Digamma function

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: .