Editing Cauchy's integral theorem (section)

==Proof==
If one assumes that the partial derivatives of a holomorphic function are continuous, the Cauchy integral theorem can be proven as a direct consequence of [[Green's theorem]] and the fact that the real and imaginary parts of <math>f=u+iv</math> must satisfy the [[Cauchy–Riemann equations]] in the region bounded by {{nowrap|<math>\gamma</math>,}} and moreover in the open neighborhood {{mvar|U}} of this region. Cauchy provided this proof, but it was later proven by Goursat without requiring techniques from [[vector calculus]], or the continuity of partial derivatives.

We can break the integrand {{nowrap|<math>f</math>,}} as well as the differential <math>dz</math> into their real and imaginary components:

<math display="block"> f=u+iv </math>
<math display="block"> dz=dx+i\,dy </math>

In this case we have
<math display="block">\oint_\gamma f(z)\,dz = \oint_\gamma (u+iv)(dx+i\,dy) = \oint_\gamma (u\,dx-v\,dy) +i\oint_\gamma (v\,dx+u\,dy)</math>

By [[Green's theorem]], we may then replace the integrals around the closed contour <math>\gamma</math> with an area integral throughout the domain <math>D</math> that is enclosed by <math>\gamma</math> as follows:

<math display="block">\oint_\gamma (u\,dx-v\,dy) = \iint_D \left( -\frac{\partial v}{\partial x} -\frac{\partial u}{\partial y} \right) \,dx\,dy </math>
<math display="block">\oint_\gamma (v\,dx+u\,dy) = \iint_D \left(  \frac{\partial u}{\partial x} -\frac{\partial v}{\partial y} \right) \,dx\,dy </math>

But as the real and imaginary parts of a function holomorphic in the domain {{nowrap|<math>D</math>,}} <math>u</math> and <math>v</math> must satisfy the [[Cauchy–Riemann equations]] there:
<math display="block">\frac{ \partial u }{ \partial x } = \frac{ \partial v }{ \partial y } </math>
<math display="block">\frac{ \partial u }{ \partial y } = -\frac{ \partial v }{ \partial x } </math>

We therefore find that both integrands (and hence their integrals) are zero

<math display="block">\iint_D \left( -\frac{\partial v}{\partial x} -\frac{\partial u}{\partial y} \right )\,dx\,dy = \iint_D \left( \frac{\partial u}{\partial y} - \frac{\partial u}{\partial y} \right ) \, dx \, dy =0</math>
<math display="block">\iint_D \left( \frac{\partial u}{\partial x}-\frac{\partial v}{\partial y} \right )\,dx\,dy = \iint_D \left( \frac{\partial u}{\partial x} - \frac{\partial u}{\partial x} \right ) \, dx \, dy = 0</math>

This gives the desired result
<math display="block">\oint_\gamma f(z)\,dz = 0</math>