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:
“Friendship is learned by watching and listening to you. If she sees that your friends are people you like and trust and don’t pretend with—people who suit you—she probably won’t pick friends who just pass by, or people who can help her or improve her status. If you treat friends in a special way, if you are kinder, more generous, more sympathetic, more forgiving with friends, she probably will be, too.”
—Stella Chess (20th century)
“Autonomy means women defining themselves and the values by which they will live, and beginning to think of institutional arrangements which will order their environment in line with their needs.... Autonomy means moving out from a world in which one is born to marginality, to a past without meaning, and a future determined by others—into a world in which one acts and chooses, aware of a meaningful past and free to shape one’s future.”
—Gerda Lerner (b. 1920)