Cauchy integral formula
The formulas.
Let be an open disk in the
complex plane, and let be a holomorphic11It is
necessary to draw a distinction between holomorphic functions (those having
a complex derivative) and analytic functions
(those representable by
power series
). The two concepts are, in fact, equivalent
, but
the standard proof of this fact uses the Cauchy Integral Formula
with the (apparently) weaker holomorphicity hypothesis
. function
defined on some open domain that contains and its boundary. Then,
for every we have
Here is the corresponding circular boundary contour, oriented counterclockwise, with the most obvious parameterization given by
Discussion.
The first of the above formulas underscores the “rigidity” of
holomorphic functions. Indeed, the values of the holomorphic function
inside a disk are completely specified by its values on the
boundary of the disk. The second formula is useful, because it gives
the derivative
in terms of an integral, rather than as the outcome of
a limit process.
Generalization.
The following technical generalization of the formula is needed for
the treatment of removable singularities. Let be a finite subset
of , and suppose that is holomorphic for all , but also that
is bounded
near all . Then, the
above formulas are valid for all .
Using the Cauchy residue theorem, one can further generalize the integral formula to the situation where is any domain and is any closed rectifiable curve in ; in this case, the formula becomes
where denotes the winding number of . It is valid for all points which are not on the curve .
Title | Cauchy integral formula |
---|---|
Canonical name | CauchyIntegralFormula |
Date of creation | 2013-03-22 12:04:46 |
Last modified on | 2013-03-22 12:04:46 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 25 |
Author | djao (24) |
Entry type | Theorem |
Classification | msc 30E20 |