若干个著名的积分及文献
1.Ising Integrals
1.Integrals of the Ising class (D.H. Bailey J.M. Borwein R.E.Crandall) 2.Hypergeometric forms for Ising-class integrals (D.H. Bailey, D. Borwein, J.M. Borwein,R.E. Crandall) 3. Finding General Explicit Formulas for Ising Integral Recursions (D.H. Bailey J.M. Borwein) 4.On Recurrences for Ising Integrals (Johannes Kepler University Linz, Austria)
2.watson Integrals
1.THREE TRIPLE INTEGRALS (G. N. WATSON) 2.WATSON'S THIRD INTEGRAL (Hannah Cairns) 3.ON THE EVALUATION OF GENERALIZED WATSON INTEGRALS (G. S. JOYCE AND I. J. ZUCKER)
\begin{align*}&{I_1} = \frac{1}{{{\pi ^3}}}\int_0^\pi {\int_0^\pi {\int_0^\pi {\frac{{dudvdw}}{{1 - \cos u\cos v\cos w}}} } } = \frac{{4{{\left[ {K\left( {\frac{1}{2}\sqrt 2 } \right)} \right]}^2}}}{{{\pi ^2}}} = \frac{{{\Gamma ^4}\left( {\frac{1}{4}} \right)}}{{4{\pi ^3}}}\\&{I_2} = \frac{1}{{{\pi ^3}}}\int_0^\pi {\int_0^\pi {\int_0^\pi {\frac{{dudvdw}}{{3 - \cos u\cos v - \cos w\cos u - \cos u\cos v}}} } } \\&= \frac{{\sqrt 3 {{\left[ {K\left( {\frac{1}{4}\left( {\sqrt 6 - \sqrt 2 } \right)} \right)} \right]}^2}}}{{{\pi ^2}}} = \frac{{3{\Gamma ^6}\left( {\frac{1}{3}} \right)}}{{{2^{14/3}}{\pi ^4}}}\\&{I_3} = \frac{1}{{{\pi ^3}}}\int_0^\pi {\int_0^\pi {\int_0^\pi {\frac{{dudvdw}}{{3 - \cos u - \cos v - \cos w}}} } } \\&= \frac{{4\left( {18 + 12\sqrt 2 - 10\sqrt 3 - 7\sqrt 6 } \right){{\left[ {K\left( {\left( {2 - \sqrt 3 } \right)\left( {\sqrt 3 - \sqrt 2 } \right)} \right)} \right]}^2}}}{{{\pi ^2}}}\\&= \frac{{\sqrt 6 }}{{96{\pi ^3}}}\Gamma \left( {\frac{1}{{24}}} \right)\Gamma \left( {\frac{5}{{24}}} \right)\Gamma \left( {\frac{7}{{24}}} \right)\Gamma \left( {\frac{{11}}{{24}}} \right).\end{align*}
3.Box Integrals
1.Box integrals (D.H. Bailey J.M. Borwein R.E. Crandall) 2.Higher-dimensional box integrals (Jonathan M. Borwein O-Yeat Chan y R. E. Crandall) 3.ADVANCES IN THE THEORY OF BOX INTEGRALS (D. H. BAILEY, J. M. BORWEIN, AND R. E. CRANDALL)
4.spin integrals
5.Lattice Sum
6.Euler Sum
1.Euler Sums and Contour Integral Representations (Philippe Flajolet and Bruno Salvy) 2.Experimental evaluation of Euler sums (D.H.Bailey J.M.Borwein andR.Girgensohn) 3.Evaluation of triple euler sums (Jonathan M. Borwein) 4.Harmonic sums,Mellin transforms and Integrals (J.A.M.Vermaseren,NIKHEF)
7.Meijer G function
8.Ramanujan-type series
9.PSLQ
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