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==Gauss's digamma theorem==
For positive integers {{mvar|r}} and {{mvar|m}} ({{math|''r'' < ''m''}}), the digamma function may be expressed in terms of [[Euler's constant]] and a finite number of [[elementary function]]s<ref>{{cite journal|first1=Junesang|last1=Choi|first2=Djurdje|last2=Cvijovic|title=Values of the polygamma functions at rational arguments|journal=Journal of Physics A|year=2007|volume=40|pages=15019|doi=10.1088/1751-8113/40/50/007|number=50|bibcode=2007JPhA...4015019C |s2cid=118527596 }}</ref>
<ref>{{cite journal|journal=Ann. Math.|year=1916|first1= J. L. W. V.|last1=Jensen|first2=T. H.|last2=Gronwall|title=An elementary exposition of the theory of the Gamma Function|volume=17|number=3|pages=124-166|doi=10.2307/2007272}}</ref>
:<math>\psi\left(\frac{r}{m}\right) = -\gamma -\ln(2m) -\frac{\pi}{2}\cot\left(\frac{r\pi}{m}\right) +2\sum_{n=1}^{\left\lfloor \frac{m-1}{2} \right\rfloor} \cos\left(\frac{2\pi nr}{m} \right) \ln\sin\left(\frac{\pi n}{m}\right) </math>
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:<math>\operatorname{Im} \psi(\tfrac{1}{2}+bi) = \frac{\pi}{2}\tanh (\pi b).</math>
Apart from Gauss's digamma theorem, no such closed formula is known for the real part in general. We have, for example, at the [[imaginary unit]] the numerical approximation {{OEIS2C|A248177}}
:<math>\operatorname{Re} \psi(i) = -\gamma-\sum_{n=0}^\infty\frac{n-1}{n^3+n^2+n+1} \approx 0.09465.</math>
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