In physics, the Fresnel integrals are most often defined by

(1) |

(2) |

(3) |

(4) | |||

(5) |

Related functions are defined as

(6) | |||

(7) | |||

(8) | |||

(9) |

An asymptotic expansion for gives

(10) |

(11) |

(12) |

(13) |

(14) |

(15) |

(16) | |||

(17) |

where is a Spherical Bessel Function of the Second Kind

(18) | |||

(19) |

**References**

Abramowitz, M. and Stegun, C. A. (Eds.). ``Fresnel Integrals.''
§7.3 in *Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 300-302, 1972.

Leonard, I. E. ``More on Fresnel Integrals.'' *Amer. Math. Monthly* **95**, 431-433, 1988.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T.
``Fresnel Integrals, Cosine and Sine Integrals.'' §6.79 in
*Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.* Cambridge, England:
Cambridge University Press, pp. 248-252, 1992.

Spanier, J. and Oldham, K. B. ``The Fresnel Integrals and .''
Ch. 39 in *An Atlas of Functions.* Washington, DC: Hemisphere, pp. 373-383, 1987.

© 1996-9

1999-05-26