Thursday, September 29, 2016
Entry 29
The first Watson triple integral was discussed in Enry 26. It turns out all three integrals can be expressed simply by the Dedekind eta function η(τ)I1=1π3∫π0∫π0∫π0dxdydz1−cosxcosycosz=4η4(i)I2=1π3∫π0∫π0∫π0dxdydz3−cosxcosy−cosxcosz−cosycosz=41/3√3η4(√−3)I3=1π3∫π0∫π0∫π0dxdydz1−cosx−cosy−cosz=2√6(1+√2)1/3η4(√−6)In terms of the gamma functionI1=Γ4(14)4π3I2=3Γ6(13)214/3π4I3=Γ(124)Γ(524)Γ(724)Γ(1124)16√6π3. There are infinitely many Ramanujan-type formulas for the Ii a few of which are (1+√2)Γ(38)4(2π)5/2√I1=∞∑n=0(2n)!3n!61(−26)n1√2I1=∞∑n=0(2n)!3n!61(−29)n4√3I2=∞∑n=0(2n)!3n!61(28)n1+√2+√6√6I3=∞∑n=0(2n)!3n!61(23(1+√2+√6)3(1+√2))n There is one other formula that belongs to the family with denominators as powers of 216η(√−7)4=∞∑n=0(2n)!3n!61(212)nbut I don't know if this has an equivalent and analogous integral.
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