A discontinuous ``step'' function, also called the Unit Step, and defined by

(1) |

(2) |

(3) |

Bracewell (1965) gives many identities, some of which include the following. Letting denote the Convolution,

(4) |

(5) | |||

(6) |

Additional identities are

(7) |

(8) |

The step function obeys the integral identities

(9) | |||

(10) |

(11) |

The Heaviside step function can be defined by the following limits,

(12) | |||

(13) | |||

(14) | |||

(15) | |||

(16) | |||

(17) | |||

(18) |

where is the Erfc function, is the Sine Integral, is the Sinc Function, and is the one-argument Triangle Function and

The Fourier Transform of the Heaviside step function is given by

(19) |

**References**

Bracewell, R. *The Fourier Transform and Its Applications.* New York: McGraw-Hill, 1965.

Spanier, J. and Oldham, K. B. ``The Unit-Step and Related Functions.''
Ch. 8 in *An Atlas of Functions.* Washington, DC: Hemisphere, pp. 63-69, 1987.

© 1996-9

1999-05-25