Editing Morera's theorem (section)

==Proof==
[[Image:Morera's Theorem Proof.png|thumb|right|The integrals along two paths from ''a'' to ''b'' are equal, since their difference is the integral along a closed loop.]]

There is a relatively elementary proof of the theorem.  One constructs an anti-derivative for ''f'' explicitly.

Without loss of generality, it can be assumed that ''D'' is [[connected space|connected]]. Fix a point ''z''<sub>0</sub> in ''D'', and for any <math>z\in D</math>, let <math>\gamma: [0,1]\to D</math> be a piecewise ''C''<sup>1</sup> curve such that <math>\gamma(0)=z_0</math> and <math>\gamma(1)=z</math>. Then define the function ''F'' to be
<math display="block">F(z) = \int_\gamma f(\zeta)\,d\zeta.</math>

To see that the function is well-defined, suppose <math>\tau: [0,1]\to D</math> is another piecewise ''C''<sup>1</sup> curve such that <math>\tau(0)=z_0</math> and <math>\tau(1)=z</math>. The curve <math>\gamma \tau^{-1}</math> (i.e. the curve combining <math>\gamma</math> with <math>\tau</math> in reverse) is a closed piecewise ''C''<sup>1</sup> curve in ''D''. Then,
<math display="block">\int_{\gamma} f(\zeta)\,d\zeta + \int_{\tau^{-1}} f(\zeta) \, d\zeta =\oint_{\gamma \tau^{-1}} f(\zeta)\,d\zeta = 0.</math>

And it follows that
<math display="block">\int_\gamma f(\zeta)\,d\zeta = \int_\tau f(\zeta)\,d\zeta.</math>

Then using the continuity of ''f'' to estimate difference quotients, we get that ''F''′(''z'')&nbsp;=&nbsp;''f''(''z'').  Had we chosen a different ''z''<sub>0</sub> in ''D'', ''F'' would change by a constant: namely, the result of integrating ''f'' along ''any'' piecewise regular curve between the new  ''z''<sub>0</sub> and the old, and this does not change the derivative.

Since ''f'' is the derivative of the holomorphic function ''F'', it is holomorphic. The fact that derivatives of holomorphic functions are holomorphic can be proved by using the fact that [[analyticity of holomorphic functions|holomorphic functions are analytic]], i.e. can be represented by a convergent [[power series]], and the fact that power series may be differentiated term by term.  This completes the proof.