatives you can then "draw" the function.

In-phony-tesimal, I like that word.

Your explanation is a nice restatement or interpretation for what the equation f(x)=f(0)+f'(0)x+.... is saying, but it doesn't give any insight for why the equation is true.

     and as there is no additional change that comes "out of nowhere", so to speak,

The whole explanation rests on this key point, and in fact this is the unintuitive part. In fact, it's not even always true. It's a "miracle" that it sometimes is true.

That's an interesting reaction... imho, the idea that change can't come out of nowhere is the most intuitive part. "How can something arise from nothing?" is a puzzling philosophical question precisely because our intuition is that it's impossible, that everything has to come from somewhere (and conversely, nothing can simply disappear). Functions where this expectation is violated -- e.g. non-constant continuous functions where all derivatives vanish to zero at a certain point -- feel like bizarre exceptions to me.

Fair enough. I do see how you could think that it's intuitive. I've even used your explanation when I'm teaching. Although I can't help but think there's some sort of circularity going on here, at least for me. My intuition on the subject is shaped by my knowledge of the theorem, and the functions that I tend to work with.

To know everything in derivatives you need to find the one where it's constant or a closed form series and its limit at infinity, otherwise you know nothing about how it's changing.

More than not always true, it's mostly not true. It's just true for a lot of functions we're interested in.

This is how I think of it too, and it makes some part of me want to believe that means I have an intuitive grasp do what’s going on. Except I don’t think that’s how math works. Rigor is necessary because an intuitive hand-wavey understanding is often wrong. I think there’s just as convincing a hand-wave available that dismisses the idea.

1) We have one point of the function.

2) We know one point of its derivative. We still don’t know any other points of the function.

3) We know one point of its second derivative. We still don’t know any other points of the function or its derivatives.

...

N) Same deal.

N+1) Same deal.

...

Inf) Now we know all the points in the function and also all the points in all the derivatives.

It looks absurd like this. Not to mention that we start with a countable set of points and derive a continuum from each.

It's more like at each step from n -> n+1, our understanding grows by some amount. The way you write it seems to imply that no information is gained until some arbitrary point when n is extremely large.

The moment we start taking derivatives we leave the world of countable sets and move to limits and epsilon neighborhoods. Evaluating a derivative at a single point by definition tells us something about the neighborhood of that point. Mentally expanding the series terms to their limit forms helps see that it's not just a random polynomial. I guess the real "miracle" is that for a large class of useful functions those ugly limits evaluate to simple forms.

1) We have one point of the function.

2) We know one point of its derivative. This means we know 2 more points of the function (one step in either direction).

3) We know one point of its second derivative. This means we know 2 more points of the derivative, which in turn gives us 2 more points of the function.

etc.

Or at least, that's the impression I get from this thread, since I wasn't familiar with the series before that.

That's not actually true. There exist functions of a single real variable that have derivatives at the origin that are all zero, yet are nonzero. For an example, see the following:

https://en.wikipedia.org/wiki/Taylor_series#Analytic_functio...

Nevertheless, it is actually true that all complex differentiable functions satisfy this property, which is miraculous.

Well, sure. Not all functions are described by their Taylor expansions. But this very large class is. I agree, if the word “miraculous” applies at all in mathematics, then this surely is one.

> There exist functions of a single real variable that have derivatives at the origin that are all zero, yet are nonzero

The function that is equal to 0 for x<1 and equal to 1 otherwise also satisfies this.

The more interesting statement is:

> There exist smooth functions of a single real variable that have derivatives at the origin that are all zero, yet are nonzero

https://en.wikipedia.org/wiki/Non-analytic_smooth_function

fyi, that's the heavyside function

This seems pretty tautological to me.

  this is not true of all functions... But it is true for very large classes of functions

Which classes of functions? It sounds like you're saying that a function is determined by its derivatives if it's determined by its derivatives.

Not really. Sure, the technical term is an “analytic function”, the definition of which is, essentially, “any function that is described by it’s Taylor series”. The surprising thing is how many functions are analytic. All polynomials. The exponential, and sums of exponentials. The trigonometric functions. Bessel functions. All complex, differentiable functions. Combinations of the above.

Nearly any function you are likely to encounter in a physics class, to the best of my knowledge, is analytic (IANAP).

Another way to look at this is: “almost all” smooth functions are “nearly” polynomials. Why should trigonometric functions be “nearly” polynomials? Why should exponentials? Why should there be any connection at all between the trig functions and the polynomials? I still say this is a surprising result.

Let's take "analytic" to mean "determined by its maclaurin series". It's not surprising that polynomials are analytic. Look at the definition of a polynomial. It's already in "power series" form, in fact there are only finitely many terms!

As for the other examples, here is one explanation for why they are ubiquitous in physics. Essentially, the class of analytic functions is closed under "solving differential equations".

So that's why sin/cos/exp/bessel are analytic - they are solutions to differential equations with constant/polynomial coefficients (we already know that constants and polynomials are analytic). That's why many functions found in physics are analytic - they are created from other analytic functions via a differential equation.

https://math.stackexchange.com/a/190167

P.S: Regarding your last statement "almost all smooth functions are almost polynomial", something much stronger is true: Every continuous function is almost a polynomial.

https://en.wikipedia.org/wiki/Stone–Weierstrass_theorem

Polynomials and complex numbers are connected via Fundamental Theorem of Algebra. Complex numbers can be represented in polar form - trigonometry/geometry, which can be connected via calculus to exponential form - Euler's formula.

http://tutorial.math.lamar.edu/Extras/ComplexPrimer/Forms.as...

https://en.m.wikipedia.org/wiki/Euler's_formula

https://en.m.wikipedia.org/wiki/Fundamental_theorem_of_algeb...

I find integrals even more magical. Just by evaluating the antiderivative of a function at a single point we get a summary of the behavior of the function everywhere between that point and minus infinity!

And like proper miracles, it often takes a ton of work and some divine insight to find an integral analytically.

Which is funny, in the analytical world differentiation is easy and integration is hard, but in the numerical world it's the other way around.

> it is true for very large classes of functions.

While it is, yes, from the point of view of a human counting them, very large, measure-theoretically it is a meager set. If you were picking functions at random from the space of all possible functions, you would never (i.e. with probability 0) pick one with a useful Maclaurin series.

It doesn't appear miraculous because knowing all derivatives for a function is quite a lot of information about a function. A polynomial of degree six can be described with six derivatives but it can also be described with six factors. There is no information advantage.

If you define a polynomial by its coefficients, and then take its derivatives at a point and forget the coefficients, it would be reasonable to wonder whether you had lost some information. The miracle is that you haven't (if you also remember that the original function is a polynomial).

It's true for analytical functions I think?

Analytic functions are usually defined to be those functions which can be represented by their Taylor series. This is what I meant by `tautological'.

It's one of the definitions, and perhaps not the most fundamental.

In the complex plane (and all real analytic functions can be analytically extended into the complex plane), you can define the equivalent property as saying a (first!) derivative has the some value no matter how you chose to take the limit.

So just by virtue of having a well defined first derivative in a particular region you get all the Maclaurin series magic.

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