Kummer's function
Encyclopedia
In mathematics
, there are several functions known as Kummer's function. One is known as the confluent hypergeometric function
of Kummer. Another one, defined below, is related to the polylogarithm. Both are named for Ernst Kummer
.
Kummer's function is defined by
The duplication formula is
.
Compare this to the duplication formula for the polylogarithm:
An explicit link to the polylogarithm is given by
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, there are several functions known as Kummer's function. One is known as the confluent hypergeometric function
Confluent hypergeometric function
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity...
of Kummer. Another one, defined below, is related to the polylogarithm. Both are named for Ernst Kummer
Ernst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
.
Kummer's function is defined by
The duplication formula is
.Compare this to the duplication formula for the polylogarithm:
An explicit link to the polylogarithm is given by