Preamble
Now let U ⊆ ℂ be an open set, and let F : U → ℂ be a function defined on points in U for which there is a path ̲ from A to B. Show that ∫̲ F'(z) dz = F(B) - F(A).
Holomorphic function: Let A and B belong to U.
So under these circumstances, the integral of F' along any piecewise C^1 curve doesn't depend upon the curve. In particular, taking ̲ = C, it follows that ∫_C F'(z) dz = 0 for any piecewise C^1 oriented closed curve in U.
Problem: Let C ⊆ ℂ be the circle of radius R centered at 0 oriented counterclockwise. Evaluate ∫_C 1/z dz and draw conclusions about the function f(z) = 1/z.
Problem: Let U ⊆ ℂ be a connected open set, and let f : U → ℂ be a continuous function with the property that ∫_C f(z) dz = 0 for any piecewise C^1 oriented closed curve in U. If z0 ∈ U and z1 ∈ U is any other point, the fact that U is connected implies that there is a path from z0 to z1. The assumption holds implies that no matter what such path we choose, the integral ∫_C f(z) dz is independent of that choice.
Exercise: Verify the claim just made.
Problem: Thus, we can define the function F : U → ℂ as F(z1) := ∫_C f(z) dz, where C is any piecewise oriented curve from z0 to z1. Show that F is holomorphic and F' = f.