Digamma Function

Two notations are used for the digamma function. The $\Psi(z)$ digamma function is defined by

$$\Psi(z)\equiv {d\over dz} \ln\Gamma(z) = {\Gamma'(z)\over\Gamma(z)},$$ (1)

where $\Gamma$ is the Gamma Function, and is the function returned by the function PolyGamma[z] in Mathematica ${}^{\scriptstyle\circledRsymbol}$ (Wolfram Research, Champaign, IL). The $F$ digamma function is defined by

$$F(z) \equiv {d\over dz} \ln z!$$ (2)

and is equal to

$$F(z)=\Psi(z+1).$$ (3)

From a series expansion of the Factorial function,

$$F(z) = {d\over dz} \lim_{n\to \infty} [\ln n!+z\ln n-\ln (z+1)-\ln (z+2)-\ldots -\ln (z+n)]$$ (4)

$$= \lim_{n\to\infty}\left({\ln n-{1\over z+1}-{1\over z+2}-\ldots-{1\over z+n}}\right)$$ (5)

$$= -\gamma - \sum_{n=1}^\infty \left({{1\over z+n} - {1\over n}}\right)$$ (6)

$$= -\gamma + \sum_{n=1}^\infty {z\over n(n+z)}$$ (7)

$$= \ln z + {1\over 2z} - \sum_{n=1}^\infty {B_{2n}\over 2nz^{2n}},$$ (8)

where $\gamma$ is the Euler-Mascheroni Constant and $B_{2n}$ are Bernoulli Numbers.

The $n$th Derivative of $\Psi(z)$ is called the Polygamma Function and is denoted $\psi_n(z)$. Since the digamma function is the zeroth derivative of $\Psi(z)$ (i.e., the function itself), it is also denoted $\psi_0(z)$.

The digamma function satisfies

$$\Psi(z) = \int_0^\infty \left({{e^{-t}\over t}-{e^{-zt}\over 1-e^{-t}}}\right)\,dt.$$ (9)

For integral $z\equiv n$,

$$\Psi(n)=-\gamma+\sum_{k=1}^{n-1}{1\over k}=-\gamma+H_{n-1},$$ (10)

where $\gamma$ is the Euler-Mascheroni Constant and $H_n$ is a Harmonic Number. Other identities include

$${d\Psi\over dz} = \sum_{n=0}^\infty {1\over(z+n)^2}$$ (11)

$$\Psi(1-z)-\Psi(z)=\pi \cot(\pi z)$$ (12)

$$\Psi(z+1)=\Psi(z)+{1\over z}$$ (13)

$$\Psi(2z)={\textstyle{1\over 2}}\Psi(z)+{\textstyle{1\over 2}}\Psi(z+{\textstyle{1\over 2}})+\ln 2.$$ (14)

Special values are

$$\Psi({\textstyle{1\over 2}}) = -\gamma-2\ln 2$$ (15)

$$\Psi(1) = -\gamma.$$ (16)

At integral values,

$$\psi_0(n+1)=-\gamma+\sum_{k=1}^n {1\over k},$$ (17)

and at half-integral values,

$$\psi_0({\textstyle{1\over 2}}\pm n)=-\ln(4\gamma)+2\sum_{k=1}^n {1\over 2k-1}.$$ (18)

At rational arguments, $\psi_0(p/q)$ is given by the explicit equation

$$\psi_0\left({p\over q}\right)=-\gamma-\ln(2q)-{\textstyle{1\... ...ver q}\right)\ln\left[{\sin\left({\pi k\over q}\right)}\right]$$ (19)

for $0<p<q$ (Knuth 1973). These give the special values

$$\psi_0({\textstyle{1\over 2}}) = -\gamma-2\ln 2$$ (20)

$$\psi_0({\textstyle{1\over 3}}) = {\textstyle{1\over 6}}(-6\gamma-\pi\sqrt{3}-9\ln 3)$$ (21)

$$\psi_0({\textstyle{2\over 3}}) = {\textstyle{1\over 6}}(-6\gamma+\pi\sqrt{3}-9\ln 3)$$ (22)

$$\psi_0({\textstyle{1\over 4}}) = -\gamma-{\textstyle{1\over 2}}\pi-3\ln 2$$ (23)

$$\psi_0({\textstyle{3\over 4}}) = {\textstyle{1\over 2}}(-2\gamma+\pi-6\ln 2)$$ (24)

$$\psi_0(1) = -\gamma,$$ (25)

where $\gamma$ is the Euler-Mascheroni Constant. Sums and differences of $\psi_1(r/s)$ for small integral $r$ and $s$ can be expressed in terms of Catalan's Constant and $\pi$.

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 $\psi(x)$.'' Ch. 44 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 423-434, 1987.