Digamma function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:[1][2]
It is the first of the polygamma functions.
The digamma function is often denoted as ψ_{0}(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:
or:
Since the harmonic numbers are defined for positive integers n as
the digamma function is related to them by
where H_{0} = 0, and γ is the Euler–Mascheroni constant. For half-integer arguments the digamma function takes the values
Integral representations
If the real part of z is positive then the digamma function has the following integral representation due to Gauss:[3]
Combining this expression with an integral identity for the Euler–Mascheroni constant gives:
The integral is Euler's harmonic number , so the previous formula may also be written
A consequence is the following generalization of the recurrence relation:
An integral representation due to Dirichlet is:[3]
Gauss's integral representation can be manipulated to give the start of the asymptotic expansion of .[4]
This formula is also a consequence of Binet's first integral for the gamma function. The integral may be recognized as a Laplace transform.
Binet's second integral for the gamma function gives a different formula for which also gives the first few terms of the asymptotic expansion:[5]
From the definition of and the integral representation of the Gamma function, one obtains
with .[6]
Infinite product representation
The function is an entire function,[7] and it can be represented by the infinite product
Here is the kth zero of (see below), and is the Euler–Mascheroni constant.
Note: This is also equal to due to the definition of the digamma function: .
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 u_{n} 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
provided the series on the left converges.
Taylor series
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.
Newton series
The Newton series for the digamma, sometimes referred to as Stern series,[8][9] reads
where (^{s}
_{k}) is the binomial coefficient. It may also be generalized to
where m = 2,3,4,...[9]
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 coefficients G_{n} is
where (v)_{n} is the rising factorial (v)_{n} = v(v+1)(v+2) ... (v+n-1), G_{n}(k) are the Gregory coefficients of higher order with G_{n}(1) = G_{n}, Γ is the gamma function and ζ is the Hurwitz zeta function.[10][9] Similar series with the Cauchy numbers of the second kind C_{n} reads[10][9]
A series with the Bernoulli polynomials of the second kind has the following form[9]
where ψ_{n}(a) are the Bernoulli polynomials of the second kind defined by the generating equation
It may be generalized to
where the polynomials N_{n,r}(a) are given by the following generating equation
so that N_{n,1}(a) = ψ_{n}(a).[9] Similar expressions with the logarithm of the gamma function involve these formulas[9]
and
Reflection formula
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
for . Another series expansion is:
- ,
where are the Bernoulli numbers. This series diverges for all z and is known as the Stirling series.
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.[11][12] More complicated formulas, such as
are due to works of certain modern authors (see e.g. Appendix B in Blagouchine (2014)[13]).
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.
Asymptotic expansion
The digamma function has the asymptotic expansion
where B_{k} is the kth Bernoulli number and ζ is the Riemann zeta function. The first few terms of this expansion are:
Although the infinite sum does not converge for any z, any finite partial sum becomes increasingly accurate as z increases.
The expansion can be found by applying the Euler–Maclaurin formula to the sum[14]
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.[15] Alzer also proved that, for s ∈ (0, 1),[16]
Related bounds were obtained by Elezovic, Giordano, and Pecaric, who proved that, for x > 0 ,
where is the Euler–Mascheroni constant.[17] The constants appearing in these bounds are the best possible.[18]
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.[19]
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:
for
Equality holds if and only if .[20]
Computation and approximation
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[21] suggests using the above recurrence to shift x to a value greater than 6 and then applying the above expansion with terms above x^{14} 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.[22] (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.
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 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 x_{0} = 1.461632144968.... All others occur single between the poles on the negative axis:
Already in 1881, Charles Hermite observed[23] 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(x_{n}) 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:[7]
Regarding the zeros, the following infinite sum identities were recently proved by István Mező and Michael Hoffman[7]
In general, the function
can be determined and it is studied in detail by the cited authors.
The following results[7]
also hold true.
Here γ is the Euler–Mascheroni constant.
Regularization
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
See also
- Polygamma function
- Trigamma function
- Chebyshev expansions of the digamma function in Wimp, Jet (1961). "Polynomial approximations to integral transforms". Math. Comp. 15 (74): 174–178. doi:10.1090/S0025-5718-61-99221-3.
References
- 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.
- Whittaker and Watson, 12.3.
- Whittaker and Watson, 12.31.
- Whittaker and Watson, 12.32, example.
- "NIST. Digital Library of Mathematical Functions. DLMF, 5.9".
- 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.
- Nörlund, N. E. (1924). Vorlesungen über Differenzenrechnung. Berlin: Springer.
- Blagouchine, Ia. V. (2018). "Three Notes on Ser's and Hasse's Representations for the Zeta-functions" (PDF). INTEGERS: The Electronic Journal of Combinatorial Number Theory. 18A: 1–45. arXiv:1606.02044. Bibcode:2016arXiv160602044B.
- 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.
- 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.
- Bernardo, José M. (1976). "Algorithm AS 103 psi(digamma function) computation" (PDF). Applied Statistics. 25: 315–317. doi:10.2307/2347257. JSTOR 2347257.
- H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (217) (1997) 373–389.
- Ibid.
- N. Elezovic, C. Giordano and J. Pecaric, The best bounds in Gautschi’s inequality, Math. Inequal. Appl. 3 (2000), 239–252.
- F. Qi and B.-N. Guo, Sharp inequalities for the psi function and harmonic numbers, arXiv:0902.2524.
- A. Laforgia, P. Natalini, Exponential, gamma and polygamma functions: Simple proofs of classical and new inequalities, J. Math. Anal. Appl. 407 (2013) 495–504.
- 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.
- 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.