Digamma function

In mathematics, the digamma function is defined by

${\displaystyle \psi (x)={\frac {d}{dx}}\ln {\Gamma (x)}={\frac {\Gamma '(x)}{\Gamma (x)}}.}$

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

${\displaystyle \psi (n)=H_{n-1}-\gamma }$ 

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

and may be calculated with the integral

${\displaystyle \psi (x)=\int _{0}^{\infty }\left({\frac {e^{-t}}{t}}-{\frac {e^{-xt}}{1-e^{-t}}}\right)\,dt}$ 

Recurrence formulae

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

${\displaystyle \psi (1-x)-\psi (x)=\pi \,\!\cot {\pi x}}$ 

which cannot be used to compute ψ(1/2), which is given below. The digamma function satisfies the recurrence relation ${\displaystyle \psi (x+1)=\psi (x)+{\frac {1}{x}}}$ 

Note that this satisfies the recurrence relation of a partial sum of the harmonic series, thus implying the formula ${\displaystyle \psi (x)=H_{n-1}-\gamma }$ 

Special values

The digamma function has the following special values:

${\displaystyle \psi (1)=-\gamma \,\!}$ 

${\displaystyle \psi \left({\frac {1}{4}}\right)=-{\frac {\pi }{2}}-3\ln {2}-\gamma }$ 

${\displaystyle \psi \left({\frac {1}{3}}\right)=-{\frac {\pi {\sqrt {3}}+9\ln {3}}{6}}-\gamma }$ 

${\displaystyle \psi \left({\frac {1}{2}}\right)=-2\ln {2}-\gamma }$ 

References

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 . See section §6.3
• Wolfram Research's MathWorld by Eric Weisstein Digamma function -- from MathWorld

See also

• Gamma function
• Trigamma function
• Polygamma function
• Harmonic number
• Euler-Mascheroni constant

External links

Digamma function -- from MathWorld