## Digamma Function

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

 (1)

where is the Gamma Function, and is the function returned by the function PolyGamma[z] in Mathematica (Wolfram Research, Champaign, IL). The digamma function is defined by
 (2)

and is equal to
 (3)

From a series expansion of the Factorial function,

 (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)

For integral ,
 (10)

where is the Euler-Mascheroni Constant and is a Harmonic Number. Other identities include
 (11)

 (12)

 (13)

 (14)

Special values are

 (15) (16)

At integral values,

 (17)

and at half-integral values,
 (18)

At rational arguments, is given by the explicit equation

 (19)

for (Knuth 1973). These give the special values
 (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 .

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
1999-05-24