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Integral of the difference between a function and its floor

Elementary derivation of certian identites related to the Riemannian Zeta function and the Euler-Mascheroni ConstantContinued fraction of $gamma+1$ using recursionWhy is $-gamma = int_0^1 frace^-z-1zdz+int_1^infty frace^-zzdz$Infinite Series $sumlimits_n=1^inftyfrac(-1)^nnleftlfloorfraclog(n)log(2)rightrfloor$An integral representation for $psi$A fractional part integral giving $fracF_n-1F_n-frac(-1)^nF_n^2lnleft(!fracF_n+2-F_ngammaF_n+1-F_ngammaright)$Series for Stieltjes constants from $gamma= sum_n=1^infty left(frac2n-sum_j=n(n-1)+1^n(n+1) frac1jright)$Proving $lim_ntoinftysum_k=n^inftyBig(frac1nleftlfloorfrac knrightrfloor-frac1kBig)=gamma$On the series $sum_n=1^infty fracleft lfloor log_2 n right rfloor2n(2n+1)(2n+2)$Fractional part of the floor function Integral

1

$begingroup$

I know the following identity:
$$int_1^inftyleft(-frac1x+frac1lfloor xrfloorright)dx=gamma$$
I have a function $f(b)=0.5left(sqrt(N-b^2)-b+1right)$. Can a similar identity be derived for $$int_1^inftyleft(-f(b)+lfloor f(b)rfloorright)db$$

Thanks

eulers-constant

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asked 15 hours ago

RTnRTn

186

$endgroup$

add a comment | 

1

$begingroup$

I know the following identity:
$$int_1^inftyleft(-frac1x+frac1lfloor xrfloorright)dx=gamma$$
I have a function $f(b)=0.5left(sqrt(N-b^2)-b+1right)$. Can a similar identity be derived for $$int_1^inftyleft(-f(b)+lfloor f(b)rfloorright)db$$

Thanks

eulers-constant

share|cite|improve this question

asked 15 hours ago

RTnRTn

186

$endgroup$

add a comment | 

$begingroup$

I know the following identity:
$$int_1^inftyleft(-frac1x+frac1lfloor xrfloorright)dx=gamma$$
I have a function $f(b)=0.5left(sqrt(N-b^2)-b+1right)$. Can a similar identity be derived for $$int_1^inftyleft(-f(b)+lfloor f(b)rfloorright)db$$

Thanks

eulers-constant

share|cite|improve this question

asked 15 hours ago

RTnRTn

186

$endgroup$

share|cite|improve this question

asked 15 hours ago

RTnRTn

186

share|cite|improve this question

asked 15 hours ago

RTnRTn

186

asked 15 hours ago

RTnRTn

186

asked 15 hours ago

RTnRTn

186