It is the first of the polygamma functions.
- 1 Relation to harmonic numbers
- 2 Integral representations
- 3 Series formula
- 4 Taylor series
- 5 Newton series
- 6 Reflection formula
- 7 Recurrence formula and characterization
- 8 Some finite sums involving the digamma function
- 9 Gauss's digamma theorem
- 10 Computation and approximation
- 11 Special values
- 12 Regularization
- 13 See also
- 14 References
- 15 External links
Relation to harmonic numbers
where Hn is the n-th harmonic number, and γ is the Euler-Mascheroni constant. For half-integer values, it may be expressed as
If the real part of x is positive then the digamma function has the following integral representation
This may be written as
which follows from Euler's integral formula for the harmonic numbers.
Digamma can be computed in the complex plane outside negative integers (Abramowitz and Stegun 6.3.16), using
This can be utilized to evaluate infinite sums of rational functions, i.e.,
where p(n) and q(n) are polynomials of n.
Performing partial fraction on un in the complex field, in the case when all roots of q(n) are simple roots,
For the series to converge,
or otherwise the series will be greater than harmonic series and thus diverges.
With the series expansion of higher rank polygamma function a generalized formula can be given as
provided the series on the left converges.
The Newton series for the digamma follows from Euler's integral formula:
where is the binomial coefficient.
Recurrence formula and characterization
The digamma function satisfies the recurrence relation
Thus, it can be said to "telescope" 1/x, for one has
where γ is the Euler-Mascheroni constant.
More generally, one has
Actually, ψ is the only solution of the functional equation
that is monotone on R+ and satisfies F(1) = −γ. This fact follows immediately from the uniqueness of the Γ function given its recurrence equation and convexity-restriction. This implies the useful difference equation:
Some finite sums involving the digamma function
There are numerous finite summation formulas for the digamma function. Basic summation formulas, such as
are due to works of certain modern authors (see e.g. Appendix B in).
Gauss's digamma theorem
which holds, because of its recurrence equation, for all rational arguments.
Computation and approximation
the digamma function for x, also a real number, can be approximated by
which is the beginning of the asymptotical expansion of ψ(x). The full asymptotic series of this expansions is
where is the k-th Bernoulli number and ζ is the Riemann zeta function. Although the infinite sum converges for no x, this expansion becomes more accurate for larger values of x and any finite partial sum cut off from the full series. To compute ψ(x) for small x, the recurrence relation
can be used to shift the value of x to a higher value. Beal suggests using the above recurrence to shift x to a value greater than 6 and then applying the above expansion with terms above cut off, which yields "more than enough precision" (at least 12 digits except near the zeroes).
From the above asymptotic series for ψ you can derive asymptotic series for that contain only rational functions and constants. The first series matches the overall behaviour of well, that is, it behaves asymptotically identically for large arguments and has a zero of unbounded multiplicity at the origin, too. It can be considered a Taylor expansion of at y = 0.
The other expansion is more precise for large arguments and saves computing terms of even order.
The digamma function has values in closed form for rational numbers, as a result of Gauss's digamma theorem. Some are listed below:
Moreover, by the series representation, it can easily be deduced that at the imaginary unit
The roots of the digamma function are the saddle points of the complex-valued gamma function. Thus they lie all on the real axis. The only one on the positive real axis is the unique minimum of the real-valued gamma function on R+ at . All others occur single between the poles on the negative axis:
Already 1881 Hermite observed that
holds asymptotically. A better approximation of the location of the roots is given by
and using a further term it becomes still better
which both spring off the reflection formula via
and substituting by its not convergent asymptotic expansion. The correct 2nd term of this expansion is of course , where the given one works good to approximate roots with small index n.
The Digamma function appears in the regularization of divergent integrals
this integral can be approximated by a divergent general Harmonic series, but the following value can be attached to the series
- Polygamma function
- Trigamma function
- Chebyshev expansions of the Digamma function in Wimp, Jet (1961). "Polynomial approximations to integral transforms". Math. Comp. 15: 174–178. doi:10.1090/S0025-5718-61-99221-3.
- Abramowitz, M.; Stegun, I. A., eds. (1972). "6.3 psi (Digamma) Function.". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (10th ed.). New York: Dover. pp. 258–259.
- Weisstein, Eric W., "Digamma function", MathWorld.
- R. Campbell. Les intégrales eulériennes et leurs applications, Dunod, Paris, 1966.
- H.M. Srivastava and J. Choi. Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, the Netherlands, 2001.
- Iaroslav V. Blagouchine A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations Journal of Number Theory (Elsevier), vol. 148, pp. 537-592, 2015. arXiv PDF
- Bernardo, José M. (1976). "Algorithm AS 103 psi(digamma function) computation". Applied Statistics 25: 315–317.
- Beal, Matthew J. (2003). Variational Algorithms for Approximate Bayesian Inference (PhD thesis). The Gatsby Computational Neuroscience Unit, University College London. pp. 265–266.