Two notations are used for the digamma function. The digamma function is defined by

(1) |

(2) |

(3) |

(4) | |||

(5) | |||

(6) | |||

(7) | |||

(8) |

where is the Euler-Mascheroni Constant and are Bernoulli Numbers.

The th Derivative of is called the Polygamma Function and is denoted . Since the digamma function is the zeroth derivative of (i.e., the function itself), it is also denoted .

The digamma function satisfies

(9) |

(10) |

(11) |

(12) |

(13) |

(14) |

Special values are

(15) | |||

(16) |

At integral values,

(17) |

(18) |

(19) |

(20) | |||

(21) | |||

(22) | |||

(23) | |||

(24) | |||

(25) |

where is the Euler-Mascheroni Constant. Sums and differences of for small integral and can be expressed in terms of Catalan's Constant and .

**References**

Abramowitz, M. and Stegun, C. A. (Eds.). ``Psi (Digamma) Function.'' §6.3 in
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 258-259, 1972.

Arfken, G. ``Digamma and Polygamma Functions.'' §10.2 in *Mathematical Methods for Physicists, 3rd ed.*
Orlando, FL: Academic Press, pp. 549-555, 1985.

Knuth, D. E. *The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 2nd ed.* Reading, MA: Addison-Wesley, p. 94, 1973.

Spanier, J. and Oldham, K. B. ``The Digamma Function .''
Ch. 44 in *An Atlas of Functions.* Washington, DC: Hemisphere, pp. 423-434, 1987.

© 1996-9

1999-05-24