Editing Pi (section)

=== Cauchy's integral formula ===
[[File:Factorial05.jpg|thumb|right|Complex analytic functions can be visualized as a collection of streamlines and equipotentials, systems of curves intersecting at right angles.  Here illustrated is the complex logarithm of the Gamma function.]]

One of the key tools in [[complex analysis]] is [[contour integration]] of a function over a positively oriented ([[rectifiable curve|rectifiable]]) [[Jordan curve]] {{math|''γ''}}.  A form of [[Cauchy's integral formula]] states that if a point {{math|''z''<sub>0</sub>}} is interior to {{math|''γ''}}, then<ref>{{cite book |first=Lars |last=Ahlfors |title=Complex analysis |publisher=McGraw-Hill |year=1966 |page=115 |author-link=Lars Ahlfors}}</ref>

<math display=block>\oint_\gamma \frac{dz}{z-z_0} = 2\pi i.</math>

Although the curve {{math|''γ''}} is not a circle, and hence does not have any obvious connection to the constant {{pi}}, a standard proof of this result uses [[Morera's theorem]], which implies that the integral is invariant under [[homotopy]] of the curve, so that it can be deformed to a circle and then integrated explicitly in polar coordinates.  More generally, it is true that if a rectifiable closed curve {{math|γ}} does not contain {{math|''z''<sub>0</sub>}}, then the above integral is {{math|2π''i''}} times the [[winding number]] of the curve.

The general form of Cauchy's integral formula establishes the relationship between the values of a [[complex analytic function]] {{math|''f''(''z'')}} on the Jordan curve {{math|''γ''}} and the value of {{math|''f''(''z'')}} at any interior point {{math|''z''<sub>0</sub>}} of {{math|γ}}:<ref>{{cite book |last=Joglekar |first=S. D. |title=Mathematical Physics |publisher=Universities Press |year=2005 |page=166 |isbn=978-81-7371-422-1}}</ref>

<math display=block>\oint_\gamma { f(z) \over z-z_0 }\,dz = 2\pi i f (z_{0})</math>

provided {{math|''f''(''z'')}} is analytic in the region enclosed by {{math|''γ''}} and extends continuously to {{math|''γ''}}.  Cauchy's integral formula is a special case of the [[residue theorem]], that if {{math|''g''(''z'')}} is a [[meromorphic function]] the region enclosed by {{math|''γ''}} and is continuous in a neighbourhood of {{math|''γ''}}, then

<math display=block>\oint_\gamma g(z)\, dz =2\pi i \sum \operatorname{Res}( g, a_k ) </math>

where the sum is of the [[residue (mathematics)|residues]] at the [[pole (complex analysis)|poles]] of {{math|''g''(''z'')}}.