![Psi1](https://upload.wikimedia.org/wikipedia/commons/0/0a/Psi1.png)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the trigamma function, denoted or , is the second of the
polygamma function
In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function:
:\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z).
Thus
:\psi^(z) ...
s, and is defined by
:
.
It follows from this definition that
:
where is the
digamma function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
:\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac.
It is the first of the polygamma functions. It is strictly increasing and strict ...
. It may also be defined as the sum of the
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
:
making it a special case of the
Hurwitz zeta function
In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables with and by
:\zeta(s,a) = \sum_^\infty \frac.
This series is absolutely convergent for the given values of and and can ...
:
Note that the last two formulas are valid when is not a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
.
Calculation
A
double integral
In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...
representation, as an alternative to the ones given above, may be derived from the series representation:
:
using the formula for the sum of a
geometric series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
:\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots
is geometric, because each succ ...
. Integration over yields:
:
An asymptotic expansion as a
Laurent series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
is
:
if we have chosen , i.e. the
Bernoulli numbers
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
of the second kind.
Recurrence and reflection formulae
The trigamma function satisfies the
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
:
and the
reflection formula
In mathematics, a reflection formula or reflection relation for a function ''f'' is a relationship between ''f''(''a'' − ''x'') and ''f''(''x''). It is a special case of a functional equation, and it is very common in the literature ...
:
which immediately gives the value for ''z'' :
.
Special values
At positive half integer values we have that
:
Moreover, the trigamma function has the following special values:
:
where represents
Catalan's constant
In mathematics, Catalan's constant , is defined by
: G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots,
where is the Dirichlet beta function. Its numerical value is approximately
:
It is not known whether is irra ...
.
There are no roots on the real axis of , but there exist infinitely many pairs of roots for . Each such pair of roots approaches quickly and their imaginary part increases slowly logarithmic with . For example, and are the first two roots with .
Relation to the Clausen function
The
digamma function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
:\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac.
It is the first of the polygamma functions. It is strictly increasing and strict ...
at rational arguments can be expressed in terms of trigonometric functions and logarithm by the
digamma theorem. A similar result holds for the trigamma function but the circular functions are replaced by
Clausen's function
In mathematics, the Clausen function, introduced by , is a transcendental, special function of a single variable. It can variously be expressed in the form of a definite integral, a trigonometric series, and various other forms. It is intimate ...
. Namely,
:
Computation and approximation
An easy method to approximate the trigamma function is to take the derivative of the
asymptotic expansion of the digamma function.
:
Appearance
The trigamma function appears in this surprising sum formula:
[
:
]
See also
* Gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
* Digamma function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
:\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac.
It is the first of the polygamma functions. It is strictly increasing and strict ...
* Polygamma function
In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function:
:\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z).
Thus
:\psi^(z) ...
* Catalan's constant
In mathematics, Catalan's constant , is defined by
: G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots,
where is the Dirichlet beta function. Its numerical value is approximately
:
It is not known whether is irra ...
Notes
References
* Milton Abramowitz and Irene A. Stegun, '' Handbook of Mathematical Functions'', (1964) Dover Publications, New York. {{ISBN, 0-486-61272-4. See sectio
§6.4
* Eric W. Weisstein
Gamma and related functions