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Encyclopedia > Polygamma function

In mathematics, the polygamma function of order m is defined as the m+1 'th derivative of the logarithm of the gamma function: Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula f′/f where f′ is the derivative of f. ... The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non natural numbers (where it is defined). ...

Here

is the digamma function and Γ(z) is the gamma function. The function ψ(1)(z) is sometimes called the trigamma function. In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: It is the first of the polygamma functions. ... In mathematics, the trigamma function, denoted ψ1(z), is the second of the polygamma functions, and is defined by . It follows from this definition that where ψ(z) is the digamma function. ...

## Contents

### Integral representation

The polygamma function may be represented as

which holds for Re z >0.

### Recurrence relation

It has the recurrence relation In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

### Series representation

The polygamma function has the series representation

which holds for any complex z not equal to a negative integer. This representation can be written more compactly in terms of the Hurwitz zeta function as In mathematics, the Hurwitz zeta function is one of the many zeta functions. ...

alternately, the Hurwitz zeta can be understood to generalize the polygamma to arbitrary, non-integer order.

### Taylor's series

The Taylor series at z=1 is As the degree of the Taylor series rises, it approaches the correct function. ...

,

which converges for |z|<1. Here, ζ(n) is the Riemann zeta function. This series is easily derived from the corresponding Taylor's series for the Hurwitz zeta function. This series may be used to derive a number of rational zeta series. In mathematics, the Riemann zeta function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... In 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. ... Results from FactBites:

 Hurwitz zeta function - Wikipedia, the free encyclopedia (524 words) In mathematics, the Hurwitz zeta function is one of the many zeta functions. The functional equation relates values of the zeta on the left- and right-hand sides of the complex plane. Note that this later form is the functional equation for the Riemann zeta, as originally given by Riemann.
 Reference.com/Encyclopedia/Gamma function (653 words) In mathematics, the Gamma function is a function that extends the concept of factorial to the complex numbers. The derivatives of the Gamma function are described in terms of the polygamma function. The derivative of the logarithm of the Gamma function is called the digamma function; higher derivatives are the polygamma functions.
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