Laurent series split into regular and principal parts

Let $0<R_0<R$ and choose $\sigma$ with $R_0<\sigma <R$; the coefficient formula gives $\mathrm{\mid }{c}_n\mathrm{\mid }\le M_{\sigma }\mathrm{/}{\sigma }^n$ for $n\ge 0$, where $M_{\sigma }={\max }_{\mathrm{\mid }\zeta -a\mathrm{\mid }=\sigma }\mathrm{\mid }f(\zeta )\mathrm{\mid }$, so $\mathrm{\mid }{c}_n(z-a{)}^n\mathrm{\mid }\le M_{\sigma }(R_0\mathrm{/}\sigma {)}^n$ for $\mathrm{\mid }z-a\mathrm{\mid }\le R_0$.

Let $\rho >r$ and choose ${\rho }_0$ with $r<{\rho }_0<\rho$; the coefficient formula gives $\mathrm{\mid }{c}_{-m}\mathrm{\mid }\le M_{{\rho }_0}{\rho }_0^m$ for $m\ge 1$, where $M_{{\rho }_0}={\max }_{\mathrm{\mid }\zeta -a\mathrm{\mid }={\rho }_0}\mathrm{\mid }f(\zeta )\mathrm{\mid }$, so $\mathrm{\mid }{c}_{-m}(z-a{)}^{-m}\mathrm{\mid }\le M_{{\rho }_0}({\rho }_0\mathrm{/}\rho {)}^m$ for $\mathrm{\mid }z-a\mathrm{\mid }\ge \rho$.

On the original annulus, [L1] gives that the Laurent series converges to $f$, and by definition that series is the sum of its nonnegative-power and negative-power subseries. So $f=f_{\mathrm{r}\mathrm{e}\mathrm{g}}+f_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}}$ there.

Uniqueness of the Laurent coefficients from [L2] makes both subseries unique term by term.

The geometric majorant in step 1.1 converges, so the regular part converges uniformly on $\mathrm{\mid }z-a\mathrm{\mid }\le R_0$; since $R_0<R$ was arbitrary, the convergence is locally uniform on the disc of radius $R$, and when $R=\mathrm{\infty }$ it is locally uniform on every bounded disc.

The geometric majorant in step 1.2 converges, so the principal part converges uniformly on $\mathrm{\mid }z-a\mathrm{\mid }\ge \rho$; since $\rho >r$ was arbitrary, the convergence is locally uniform on the exterior region $\mathrm{\mid }z-a\mathrm{\mid }>r$.

Steps 2.1, 2.2, 1.3, and 1.4 are exactly the claimed decomposition.