The color representation of the digamma function,
, in a rectangular region of the complex plane
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
It is the first of the polygamma functions.
The digamma function is often denoted as ψ0(x), ψ(0)(x) or Ϝ (the uppercase form of the archaic Greek consonant digamma meaning double-gamma).
Relation to harmonic numbers
The gamma function obeys the equation
Taking the derivative with respect to z gives:
Dividing by Γ(z + 1) or the equivalent zΓ(z) gives:
Since the harmonic numbers are defined as
the digamma function is related to it by:
where Hn is the nth harmonic number, and γ is the Euler–Mascheroni constant. For half-integer values, it may be expressed as
Harmonic Mean Value Inequality
Inspired by the harmonic mean value inequality for the classical gamma function, Horzt Alzer and Graham Jameson proved, among other things, a harmonic mean-value inequality for the digamma function:
Equality holds if and only if 
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 Leonhard Euler's integral formula for the harmonic numbers.
Infinite product representation
The function is an entire function, and it can be represented by the infinite product
Here is the kth zero of (see below), and is the Euler–Mascheroni constant.
The digamma function 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,
otherwise the series will be greater than the harmonic series and thus diverge. Hence
With the series expansion of higher rank polygamma function a generalized formula can be given as
provided the series on the left converges.
The digamma has a rational zeta series, given by the Taylor series at z = 1. This is
which converges for |z| < 1. Here, ζ(n) is the Riemann zeta function. This series is easily derived from the corresponding Taylor's series for the Hurwitz zeta function.
The Newton series for the digamma, sometimes referred to as Stern series, reads
k) is the binomial coefficient. It may also be generalized to
where m = 2,3,4,...
The digamma function satisfies a reflection formula similar to that of the gamma function:
Recurrence formula and characterization
The digamma function satisfies the recurrence relation
Thus, it can be said to "telescope" 1 / x, for one has
where Δ is the forward difference operator. This satisfies the recurrence relation of a partial sum of the harmonic series, thus implying the formula
where γ is the Euler–Mascheroni constant.
More generally, one has
Another series expansion is:
where are the Bernoulli numbers.
Actually, ψ is the only solution of the functional equation
that is monotonic 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 Gauss. More complicated formulas, such as
are due to works of certain modern authors (see e.g. Appendix B in Blagouchine (2014)).
Gauss's digamma theorem
For positive integers r and m (r < m), the digamma function may be expressed in terms of Euler's constant and a finite number of elementary functions
which holds, because of its recurrence equation, for all rational arguments.
Computation and approximation
According to the Euler–Maclaurin formula applied to
the digamma function for x, 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 Bk is the kth Bernoulli number and ζ is the Riemann zeta function. Although the infinite sum does not converge for any 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 x14 cut off, which yields "more than enough precision" (at least 12 digits except near the zeroes).
As x goes to infinity, ψ(x) gets arbitrarily close to both ln(x − 1/2) and ln x. Going down from x + 1 to x, ψ decreases by 1 / x, ln(x − 1/2) decreases by ln (x + 1/2) / (x − 1/2), which is more than 1 / x, and ln x decreases by ln (1 + 1 / x), which is less than 1 / x. From this we see that for any positive x greater than 1/2,
or, for any positive x,
The exponential exp ψ(x) is approximately x − 1/2 for large x, but gets closer to x at small x, approaching 0 at x = 0.
For x < 1, we can calculate limits based on the fact that between 1 and 2, ψ(x) ∈ [−γ, 1 − γ], so
From the above asymptotic series for ψ, one can derive an asymptotic series for exp(−ψ(x)). The series matches the overall behaviour well, that is, it behaves asymptotically as it should for large arguments, and has a zero of unbounded multiplicity at the origin too.
This is similar to a Taylor expansion of exp(−ψ(1 / y)) at y = 0, but it does not converge. (The function is not analytic at infinity.) A similar series exists for exp(ψ(x)) which starts with
If one calculates the asymptotic series for ψ(x+1/2) it turns out that there are no odd powers of x (there is no x−1 term). This leads to the following asymptotic expansion, which 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,
Roots of the digamma function
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 x0 = 632144968... 1.461. All others occur single between the poles on the negative axis:
Already in 1881, Charles 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 ψ(xn) by its not convergent asymptotic expansion. The correct second term of this expansion is 1 / 2n, where the given one works good to approximate roots with small n.
Another improvement of Hermite's formula can be given:
Regarding the zeros, the following infinite sum identities were recently proved by István Mező and Michael Hoffman
In general, the function
can be determined and it is studied in detail by the cited authors.
The following results
also hold true.
Here γ is the Euler–Mascheroni constant.
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
- ^ a b
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.
- ^ Alzer, Horst; Jameson, Graham (2017). "A harmonic mean inequality for the digamma function and related results" (PDF). Rendiconti del Seminario Matematico della Università di Padova. 70 (201): 203–209. doi:10.4171/RSMUP/137-10. ISSN 0041-8994. LCCN 50046633. OCLC 01761704.
- ^ a b c d Mező, István; Hoffman, Michael E. (2017). "Zeros of the digamma function and its Barnes G-function analogue" (28): 846–858. doi:10.1080/10652469.2017.1376193.
- ^ Nörlund, N. E. (1924). Vorlesungen über Differenzenrechnung. Berlin: Springer.
- ^ a b Blagouchine, Ia. V. (2018). "Three Notes on Ser's and Hasse's Representations for the Zeta-functions". Integers (Electronic Journal of Combinatorial Number Theory). 18A: 1–45. arXiv:1606.02044 . Bibcode:2016arXiv160602044B.
- ^ 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.
- ^ Blagouchine, Iaroslav V. (2014). "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. 148: 537–592. arXiv:1401.3724 . doi:10.1016/j.jnt.2014.08.009.
- ^ Bernardo, José M. (1976). "Algorithm AS 103 psi(digamma function) computation" (PDF). Applied Statistics. 25: 315–317.
- ^ Beal, Matthew J. (2003). Variational Algorithms for Approximate Bayesian Inference (PDF) (PhD thesis). The Gatsby Computational Neuroscience Unit, University College London. pp. 265–266.
- ^ If it converged to a function f(y) then ln(f(y) / y) would have the same Maclaurin series as ln(1 / y) − φ(1 / y). But this does not converge because the series given earlier for φ(x) does not converge.
- ^ Hermite, Charles (1881). "Sur l'intégrale Eulérienne de seconde espéce,". Journal für die reine und angewandte Mathematik (90): 332–338.
- A047787 psi(1/3), A200064 psi(2/3), A020777 psi(1/4), A200134 psi(3/4), A200135 to A200138 psi(1/5) to psi(4/5).