Digamma function

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In mathematics, the digamma function is defined by

It is the first of the polygamma functions.

Calculation

The digamma function, often denoted also ψ0(x) or even ψ0(x), is related to the harmonic numbers in that

 

where Hn−1 is the (n−1)th harmonic number, and γ is the well-known Euler-Mascheroni constant.

and may be calculated with the integral

 

Recurrence formulae

The digamma function satisfies a reflection formula similar to that of the Gamma function,

 

which cannot be used to compute ψ(1/2), which is given below. The digamma function satisfies the recurrence relation  

Note that this satisfies the recurrence relation of a partial sum of the harmonic series, thus implying the formula  

Special values

The digamma function has the following special values:

 
 
 
 

References

Also see

Digamma function -- from MathWorld