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11: 3.11 Approximation Techniques

… ►Furthermore, if $f\in C^{\mathrm{\infty }}[-1,1]$, then the convergence of (3.11.11) is usually very rapid; compare (1.8.7) with $k$ arbitrary. … ►Let $f$ be continuous on a closed interval $[a,b]$ and $w$ be a continuous nonvanishing function on $[a,b]$: $w$ is called a weight function. …

12: 1.9 Calculus of a Complex Variable

… ►

Continuity

►A function $f(z)$ is continuous at a point $z_0$ if $\underset{z\to z_0}{lim}f(z)=f(z_0)$. … ►A function $f(z)$ is continuous on a region $R$ if for each point $z_0$ in $R$ and any given number $ϵ$ ($>0$) we can find a neighborhood of $z_0$ such that for all points $z$ in the intersection of the neighborhood with $R$. … ►Let $(a,b)$ be a finite or infinite interval, and $f_0(t),f_1(t),\mathrm{\dots }$ be real or complex continuous functions, $t\in (a,b)$. …

13: 1.6 Vectors and Vector-Valued Functions

… ►The path integral of a continuous function $f(x,y,z)$ is …If $h(a)=a^{\prime }$ and $h(b)=b^{\prime }$, then the reparametrization is called orientation-preserving, and … ►The integral of a continuous function $f(x,y,z)$ over a surface $S$ is …

14: 1.10 Functions of a Complex Variable

… ►Assume that for each $t\in [a,b]$, $f(z,t)$ is an analytic function of $z$ in $D$, and also that $f(z,t)$ is a continuous function of both variables. … ►For each $t\in [a,b)$, $f(z,t)$ is analytic in $D$; $f(z,t)$ is a continuous function of both variables when $z\in D$ and $t\in [a,b)$; the integral (1.10.18) converges at $b$, and this convergence is uniform with respect to $z$ in every compact subset $S$ of $D$. …

15: 1.13 Differential Equations

… ► $u$ and $z$ belong to domains $U$ and $D$ respectively, the coefficients $f(u,z)$ and $g(u,z)$ are continuous functions of both variables, and for each fixed $u$ (fixed $z$) the two functions are analytic in $z$ (in $u$). …

16: 15.6 Integral Representations

… ►In all cases the integrands are continuous functions of $t$ on the integration paths, except possibly at the endpoints. …

17: 28.31 Equations of Whittaker–Hill and Ince

… ►ambiguities in sign being resolved by requiring $C_p^m(x,\xi )$ and $S_p^m{}^{\prime }(x,\xi )$ to be continuous functions of $x$ and positive when $x=0$. …

18: 31.9 Orthogonality

… ►The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning. …

19: 28.30 Expansions in Series of Eigenfunctions

… ►Then every continuous $2\pi$-periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi ]$ can be expanded in a uniformly and absolutely convergent series …

20: 1.16 Distributions

… ► $\mathrm{\Lambda }:𝒟(I)\to ℂ$ is called a distribution if it is a continuous linear functional on $𝒟(I)$, that is, it is a linear functional and for every ${\varphi }_n\to \varphi$ in $𝒟(I)$, … ►A tempered distribution is a continuous linear functional $\mathrm{\Lambda }$ on $𝒯$. … ►A distribution in ${ℝ}^n$ is a continuous linear functional on ${𝒟}_n$. … ►Tempered distributions are continuous linear functionals on this space of test functions. …