Two notations are used for the digamma function. The digamma function is defined by
(1) |
(2) |
(3) |
(4) | |||
(5) | |||
(6) | |||
(7) | |||
(8) |
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) |
See also Gamma Function, Harmonic Number, Hurwitz Zeta Function, Polygamma Function
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 Eric W. Weisstein