A function also called the Sampling Function and defined by

(1) |

(2) |

The sinc function can be written as a complex Integral by noting that

(3) |

The sinc function can also be written as the Infinite Product

(4) |

(5) | |||

(6) | |||

(7) | |||

(8) | |||

(9) |

These are all special cases of the amazing general result

(10) |

(11) |

(12) |

The half-infinite integral of
can be derived using Contour Integration.
In the above figure, consider the path
. Now write
. On an arc,
and on the *x*-Axis,
. Write

(13) |

(14) |

where the second and fourth terms use the identities and . Simplifying,

(15) |

where the third term vanishes by Jordan's Lemma. Performing the integration of the first term and combining the others yield

(16) |

(17) |

(18) |

(19) |

so

(20) |

(21) |

(22) |

An interesting property of is that the set of Local Extrema of corresponds to its intersections with the Cosine function , as illustrated above.

**References**

Kogan, S. ``A Note on Definite Integrals Involving Trigonometric Functions.'' http://www.mathsoft.com/asolve/constant/pi/sin/sin.html.

Morrison, K. E. ``Cosine Products, Fourier Transforms, and Random Sums.'' *Amer. Math. Monthly* **102**, 716-724, 1995.

© 1996-9

1999-05-26