Complex analysis review 2(Cauchy Integral Theory)

Theorem 1(Cauchy-Green formula, Pompeiu formula)

Suppose that U⊂C is a bounded domain, having C1 boundary. f(z)=u(x,y)+iv(x,y)∈C1(U¯), then

f(z)=12πi∫∂Uf(ξ)ξ−zdξ−12πi∬U∂f(ξ)∂ξ¯dξ¯∧dξξ−z.

We can prove this by first consider the domain Uz,ϵ=U∖D(z,ϵ). Then use Green’s formula in complex form for

f(ξ)ξ−zdξ

in Uz,ϵ. And note that

d=∂+∂¯

Finally, let ϵ→0.

Theorem 2 (Cauchy Integral Formula)

Suppose that U⊂C is a bounded domain, having C1 boundary, f(z) is analytic on U, f(z)∈C1(U¯). Then

f(z)=12πi∫∂Uf(ξ)ξ−zdz.

This is because

∂f(ξ)∂ξ¯=0.

Theorem 3 (Cauchy Integral Theorem)

Suppose that U⊂C is a bounded domain, having C1 boundary, F(z) is analytic on U, F(z)∈C1(U¯). Then

∫∂UF(ξ)dξ=0.

Note that Theorem 2 and Theorem 3 are equivalent.

Cauchy-Goursat Theorem

It is often too strong to have smooth boundary. We can refine the condition and get the following theorems.

Theorem 2’

Suppose that U⊂C is a bounded domain, and ∂U is rectifiable. f(z) is analytic on U, and continuous on U¯. Then

f(z)=12πi∫∂Uf(ξ)ξ−zdz.

Theorem 3’

Suppose that U⊂C is a bounded domain, and ∂U is rectifiable. f(z) is analytic on U, and continuous on U¯. Then

∫∂Uf(ξ)dξ=0.

Note that Theorem 2’ and Theorem 3’ are also equivalent. And the theorems are true for multi-connected fields.