On elementary complex variable theory applied to sinc and related integrals**

Abstract

An elementary application of complex variable theory demonstrates that some properties of sinc and related integrals, which have been variously described as “tricky,” “intriguing” and “deluding intuition,” arise as simple consequences of Jordan’s lemma. This approach, in offering an interesting alternative to the established treatment based upon Fourier transform theory, may thus be recommended in addition for its pedagogic value. The present treatment extends the earlier results significantly in applications to both divergent and convergent sequences, by introducing the useful concepts of “turning points” and “ranks” of the various contributions to summations over complex path integrals. Finally, a result stated as a student problem in the well-known treatise of Whittaker and Watson (1927) is similarly extended in its application to sequences of both types.

Keywords:

• sinc and sinc.cosine integrals
• complex variable integration
• Jordan's lemma
• divergent and convergent sequences

ACKNOWLEDGEMENTS

The author is indebted to Dr Johnathan K. Burchill of the Department of Physics & Astronomy of the University of Calgary for performing independent computations of the numerical results tabulated in this paper, which proved invaluable for preparing the prime number classification. Warm thanks are due to Miss Lynne K. Hey for drafting the figures.