다이감마 함수(digamma function)

간단한 소개

 

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

 

 

반사공식

\(\psi(1 - x) - \psi(x) = \pi\,\!\cot{ \left ( \pi x \right ) }\)

 

차분방정식

\(\psi(x + 1) = \psi(x) + \frac{1}{x}\)

 

 

덧셈공식

 

\(\Gamma(z) \; \Gamma\left(z + \frac{1}{2}\right) = 2^{\frac{1}{2}-2z} \; \sqrt{2\pi} \; \Gamma(2z) \,\!\) 의 로그미분을 통해서 다음을 얻을 수 있음.

\(\psi(2x)=\psi(x)+\psi(x+{1\over2})+\ln 2\)

 

 

가우스의 Digamma 정리

 

\(\psi\left(\frac{m}{k}\right) = -\gamma -\ln(2k) -\frac{\pi}{2}\cot\left(\frac{m\pi}{k}\right) +2\sum_{n=1}^{\lceil (k-1)/2\rceil} \cos\left(\frac{2\pi nm}{k} \right) \ln\left(\sin\left(\frac{n\pi}{k}\right)\right)\)

 

 

special values

 

\(\psi(1) = -\gamma\,\!\)

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

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

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

\(\psi\left(\frac{1}{6}\right) = -\frac{\pi}{2}\sqrt{3} -2\ln{2} -\frac{3}{2}\ln(3) - \gamma\)

\(\psi\left(\frac{1}{8}\right) = -\frac{\pi}{2} - 4\ln{2} - \frac{1}{\sqrt{2}} \left\{\pi + \ln(2 + \sqrt{2}) - \ln(2 - \sqrt{2})\right\} - \gamma\)

 

 

 

 

 

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참고할만한 자료

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• http://en.wikipedia.org/wiki/Digamma_function
• http://www76.wolframalpha.com/input/?i=Digamma+function
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
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