Kevin Buzzard
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 Apr3 awarded Popular Question Aug13 awarded Yearling Jul2 awarded Curious May20 comment Proving that sin(x)/x=(1-x^2/pi^2)(1-x^2/4pi^2)(…) @abel: my gut feeling is that any attempt to prove that the RHS satisfies the differential equation will get bogged down in trying to e.g. relate sum 1/n^2 to sum 1/n^4 (when one is checking coefficients of the power series match up) and this is not elementary. May20 asked Proving that sin(x)/x=(1-x^2/pi^2)(1-x^2/4pi^2)(…) May15 awarded Nice Question Mar14 awarded Nice Question Feb11 awarded Popular Question Dec12 awarded Popular Question Oct24 comment Is this an undergraduate-level proof of conservation of energy, or an arbitrary manipulation of symbols that happens to give the right answer? Sure -- the issue is not proving the assertion, the issue is deciding whether the argument I give in the original question is a valid proof or not. Aug13 awarded Yearling Mar22 comment Is this an undergraduate-level proof of conservation of energy, or an arbitrary manipulation of symbols that happens to give the right answer? @Rhys: Consider the equation $x^2+y^2=7$. Would you be happy to say that $2xdx+2ydy=0$? Would you be happy to say that $dy/dx=-x/y$? I am absolutely convinced that on some level that I can make completely rigorous, the assertion $dy/dx=-x/y$ is true -- I could talk about modules of Kaehler differentials in algebraic geometry, for example. However $y$ is not a function of $x$. Hence I've now decided that I don't buy your objection raised in the first comment. I am genuinely confused. Mar22 comment Is this an undergraduate-level proof of conservation of energy, or an arbitrary manipulation of symbols that happens to give the right answer? I have now found two respectable mathematicians in my department, one of whom claims the solution deserves full marks and the other one claims that it does not. My conclusion is that I finally now realise why I struggled with applied mathematics -- unlike pure mathematics at this level it still seems to be the case that whether or not an answer is correct is a matter of opinion! Mar22 comment Is this an undergraduate-level proof of conservation of energy, or an arbitrary manipulation of symbols that happens to give the right answer? @Rhys: I don't believe your last comment. $x$ is defined as the solution to a differential equation which mentions a function $V$ about which we know nothing other than the implicit assertion that it's continuously differentiable. Mar21 comment Is this an undergraduate-level proof of conservation of energy, or an arbitrary manipulation of symbols that happens to give the right answer? What does it mean, on a formal level, for $\dot{x}$ to change sign? I am thinking of a continuous function which one might argue changes sign on the Cantor set (for some definition of "changes sign") and that is a long way from "$t_1,t_2,\ldots,t_k$". Does this worry you, or can you rule it out, or are your methods robust enough not to care? I'm pretty sure the UG will only recently have seen a rigorous definition of Riemann integral at this point in his education ;-) Mar21 asked Is this an undergraduate-level proof of conservation of energy, or an arbitrary manipulation of symbols that happens to give the right answer? Jan10 accepted Why isn't the inverse of the function $x\mapsto x+\sin(x)$ expressible in terms of “the functions one finds on a calculator”? Jan10 answered Why isn't the inverse of the function $x\mapsto x+\sin(x)$ expressible in terms of “the functions one finds on a calculator”? Jan4 comment Why isn't the inverse of the function $x\mapsto x+\sin(x)$ expressible in terms of “the functions one finds on a calculator”? Here is one last comment. A theorem of Liouville gives a necessary and sufficient criterion for a function to be integrable in elementary terms. However the criterion is, in my mind, tough to verify in practice. But a paper by Brian Conrad called "impossibility theorems for elementary integration" states Liouville's result (Theorem 4.1) but then also deduces Theorem 4.4, which is a practical test for impossibility. Conrad uses Theorem 4.4 to prove that $Li(x)$ and $erf(x)$ aren't expressible in elementary terms. I guess I need a practical consequence of Ritt's work but don't know one. Jan4 comment Why isn't the inverse of the function $x\mapsto x+\sin(x)$ expressible in terms of “the functions one finds on a calculator”? Yes, I'm looking for a proof that the inverse function is not an elementary function. Thanks.