Digamma function
Encyclopedia
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the digamma function is defined as the logarithmic derivative
Logarithmic derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formulawhere f ′ is the derivative of f....
of the gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
:
It is the first of the polygamma functions.
Relation to harmonic numbers
The digamma functionFunction (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
, often denoted also as ψ0(x), ψ0(x) or
(after the shape of the archaic Greek letter Ϝ digammaDigamma
Digamma is an archaic letter of the Greek alphabet which originally stood for the sound /w/ and later remained in use only as a numeral symbol for the number "6"...
), is related to the harmonic numbers in that
where Hn is the n 'th harmonic number, and γ is the Euler-Mascheroni constant
Euler-Mascheroni constant
The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
. For half-integer values, it may be expressed as
Integral representations
It has the integralIntegral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
representation
valid if the real part of
is positive. This may be written as
which follows from Euler's integral formula for the harmonic numbers.
Series formula
Digamma can be computed in the complex plane outside negative integers (Abramowitz and Stegun 6.3.16), using
or
This can be utilized to evaluate infinite sums of rational functions, i.e.,
, where p(n) and q(n) are polynomials of n.Performing partial fraction
Partial fraction
In algebra, the partial fraction decomposition or partial fraction expansion is a procedure used to reduce the degree of either the numerator or the denominator of a rational function ....
on un in the complex field, in the case when all roots of q(n) are simple roots,
For the series to converge,
or otherwise the series will be greater than harmonic series
Harmonic series (mathematics)
In mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength...
and thus diverges.
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 seriesRational zeta seriesIn mathematics, a rational zeta series is the representation of an arbitrary real number in terms of a series consisting of rational numbers and the Riemann zeta function or the Hurwitz zeta function...
, given by the Taylor seriesTaylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
at z=1. This is
,
which converges for |z|<1. Here,
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 follows from Euler's integral formula:
where
is the binomial coefficientBinomial coefficientIn mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...
.
Reflection formula
The digamma function satisfies a reflection formulaReflection formulaIn mathematics, a reflection formula or reflection relation for a function f is a relationship between f and f...
similar to that of the Gamma functionGamma functionIn mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
,
Recurrence formula and characterization
The digamma function satisfies the recurrence relationRecurrence relationIn mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....
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 seriesHarmonic series (mathematics)In mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength...
, thus implying the formula
where
is the Euler-Mascheroni constantEuler-Mascheroni constantThe Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
.
More generally, one has
Actually,
is the only solution of the functional equation 
that is monotone on 
and such that 
.
Gaussian sum
The digamma has a Gaussian sum of the form
for integers
. Here, ζ(s,q) is the Hurwitz zeta function and 
is a Bernoulli polynomial. A special case of the multiplication theoremMultiplication theoremIn mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; thus the name...
is
and a neat generalization of this is
in which it is assumed that q is a natural number, and that 1-qa is not.
Gauss's digamma theorem
For positive integers m and k (with m < k), the digamma function may be expressed in terms of elementary functions as:
Computation & approximation
According to J.M. Bernardo AS 103 algorithm the digamma function for x, a real number, can be approximated by
or the asymptotic series
n as integer, where B(n) is the nth Bernoulli number for and
is the Riemann zeta function.
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:
External links
- Cephes - C and C++ language special functions math library psi(1/2), psi(1/3), psi(2/3), psi(1/4), psi(3/4), to psi(1/5) to psi(4/5).
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