The `1/9'-problem for the $\phi$-functions

The functions ${\phi }_{\mathrm{\ell }}(z)=(e^z-s_{\mathrm{\ell }}(z))\mathrm{/}{z}^{\mathrm{\ell }}$, with $s_{\mathrm{\ell }}$ the Taylor polynomial of $e^z$ of degree $\mathrm{\ell }-1$, are what exponential integrators evaluate, and they evaluate them through rational approximations on the negative real axis ${\mathbb{R}}_{-}$. Schmelzer and Trefethen (2007/08) conjectured that the best such approximations converge at the rate of the classical `1/9’-problem: $E_n({\phi }_{\mathrm{\ell }}{)}^{1\mathrm{/}n}\to H$, Halphen’s constant, for every $\mathrm{\ell }$. We prove it. The theorem behind it gives the same rate for $f=u_0+u_1\exp$ with $u_0$ rational and $u_1\equiv ̸0$ meromorphic with finitely many poles and $\log \mathrm{\mid }{u}_1(z)\mathrm{\mid }=o(\mathrm{\mid }z\mathrm{\mid })$, the lower bound required only away from ${\mathbb{R}}_{-}$, whenever $f$ has no pole on ${\mathbb{R}}_{-}$ (Theorem~4.4). For rational $u_1$ this is Theorem 1 of Stahl and Schmelzer (2009), announced with its proof deferred to a manuscript that never appeared; the hypotheses admit, for instance, $u_1(z)=\sinh (\pi \sqrt{z})\mathrm{/}(\pi \sqrt{z})$, whose zeros are infinite in number. The proof is a reading of Gonchar and Rakhmanov (1989) rather than an extension of them. Their theorem concerns a sequence given by a Cauchy-type integral with a varying weight ${\mathrm{\Phi }}_n$, and a contour identity of Schmelzer and Trefethen presents $u_1\exp$, once its principal parts are removed, in exactly that form, with ${\mathrm{\Phi }}_n(t)=u_1(-nt)e^{-nt}$. The factor $u_1(-nt)$ grows subexponentially in $n$, and their hypothesis on ${\mathrm{\Phi }}_n$ divides by $2n$, so it drops out of the external field: field, extremal arc, $S$-property and constant are all theirs, unchanged. For ${\phi }_{\mathrm{\ell }}$ the factor is $t^{-\mathrm{\ell }}$, whose pole sits at the origin, a point of the approximation set and hence at positive distance from every admissible contour.