Abstract
PurposeThe authors propose a rather elementary method to compute a family of integrals on the half line, involving positive powers of sin x and negative powers of x, depending on the integer parameters n≥q≥1.Design/methodology/approachCombinatorics, sine and cosine integral functions.FindingsThe authors prove an explicit formula to evaluate sinc-type integrals.Originality/valueThe proof is not present in the current literature, and it could be of interest for a large audience.
Highlights
The authors propose a rather elementary method to compute a family of integrals on the half line, involving positive powers of sin x and negative powers of x, depending on the integer parameters n ≥ q ≥ 1
We consider the family of integrals ðsin xÞn xq dx: Theorem 1
2kÞ: The formulae above are recorded in the Wolfram MathWorld web page titled Sinc Function [1], which refers to the result as “amazing” and “spectacular”
Summary
Let n ≥ q ≥ 1 be any two given integers. The symbol b:c will stand, as usual, for the integer part. The following formulae hold (i) If n þ q is even, q−n ð−1Þ 2 π 2nðq À 1Þ!. 2kÞ: The formulae above are recorded in the Wolfram MathWorld web page titled Sinc Function [1], which refers to the result as “amazing” and “spectacular”. The web page omits the proof, citing a 20-year-old online paper that seems not to be available any longer. The full terms of this licence may be seen at http:// creativecommons.org/licences/by/4.0/legalcode
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