“Digamma Function”的概念、定义、习题解题方法及翻译-数学辞典

单词 Digamma Function

释义

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] inMathematica

(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 thedigamma 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 aHarmonic 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.

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