section_path: Role and characterizations in mathematics/Cauchy's integral formula
section_name: Cauchy's integral formula
paragraph_text: Cauchy's integral formula
  
  One of the key tools in complex analysis is contour integration of a function over a positively oriented (rectifiable) Jordan curve γ. A form of Cauchy's integral formula states that if a point z is interior to γ, then ∮ γ d z z − z 0 = 2 π i. {\displaystyle \oint _{\gamma }{\frac {dz}{z-z_{0}}}=2\pi i.}
  
  Although the curve γ is not a circle, and hence does not have any obvious connection to the constant π, 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 γ does not contain z, then the above integral is 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 f (z) on the Jordan curve γ and the value of f (z) at any interior point z of γ: ∮ γ f (z) z − z 0 d z = 2 π i f (z 0) {\displaystyle \oint _{\gamma }{f(z) \over z-z_{0}}\,dz=2\pi if(z_{0})} provided f (z) is analytic in the region enclosed by γ and extends continuously to γ. Cauchy's integral formula is a special case of the residue theorem, that if g (z) is a meromorphic function the region enclosed by γ and is continuous in a neighbourhood of γ, then ∮ γ g (z) d z = 2 π i ∑ Res (g, a k) {\displaystyle \oint _{\gamma }g(z)\,dz=2\pi i\sum \operatorname {Res} (g,a_{k})} where the sum is of the residues at the poles of g (z).