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Building on the nice answer of Guillaume: The integral $$ \int_{[-1,1]^n} \prod_{i < j} \left| x_i^2 - x_j^2 \right| \, dx_1 \dots dx_n $$ has the closed-form evaluation $$ 4^n \prod_{k \leq n} \bi …

On less practical terms, you can assign a(n extended) limit to any bounded sequence once you have an ultrafilter (on the natural numbers) at hand: Let F be your ultrafilter (that's what makes it less …

Let $P = a_n(x) D_x^n + a_{n-1}(x) D_x^{n-1} + \ldots + a_0(x)$ be a linear ordinary differential operator with polynomial (or real analytic) coefficients $a_j(x)$. Suppose that $a_n(x)$ doesn't vanis …

Using Mathematica and using reflection formulae for Gamma one finds: x[n,b] = (b+1) n/(n+b) G[n+b+1]/G[n+2b+2] / ( G[b+1]/G[2b+2] - 2 G[n+b+1]/G[n+2b+2] ) Now, observe that for b<-1 the quotients G[ …

Concerning the 2x2 case: As Mike points out, you can write down an explicit formula for the norm of the matrix {{a,b},{c,d}}. It takes a good while but Mathematica can then compute the volume you're …

A complex function is analytic if and only if locally it can be represented by a power series. This means that (at least locally) an analytic function is determined by countable data (namely, the Tay …