A386718 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} {1/(x*y*z)} dx dy dz, where {} denotes fractional part.
5, 0, 0, 4, 4, 5, 3, 6, 2, 1, 7, 8, 5, 8, 0, 0, 2, 3, 4, 9, 6, 3, 3, 9, 4, 7, 8, 8, 1, 0, 1, 0, 5, 1, 5, 2, 7, 7, 5, 1, 0, 9, 9, 0, 5, 4, 4, 5, 0, 8, 4, 7, 2, 8, 7, 3, 3, 5, 9, 0, 0, 0, 7, 5, 8, 2, 4, 5, 9, 0, 8, 4, 4, 8, 4, 9, 8, 7, 0, 2, 1, 0, 2, 7, 1, 2, 8, 9, 6, 3, 6, 4, 3, 7, 8, 4, 5, 3, 3, 7, 4, 9, 0, 8, 8
Offset: 0
Examples
0.50044536217858002349633947881010515277510990544508...
References
- Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See section 2.43, page 106.
Links
- Yaming Yu, A Multiple Integral in Terms of Stieltjes Constants, SIAM Problems and Solutions, Classical Analysis, Integrals, Problem 07-002 (2007).
Crossrefs
Programs
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Mathematica
With[{m = 2}, RealDigits[1 - Sum[StieltjesGamma[k]/k!, {k, 0, 2}], 10, 120][[1]]]
Formula
Equals 1 - gamma - gamma_1 - gamma_2/2, where gamma_k is the k-th Stieltjes constant.
In general, for m >= 1, Integral_{x_1=0..1} ... Integral_{x_m=0..1} {1/(x_1*...*x_m)} dx_1 ... dx_m = 1 - Sum_{k=0..m-1} gamma_k/k!, where gamma_0 = gamma is Euler's constant.