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Concept
Digamma function
Summary
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
It is the first of the polygamma functions. This function is strictly increasing and strictly concave on , and it asymptotically behaves as
for large arguments () in the sector with some infinitesimally small positive constant .
The digamma function is often denoted as or Ϝ (the uppercase form of the archaic Greek consonant digamma meaning double-gamma).
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 H0 = 0, and γ is the Euler–Mascheroni constant. For half-integer arguments the digamma function takes the values
If the real part of z is positive then the digamma function has the following integral representation due to Gauss:
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:
Gauss's integral representation can be manipulated to give the start of the asymptotic expansion of .
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:
From the definition of and the integral representation of the gamma function, one obtains
with .
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.
Note: This is also equal to due to the definition of the digamma function: .
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Polygamma function
In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers defined as the (m + 1)th derivative of the logarithm of the gamma function: Thus holds where ψ(z) is the digamma function and Γ(z) is the gamma function. They are holomorphic on . At all the nonpositive integers these polygamma functions have a pole of order m + 1. The function ψ(1)(z) is sometimes called the trigamma function. Digamma function#Integral representations When m > 0 and Re z > 0, the polygamma function equals where is the Hurwitz zeta function.
Hurwitz zeta function
In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables s with Re(s) > 1 and a ≠ 0, −1, −2, ... by This series is absolutely convergent for the given values of s and a and can be extended to a meromorphic function defined for all s ≠ 1. The Riemann zeta function is ζ(s,1). The Hurwitz zeta function is named after Adolf Hurwitz, who introduced it in 1882. The Hurwitz zeta function has an integral representation for and (This integral can be viewed as a Mellin transform.
Harmonic number
In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: Starting from n = 1, the sequence of harmonic numbers begins: Harmonic numbers are related to the harmonic mean in that the n-th harmonic number is also n times the reciprocal of the harmonic mean of the first n positive integers. Harmonic numbers have been studied since antiquity and are important in various branches of number theory.