Complex analysis review 2(Cauchy Integral Theory)

时间:2022-03-21 16:21:16

Theorem 1(Cauchy-Green formula, Pompeiu formula)

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

f(z)=12πiUf(ξ)ξzdξ12πiUf(ξ)ξ¯dξ¯dξξz.

We can prove this by first consider the domain Uz,ϵ=UD(z,ϵ) . Then use Green’s formula in complex form for
f(ξ)ξzdξ

in Uz,ϵ . And note that
d=+¯

Finally, let ϵ0 .

Theorem 2 (Cauchy Integral Formula)

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

f(z)=12πiUf(ξ)ξzdz.

This is because
f(ξ)ξ¯=0.

Theorem 3 (Cauchy Integral Theorem)

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

f(z)=12πiUf(ξ)ξzdz.

Theorem 3’

Suppose that UC 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.