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Lattice Green's Function for the Simple Cubic Lattice in Terms of a Mellin‐Barnes Type Integral

### Abstract

The lattice Green's function for the simple cubic latticefor *a* > 3 is expressed as a Mellin‐Barnes type integral. The analytic continuation gives simple and useful expressions in series for the numerical calculation of the real part *I* _{R}(*a*) and the imaginary part *I* _{I}(*a*) of the integral for 0 < *a* < 1 and 1 < *a* < 3. The values at *a* = 1, *a* = 0, and are obtained exactly: *I* _{R}(1) = (π/2)[Γ(5/8)Γ(7/8)]^{−2}, and *I* _{I}(0) = 3·2^{−11/3}π^{−4}[(Γ(1/3)]^{6}.

© 1971 The American Institute of Physics

Received 20 July 1970
Published online 28 October 2003