The Ramanujan Journal

, Volume 5, Issue 1, pp 73–89 | Cite as

Some Remarkable Properties of Sinc and Related Integrals

  • David Borwein
  • Jonathan M. Borwein

Abstract

Using Fourier transform techniques, we establish inequalities for integrals of the form\(\smallint _0^x \prod\limits_{k = 0}^n {\frac{{\sin (akx)_{} }}{{akx}}dx.} \)

We then give quite striking closed form evaluations of such integrals and finish by discussing various extensions and applications.

sinc integrals Fourier transforms convolution Parseval's theorem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L.V. Ahlfors, Complex Analysis, McGraw-Hill, New York, 1966.Google Scholar
  2. 2.
    C. Störmer, “Sur un généralisation de la formule (ie89–1),” Acta Math. 19 (1885) 341–350.Google Scholar
  3. 3.
    K.R. Stromberg, An Introduction to Modern Real Analysis, Wadsworth Inc., Belmont California, 1981.Google Scholar
  4. 4.
    E.C. Titchmarsh, The Theory of Functions, Oxford University Press, London, 1947.Google Scholar
  5. 5.
    E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th edn., Cambridge University Press, London, 1967.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • David Borwein
    • 1
  • Jonathan M. Borwein
    • 2
  1. 1.Department of MathematicsUniversity of Western OntarioLondonCanada
  2. 2.Centre for Experimental and Constructive Mathematics, Department of Mathematics and StatisticsSimon Fraser UniversityBurnabyCanada

Personalised recommendations