Will Jagy
Reputation
88/100 score
 1h comment Why are there conflicting results for integration? yes, and you have no way of knowing how a machine deals with that. As it happens, the limit at $0^+$ of $x^x$ is $1,$ so this is actually a nice integral; just not formally nice. 1h comment Why are there conflicting results for integration? why would you trust either one when there are undefined quantities involved? 7h comment Find $f(x)$ if $f(f(x)) = x^2 -1$ @anomaly thanks. I think for contests, people just want to know global answers, which are quite rare, proof by I. N. Baker. However, while difficult, there are generally real analytic solutions on intervals, usually semi-infinite beginning at the largest fixpoint regardless of the derivative there. 7h revised Find $f(x)$ if $f(f(x)) = x^2 -1$ added 103 characters in body 7h answered Find $f(x)$ if $f(f(x)) = x^2 -1$ 7h comment Find $f(x)$ if $f(f(x)) = x^2 -1$ @csx thanks. The easiest source for Schroder's method is D. S. Alexander, A History of Complex Dynamics. A more elaborate treatment in Iterative Functional Equations by Kuczma, Choczewski and Ger 7h comment Find $f(x)$ if $f(f(x)) = x^2 -1$ two notes: approximate solution $x^{\sqrt 2}$ for $x \geq 0.$ Meanwhile, there is a real analytic solution around the fixpoint at $\frac{1 + \sqrt 5}{2}$ by Schroder's method. This does extend continuously to all $\mathbb R,$ 8h comment Find $f(x)$ if $f(f(x)) = x^2 -1$ @cxz do you have a link for that? 11h comment Complete formalization of solutions to $a^2+b^2=c^2+k$ for fixed $k>0$ No proof, long, long experience. The most clear cut method for your problem is stereographic projection around one integer solution, that is easy enough, but is three parameters by nature. The fact that Pythagorean triples get down to just two parameters is a fortunate accident, explored fully by Fricke and Klein and related to Fuchsian groups. Meanwhile: your profile shows your principal interest in computers. Were you going to actually program this and work with the mathematics, or did you just want someone to assure you about low complexity? 23h answered If $\varphi$ is bounded above, increasing, and concave down, does $x\varphi'(x)$ go to zero? How fast? 1d comment If $\varphi$ is bounded above, increasing, and concave down, does $x\varphi'(x)$ go to zero? How fast? @Timkinsella fixed it 1d revised If $\varphi$ is bounded above, increasing, and concave down, does $x\varphi'(x)$ go to zero? How fast? added 89 characters in body 1d answered If $\varphi$ is bounded above, increasing, and concave down, does $x\varphi'(x)$ go to zero? How fast? 1d comment The following graph makes no sense yours is correct for $\cos ( \pi x / 10$ for all $x$ from $-5$ to $5.$ However, they wanted the odd extension of the part from $0$ to $5.$ This means you take the indicated part, $x > 0,$ and rotate that $180^\circ$ around the origin and draw that in. 1d comment Solving for a cubic polynomial's roots using Viete's Theorem They were very careful to pick coefficients so that, if you guessed the root $25,$ you could confirm it on paper without error; or, in your head, as there is a clear pattern. All you need is to notice is that $25 - 24 = 1$ 1d comment Why do we know that , besides the known idoneal numbers , there is at most one more? Take a look and see for yourself 1d comment Why do we know that , besides the known idoneal numbers , there is at most one more? The proof by Weinberger is a pdf link at the Wikipedia page. 1d comment Irreducibility of Non-monic Quartic Polynomials in Q[x] @JasonSmith rational roots theorem 1d comment Complete formalization of solutions to $a^2+b^2=c^2+k$ for fixed $k>0$ Looked into it, there is nothing with two parameters unless $k=0.$ Quickest is probably to take $k - b^2 = a^2 - c^2 = (a+c)(a-c),$ so iterate over all $b$ and whenever $k - b^2 \neq 2 \pmod 4$ find divisors $k-b^2 = uv$ with $u \equiv v \pmod 2.$ 1d awarded Yearling