Cauchy integral formula

For a continuously differentiable function on a bounded planar domain, the Cauchy-Pompeiu formula supplements the Cauchy integral formula with an area integral of its Wirtinger derivative. Apply the Generalized Stokes theorem to $f(\zeta )d\zeta /(\zeta -z)$ on a punctured domain; the small-circle integral supplies $2\pi if(z)$. If the boundary term vanishes at infinity, it gives the Cauchy-Green operator representation $f(z)={\pi }^{-1}\int f_{\bar{\zeta }}(\zeta )/(z-\zeta )dA$.

Let $g$ be a meromorphic function with simple poles ${\alpha }_{+}$ above $c$ and ${\alpha }_{-}$ below $-c$, and $zg(z)\to 0$ uniformly at infinity in $\mathrm{Im}z\ge -c$. The left-to-right line integrals $H(\lambda )=(2\pi i{)}^{-1}{\int }_{\mathbb{R}-ic}g(z)/(z-\lambda )dz$ and $J(\lambda )=-(2\pi i{)}^{-1}{\int }_{\mathbb{R}+ic}g(z)/(z-\lambda )dz$ satisfy $H=g-r_{+}/(\lambda -{\alpha }_{+})$ above $-c$ and $J=r_{+}/(\lambda -{\alpha }_{+})$ below $c$, where $r_{+}$ is the upper residue. Upper semicircle closure proves both formulas by the residue theorem; subtracting the principal part makes $H$ holomorphic at ${\alpha }_{+}$ too. Consequently $H+J=g$ on the intervening strip. No lower-half-plane decay is needed.

If $f$ is holomorphic on a neighbourhood of the closed disc $|z-a|\le R$ and $|f(z)|\le M$ on its boundary, differentiating the Cauchy integral formula gives

For an entire function $f(z)={\sum }_j{c}_j{z}^j$, equality $|c_n|={\max }_{|z|=r}|f(z)|/r^n$ at a fixed $r>0$ holds exactly when $f(z)=c{z}^n$. The coefficient is the average of $r^{-n}f(r{e}^{i\theta })e^{-in\theta }$. Equality in its modulus bound forces the continuous integrand to have constant modulus and argument, hence to be constant. The identity theorem then extends the monomial identity from the circle to the whole plane. The zero function and the constant case $n=0$ are included.