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.
Read more about this topic: Digamma Function
Famous quotes containing the words special and/or values:
“Research shows clearly that parents who have modeled nurturant, reassuring responses to infants’ fears and distress by soothing words and stroking gentleness have toddlers who already can stroke a crying child’s hair. Toddlers whose special adults model kindliness will even pick up a cookie dropped from a peer’s high chair and return it to the crying peer rather than eat it themselves!”
—Alice Sterling Honig (20th century)
“Normality highly values its normal man. It educates children to lose themselves and to become absurd, and thus to be normal. Normal men have killed perhaps 100,000,000 of their fellow normal men in the last fifty years.”
—R.D. (Ronald David)