Digamma function
The digamma function or psi- function is a function in mathematics, which is defined as:
So it is the logarithmic derivative of the gamma function. The digamma function is the first of the Polygammafunktionen. Except for their simple poles for negative integer arguments it is (just like the gamma function ) throughout holomorphic.
- 3.1 Special values
Calculation
The relationship to the harmonic series
The Digammafunktion, which is usually represented as ψ0 (x ) ψ0 (x ) or ( according to the shape of the pre-classical Greek letter digamma Ϝ ) is related to the harmonic series in the following relationship:
Wherein Hn is the n- th element of the harmonic series and the Euler Mascheroni γ constant. For half-integer values , it can be written as:
Integral representation
The Digammafunktion can be represented as the following integral:
This can also be written as:
This follows from the formula for the Euler integral for the harmonic series.
Taylor series
By the Taylor series expansion around the point z = 1 Digammafunktion can be represented as follows:
It converges for | z | < 1 Here is the Riemann ζ - function. The series can be easily derived from the corresponding Taylor series for the Hurwitz ζ - function.
Binomial series
The binomial series for the Digammafunktion follows from the Euler integral
Where the generalized binomial coefficient is.
Functional equation
The Digammafunktion satisfies the following functional equation, which can be derived directly from the logarithmic derivative of the gamma function:
This, however, can not be calculated ψ (1/ 2); this value is listed below.
Recursion and sum expressions
The digamma function satisfies the recursion
Or
Where Δ is the difference operator right-sided. This satisfies the recursion relation of the harmonic series. It follows
More generally,
From the Gaussian product representation of the gamma function is equivalent to leaves
Conclude.
Quotient relation to the gamma function
For the quotient of Digammafunktion and gamma function, the product representation provides the expression
For positive integers diverge both Digamma and gamma function at their negative values , then it follows
Using the functional equation for the Gamma function, one can even find out that the value of the ratio depending only on the argument of the gamma function, that is valid for integer finally
Gaussian sum
The Digammafunktion has a Gaussian sum of the form
For natural numbers. Here, ζ (s, q) Hurwitz ζ function and the Bernoulli polynomial. A special case of the theorem of multiplication
Gaussian digamma theorem
For integers m and k ( m < k), which can be expressed Digammafunktion elementary functions
Special values
The digamma function will include the following special values:
Derivation
The derivation of the Digammafunktion is, according to the definition Trigamma function
The second Polygammafunktion.