Error-function series geometric tail bound

For $z\mathrm{\ne }0$, every term in [F1] is nonzero, so dividing successive absolute terms gives the stated ratio. At $z=0$ all terms vanish, and the stated division-free inequality and zero tail follow directly without division. For $k\ge K+1$ it is at most $q<1$, so the tail is dominated by $\mathrm{\mid }{e}_{K+1}\mathrm{\mid }{\sum }_{j\ge 0}{q}^j=\mathrm{\mid }{e}_{K+1}\mathrm{\mid }\mathrm{/}(1-q)$. On every disk the ratio is eventually below $1$, proving local uniform convergence.

The series in [F1] has infinite radius by step 1.1, so [F3] gives ${\mathrm{erf}}^{\mathrm{\prime }}(z)=\frac{2}{\sqrt{\pi }}{\sum }_{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k{z}^{2k}}{k!}=\frac{2}{\sqrt{\pi }}\exp (-z^2),$ using [F2] and the definition of the exponential series. Thus erf is an entire primitive of the stated integrand and $\mathrm{erf}(0)=0$.

Let $\gamma :[0,1]\to \mathbb{C}$ be any piecewise-$C^1$ path with $\gamma (0)=0$ and $\gamma (1)=z$. On a smooth piece, put ${\eta }_h=\gamma (s+h)-\gamma (s)$. Complex differentiability gives $\mathrm{erf}(\gamma (s)+{\eta }_h)-\mathrm{erf}(\gamma (s))={\mathrm{erf}}^{\mathrm{\prime }}(\gamma (s)){\eta }_h+{\eta }_hϵ({\eta }_h)$ with $ϵ(\eta )\to 0$ as $\eta \to 0$; set $ϵ(0)=0$. Since ${\eta }_h\mathrm{/}h\to {\gamma }^{\mathrm{\prime }}(s)$, division by $h$ makes the remainder tend to zero and proves $(\mathrm{erf}\circ \gamma {)}^{\mathrm{\prime }}(s)={\mathrm{erf}}^{\mathrm{\prime }}(\gamma (s)){\gamma }^{\mathrm{\prime }}(s)$. At a piece endpoint the same limit with one-sided increments gives the relative derivative, and continuity of ${\gamma }^{\mathrm{\prime }}$ extends this derivative continuously to the closed piece. By [F4], the real and imaginary parts of this derivative integrate to the endpoint differences. By [F6], these parameter integrals sum to the line integral defined in [F5]. Hence ${\int }_{\gamma }\frac{2}{\sqrt{\pi }}{e}^{-t^2}dt=\mathrm{erf}(z)-\mathrm{erf}(0)=\mathrm{erf}(z)\mathrm{.}$ This proves the segment identity and path independence without interchanging a series limit with a line integral.