The digamma function , visualized using domain coloringPlots of the digamma and the next three polygamma functions along the real line (they are real-valued on the real line)
Note: This is also equal to due to the definition of the digamma function: .
Series representation
Series formula
Euler's product formula for the gamma function, combined with the functional equation and an identity for the Euler–Mascheroni constant, yields the following expression for the digamma function, valid in the complex plane outside the negative integers (Abramowitz and Stegun 6.3.16):[1]
Equivalently,
Evaluation of sums of rational functions
The above identity can be used to evaluate sums of the form
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
and
With the series expansion of higher rank polygamma function a generalized formula can be given as
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.
Series with Gregory's coefficients, Cauchy numbers and Bernoulli polynomials of the second kind
There exist various series for the digamma containing rational coefficients only for the rational arguments. In particular, the series with Gregory's coefficientsGn is
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.[16][17] More complicated formulas, such as
are due to works of certain modern authors (see e.g. Appendix B in Blagouchine (2014)[18]).
The expansion can also be derived from the integral representation coming from Binet's second integral formula for the gamma function. Expanding as a geometric series and substituting an integral representation of the Bernoulli numbers leads to the same asymptotic series as above. Furthermore, expanding only finitely many terms of the series gives a formula with an explicit error term:
Inequalities
When x > 0, the function
is completely monotonic and in particular positive. This is a consequence of Bernstein's theorem on monotone functions applied to the integral representation coming from Binet's first integral for the gamma function. Additionally, by the convexity inequality , the integrand in this representation is bounded above by . Consequently
is also completely monotonic. It follows that, for all x > 0,
This recovers a theorem of Horst Alzer.[23] Alzer also proved that, for s ∈ (0, 1),
Related bounds were obtained by Elezovic, Giordano, and Pecaric, who proved that, for x > 0 ,
The mean value theorem implies the following analog of Gautschi's inequality: If x > c, where c ≈ 1.461 is the unique positive real root of the digamma function, and if s > 0, then
Moreover, equality holds if and only if s = 1.[26]
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:
The asymptotic expansion gives an easy way to compute ψ(x) when the real part of x is large. To compute ψ(x) for small x, the recurrence relation
can be used to shift the value of x to a higher value. Beal[28] 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
or
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.[29] (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.
Another alternative is to use the recurrence relation or the multiplication formula to shift the argument of into the range and to evaluate the Chebyshev series there.[30][31]
Special values
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 taking the logarithmic derivative of or where is real-valued, it can easily be deduced that
Apart from Gauss's digamma theorem, no such closed formula is known for the real part in general. We have, for example, at the imaginary unit the numerical approximation
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 = 1.46163214496836234126.... All others occur single between the poles on the negative axis:
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 well to approximate roots with small n.
Another improvement of Hermite's formula can be given:[11]
Regarding the zeros, the following infinite sum identities were recently proved by István Mező and Michael Hoffman[11][33]
In general, the function
can be determined and it is studied in detail by the cited authors.
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References
1 2 Abramowitz, M.; Stegun, I. A., eds. (1972). "6.3 psi (Digamma) Function.". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (10thed.). New York: Dover. pp.258–259.
1 2 3 4 Mező, István; Hoffman, Michael E. (2017). "Zeros of the digamma function and its Barnes G-function analogue". Integral Transforms and Special Functions. 28 (11): 846–858. doi:10.1080/10652469.2017.1376193. S2CID126115156.
↑ Nörlund, N. E. (1924). Vorlesungen über Differenzenrechnung. Berlin: Springer.
1 2 Blagouchine, Ia. V. (2016). "Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to π−1". Journal of Mathematical Analysis and Applications. 442: 404–434. arXiv:1408.3902. Bibcode:2014arXiv1408.3902B. doi:10.1016/J.JMAA.2016.04.032. S2CID119661147.
↑ 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. 148: 537–592. arXiv:1401.3724. doi:10.1016/j.jnt.2014.08.009.
↑ Classical topi s in complex function theorey. p.46.
↑ Gradshteyn, I. S.; Ryzhik, I. M. (2015). "8.365.5". Table of integrals, series and products. Elsevier Science. ISBN978-0-12-384933-5. LCCN2014010276.
↑ 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.
↑ Mathar, R. J. (2004). "Chebyshev series expansion of inverse polynomials". Journal of Computational and Applied Mathematics. 196 (2): 596–607. arXiv:math/0403344. doi:10.1016/j.cam.2005.10.013. App. E
↑ Hermite, Charles (1881). "Sur l'intégrale Eulérienne de seconde espéce". Journal für die reine und angewandte Mathematik (90): 332–338. doi:10.1515/crll.1881.90.332. S2CID118866486.
↑ Mező, István (2014). "A note on the zeros and local extrema of Digamma related functions". arXiv:1409.2971 [math.CV].
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