binomial formula
The binomial formula gives the power series
expansion of the
power function. The power can be an integer,
rational, real, or even a complex number
. The formula
is
where denotes the falling
factorial, and where denotes the generalized binomial
coefficient.
For the power series reduces to a polynomial, and we
obtain the usual binomial theorem
. For other values of , the
radius of convergence
of the series is ; the right-hand series
converges
pointwise
for all complex to the value on the left
side. Also note that the binomial formula is valid at , but
for certain values of only. Of course, we have convergence if
is a natural number
. Furthermore, for and real , we have
absolute convergence
if , and conditional convergence if
. For we have absolute convergence for .
Title | binomial formula |
---|---|
Canonical name | BinomialFormula |
Date of creation | 2013-03-22 12:23:52 |
Last modified on | 2013-03-22 12:23:52 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 11 |
Author | rmilson (146) |
Entry type | Theorem |
Classification | msc 26A06 |
Synonym | Newton’s binomial series |
Related topic | BinomialTheorem |
Related topic | GeneralizedBinomialCoefficients |