<Dd> ∮ γ ⁡ d z z − z 0 = 2 π i . (\ displaystyle \ oint _ (\ gamma) (\ frac (dz) (z - z_ (0))) = 2 \ pi i .) </Dd> <P> 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 2πi times the winding number of the curve . </P> <P> 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: </P> <Dl> <Dd> ∮ 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))) </Dd> </Dl>

History of pi in maths for class 9