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11: Nico M. Temme

… ►Temme was a member of the original editorial committee for the DLMF project, in existence from the mid-1990’s to the mid-2010’s. …

12: 1.4 Calculus of One Variable

… ►When this limit exists $f$ is differentiable at $x$. … ►when the last limit exists. … ►when this limit exists. … ►when this limit exists. … ►whenever this integral exists. …

13: 15.17 Mathematical Applications

… ►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. …

14: 20.12 Mathematical Applications

… ►The space of complex tori $ℂ/(\mathbb{Z}+\tau \mathbb{Z})$ (that is, the set of complex numbers $z$ in which two of these numbers $z_1$ and $z_2$ are regarded as equivalent if there exist integers $m,n$ such that $z_1-z_2=m+\tau n$) is mapped into the projective space $P^3$ via the identification $z\to ({\theta }_1\left(2z\right|\tau ),{\theta }_2\left(2z\right|\tau ),{\theta }_3\left(2z\right|\tau ),{\theta }_4\left(2z\right|\tau ))$. …

15: 27.13 Functions

… ►The smallest $m$ that exists for a given $k$ is denoted by $g\left(k\right)$. … ►Lagrange (1770) proves that $g\left(2\right)=4$, and during the next 139 years the existence of $g\left(k\right)$ was shown for $k=3,4,5,6,7,8,10$. Hilbert (1909) proves the existence of $g\left(k\right)$ for every $k$ but does not determine its corresponding numerical value. … ►The existence of $G\left(k\right)$ follows from that of $g\left(k\right)$ because $G\left(k\right)\le g\left(k\right)$, but only the values $G\left(2\right)=4$ and $G\left(4\right)=16$ are known exactly. …

16: 27.21 Tables

… ►No sequel to Lehmer (1941) exists to date, but many tables of functions of number theory are included in Unpublished Mathematical Tables (1944). …

17: 1.12 Continued Fractions

… ►

§1.12(iii) Existence of Convergents

… ►The even part of $C$ exists iff $b_{2k}\ne 0$, $k=1,2,\mathrm{\dots }$, and up to equivalence is given by …The odd part of $C$ exists iff $b_{2k+1}\ne 0$, $k=0,1,2,\mathrm{\dots }$, and up to equivalence is given by …

18: 9.17 Methods of Computation

… ►In the case of the Scorer functions, integration of the differential equation (9.12.1) is more difficult than (9.2.1), because in some regions stable directions of integration do not exist. …

19: 26.2 Basic Definitions

… ►Given a finite set $S$ with permutation $\sigma$, a cycle is an ordered equivalence class of elements of $S$ where $j$ is equivalent to $k$ if there exists an $\mathrm{\ell }=\mathrm{\ell }(j,k)$ such that $j={\sigma }^{\mathrm{\ell }}(k)$, where ${\sigma }^1=\sigma$ and ${\sigma }^{\mathrm{\ell }}$ is the composition of $\sigma$ with ${\sigma }^{\mathrm{\ell }-1}$. …

20: 27.12 Asymptotic Formulas: Primes

… ►There exists a positive constant $c$ such that …The best available asymptotic error estimate (2009) appears in Korobov (1958) and Vinogradov (1958): there exists a positive constant $d$ such that …