Special Values
The digamma function has values in closed form for rational numbers, as a result of Gauss's digamma theorem. Some are listed below:
The roots of the digamma function are the saddle points of the complex-valued gamma function. Thus they lie all on the real axis. The only one on the positive real axis is the unique minimum of the real-valued gamma function on at . All others occur single between the pols on the negative axis: . Already 1881 Hermite observed that holds asymptotically . A better approximation of the location of the roots is given by
and using a further term it becomes still better
which both spring off the reflection formula via and substituting by its not convergent asymptotic expansion. The correct 2nd term of this expansion is of course, where the given one works good to approximate roots with small index n.
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