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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: .
Official source
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