# Morera's theorem

Here we follow standard texts, such as Spiegel (1964)[1] or Levinson and Redheffer (1970). [2]

If for every closed contour ${\displaystyle C}$ within a region ${\displaystyle {\mathcal {R}}}$ of the complex and

${\displaystyle \oint _{C}f(z)\;dz=0}$

then ${\displaystyle f(z)}$ is analytic everywhere in ${\displaystyle {\mathcal {R}}}$.

## Proof:

Suppose that the hypothesis is false, and that in some portion of ${\displaystyle {\mathcal {R}}}$, there exists some contour ${\displaystyle \gamma }$ such that

${\displaystyle \oint _{\gamma }f(z)\;dz=A}$

where ${\displaystyle A}$ is nonzero.

We assume that the real and imaginary parts of ${\displaystyle f(z)}$ are continuously differentiable in ${\displaystyle {\mathcal {R}}}$, (otherwise ${\displaystyle df(z)/dz}$ would not exist at some points in ${\displaystyle {\mathcal {R}}}$ and ${\displaystyle f(z)}$ would not be analytic by the definition of analyticity.)

Rewriting the integral by subsituting real and imaginary parts ${\displaystyle f(z)=u(x,y)+iv(x,y)}$ and ${\displaystyle dz=dx+idy}$ we obtain

${\displaystyle \oint _{\gamma }f(z)\;dz=\int _{\gamma }(u+iv)(dx+idy)=\oint _{\gamma }(udx-vdy)+i\oint _{\gamma }(vdx+udy)=A}$

.

Because the partial derivatives of ${\displaystyle u(x,y)}$ and ${\displaystyle v(x,y)}$ exist, we may invoke the 2D version of Green's theorem for the real and imaginary parts

${\displaystyle \int _{C}f(z)\;dz=-\left(\int _{\mathcal {R}}\int \left({\frac {\partial v(x,y)}{\partial x}}+{\frac {\partial u(x,y)}{\partial y}}\right)\;dx\;dy\right)+i\left(\int _{\mathcal {R}}\int \left({\frac {\partial u(x,y)}{\partial x}}-{\frac {\partial v(x,y)}{\partial y}}\right)\;dx\;dy\right)=A.}$

Because ${\displaystyle A=a+ib\neq 0}$ at least one of the Cauchy-Riemann equations is not satisfied, and the theorem is proven. However, this proof is less satisfactory, because we had to assume that the real and imaginary parts of ${\displaystyle f(z)}$ were differentiable.

# Alternate proof:

In general, Morera's theorem is a statement that if ${\displaystyle f(z)}$ is continuous, then it has an anti-derivative ${\displaystyle F(z)}$, which is an analytic function for all ${\displaystyle z}$ in the region ${\displaystyle {\mathcal {R}}}$. Furthermore, this forces ${\displaystyle f(z)}$ to be analytic, as well.

Suppose that we define a function ${\displaystyle F(z)=\int _{a}^{z}f(u)\;du}$ and form the derivative of ${\displaystyle F(z)}$ by the formal definition of a differentiation

${\displaystyle \lim _{\Delta z\rightarrow 0}\left|{\frac {F(z+\Delta z)-F(z)}{\Delta z}}-f(z)\right|=\lim _{\Delta z\rightarrow 0}\left|{\frac {\int _{a}^{z+\Delta z}f(u)\;du-\int _{a}^{z}f(u)\;du-f(z)\;\Delta z}{\Delta z}}\right|}$

.

This expression should vanish in the limit, if ${\displaystyle F(z)}$ is the anti-derivative of ${\displaystyle f(z)}$.

We can rewrite this limit as

${\displaystyle \lim _{\Delta z\rightarrow 0}\left|{\frac {\int _{a}^{z+\Delta z}f(u)\;du-\int _{a}^{z}f(u)\;du-f(z)\;\Delta z}{\Delta z}}\right|=\lim _{\Delta z\rightarrow 0}\left|{\frac {\int _{z}^{z+\Delta z}\left[f(u)-f(z)\right]\;du}{\Delta z}}\right|}$

.

Because ${\displaystyle f(z)}$ is assumed to be continuous, by the formal definition of continuity, whenever ${\displaystyle |f(u)-f(z)|<\epsilon }$ there exists, for all ${\displaystyle u}$ a ${\displaystyle \delta }$ such that ${\displaystyle |u-z|<\delta }$.

Given ${\displaystyle \delta >|\Delta z|}$ this is true, allowing us to estimate the bound on the integrand with ${\displaystyle \epsilon }$

${\displaystyle \lim _{\Delta z\rightarrow 0}\left|{\frac {F(z+\Delta z)-F(z)}{\Delta z}}-f(z)\right|\leq \lim _{\Delta z\rightarrow 0}{\frac {1}{\Delta z}}\left|\int _{z}^{z+\Delta z}\left[f(u)-f(z)\right]\;du\right|<\epsilon }$

.

Because ${\displaystyle \epsilon }$ is arbitrary ${\displaystyle F^{\prime }(z)=f(z)}$. ${\displaystyle F(z)}$ is analytic by the definition of analyticity as the existence of the derivative. Furthermore, because ${\displaystyle F(z)}$ is defined by an integral, this result holds for any integration path in ${\displaystyle {\mathcal {R}}}$ we have

${\displaystyle \oint _{C}f(u)\;du=\int _{a}^{z}f(u)\;du+\int _{z}^{a}f(u)\;du=0}$

for arbitrary ${\displaystyle a}$ and ${\displaystyle z}$ in ${\displaystyle {\mathcal {R}}}$, which in turn shows that ${\displaystyle f(z)}$ must be analytic for all ${\displaystyle z}$ in ${\displaystyle {\mathcal {R}}}$, proving Morera's theorem.

## Further ramifications of this second proof

It is possible to show from the Cauchy-Riemann equations that the derivative of an analytic function is, itself, analytic, and is infinitely differentiable in its region of analyticity. It is furthermore possible to construct a new function ${\displaystyle F_{0}(z)}$ that is the anti-derivative of ${\displaystyle F(z)}$. This process may also be repeated ad infinitum.

## References

1. Spiegel, Murray R. "Theory and problems of complex variables, with an introduction to Conformal Mapping and its applications." Schaum's outline series (1964).
2. Levinson, Norman, and Raymond M. Redheffer. "Complex variables." (1970), Holden-Day, New York.