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Comment by boyobo | Hacker News Reader

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.

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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...

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