According to David Jeffrey, I have promulgated a bogon. I can't find time to track down the problem, but neither do I observe any discontinuities in Subj: Integrate[Floor[t]^e*t^f, {t, 0, x}] For Element[x,Reals]&&e>0&&f>0, with integer e (respectively f): Integrate[t^f*Floor[t]^e, {t, 0, x}] == (1/(1 + f))*(x^(1 + f)*Floor[x]^e + Sum[(-1)^i*Binomial[e, i]*(-HurwitzZeta[-1 - e - f + i, 1 + Floor[x]] + Zeta[-1 - e - f + i]), {i, e}]) == -((1/(1 + f))*((-x^(1 + f))*Floor[x]^e + Floor[x]^(1 + e + f) - Sum[Binomial[1 + f, i]*(-HurwitzZeta[-1 - e - f + i, Floor[x]] + HurwitzZeta[-1 - e - f + i, 0]), {i, 1 + f}])) E.g., t->u^2 in Integrate[Floor[Sqrt[t]],{t,0,x}] gives 2*Integrate[Floor[t]*t, {t, 0, Sqrt[x]}] == Integrate[Floor[Sqrt[t]], {t, 0, x}]== x*Floor[Sqrt[x]] + HurwitzZeta[-2, 1 + Floor[Sqrt[x]]]== (-(1/6))*Floor[Sqrt[x]]* (1 - 6*x + Floor[Sqrt[x]]*(3 + 2*Floor[Sqrt[x]])) so I'm hoping these old formulae subsume and correct the bogon. --rwg rwg@sdf.lonestar.org wrote:
Yikes, that's not a simple typo. It came from plugging into more general formulae, e.g., 'integrate(t^k * floor(t)^e,t,0,x) = ((f^e * x^(k + 1) + sum(n^(k + 1) * (n^e - (n - 1)^e),n,1,f))/(k + 1)) =((f^e * x^(k + 1) - sum(n^e * ((n + 1)^(k + 1) - n^(k + 1)),n,1,f - 1) + f^(k + e + 1))/(k + 1)), f:=floor x . I'll need to track this down and send a correction to the more general formulae. --Bill
Dear Bill,
Some time ago, you posted this to math-fun. I believe there is a typing error. The second factor should be (x - etc) I assume you were intending to construct a continuous integral expression. I am writing a paper on this at the moment. Where did the integral come from? Do you have more results of this type?
David Jeffrey
rwg@sdf.lonestar.org wrote:
It's very easy to "shew that"
/ x [ I floor(sqrt(t)) dt ] / 0 (floor(sqrt(x)) + 1) (2 floor(sqrt(x)) + 1) = floor(sqrt(x)) (x + -------------------------------------------), 6 but a little surprising when you look at it.
/ x [ 3/2 I sqrt(floor(t)) dt = floor (x) + x sqrt(floor(x)) ] / 0 1 1 + hurwitz_zeta(- -, floor(x)) - zeta(- -). 2 2 --rwg