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21: 1.4 Calculus of One Variable

… ►If also $f(x)$ is continuous on the right at $x=a$, and continuous on the left at $x=b$, then $f(x)$ is continuous on the interval $[a,b]$, and we write $f(x)\in C[a,b]$. … ►If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists a point $c\in (a,b)$ such that … ►Let , and ${\xi }_j$ denote any point in $[x_j,x_{j+1}]$, $j=0,1,\mathrm{\dots },n-1$. … ►If $f(x)\in C^{n+1}[a,b]$, then … ►The overall maximum (minimum) of $f(x)$ on $[a,b]$ will either be at a local maximum (minimum) or at one of the end points $a$ or $b$. …

22: 23.20 Mathematical Applications

… ►There is a unique point $z_0\in [{\omega }_1,{\omega }_1+{\omega }_3]\cup [{\omega }_1+{\omega }_3,{\omega }_3]$ such that $\mathrm{\wp }\left(z_0\right)=0$. … ►It follows from the addition formula (23.10.1) that the points $P_j=P(z_j)$, $j=1,2,3$, have zero sum iff $z_1+z_2+z_3\in 𝕃$, so that addition of points on the curve $C$ corresponds to addition of parameters $z_j$ on the torus $ℂ/𝕃$; see McKean and Moll (1999, §§2.11, 2.14). … ►Let $T$ denote the set of points on $C$ that are of finite order (that is, those points $P$ for which there exists a positive integer $n$ with $nP=o$), and let $I,K$ be the sets of points with integer and rational coordinates, respectively. …The resulting points are then tested for finite order as follows. …If any of these quantities is zero, then the point has finite order. …

23: 1.13 Differential Equations

… ►A solution becomes unique, for example, when $w$ and $dw/dz$ are prescribed at a point in $D$. … ►

Transformation of the Point at Infinity

… ►

§1.13(vii) Closed-Form Solutions

… ►Equation (1.13.26) with $x\in [a,b]$ may be transformed to the Liouville normal form …As the interval $[a,b]$ is mapped, one-to-one, onto $[0,c]$ by the above definition of $t$, the integrand being positive, the inverse of this same transformation allows $\widehat{q}(t)$ to be calculated from $p,q,\rho$ in (1.13.31), $p,\rho \in C^2(a,b)$ and $q\in C(a,b)$. …

24: 3.11 Approximation Techniques

… ►A sufficient condition for $p_n(x)$ to be the minimax polynomial is that $\left|{ϵ}_n(x)\right|$ attains its maximum at $n+2$ distinct points in $[a,b]$ and ${ϵ}_n(x)$ changes sign at these consecutive maxima. … ►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. … ►Here $x_j$, $j=1,2,\mathrm{\dots },J$, is a given set of distinct real points and $J\ge n+1$. … ►Given $n+1$ distinct points $x_k$ in the real interval $[a,b]$, with ($a=$)($=b$), on each subinterval $[x_k,x_{k+1}]$, $k=0,1,\mathrm{\dots },n-1$, a low-degree polynomial is defined with coefficients determined by, for example, values $f_k$ and $f_k^{\prime }$ of a function $f$ and its derivative at the nodes $x_k$ and $x_{k+1}$. The set of all the polynomials defines a function, the spline, on $[a,b]$. …

25: 21.7 Riemann Surfaces

… ►Consider the set of points in ${ℂ}^2$ that satisfy the equation …Equation (21.7.1) determines a plane algebraic curve in ${ℂ}^2$, which is made compact by adding its points at infinity. …This compact curve may have singular points, that is, points at which the gradient of $\tilde{P}$ vanishes. … ►On this surface, we choose $2g$ cycles (that is, closed oriented curves, each with at most a finite number of singular points) $a_j$, $b_j$, $j=1,2,\mathrm{\dots },g$, such that their intersection indices satisfy … ►Denote the set of all branch points by $B=\{P_1,P_2,\mathrm{\dots },P_{2g+1},P_{\mathrm{\infty }}\}$. …

26: 10.2 Definitions

… ►This solution of (10.2.1) is an analytic function of $z\in ℂ$, except for a branch point at $z=0$ when $\nu$ is not an integer. The principal branch of $J_{\nu }\left(z\right)$ corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu }$ (§4.2(iv)) and is analytic in the $z$-plane cut along the interval $(-\mathrm{\infty },0]$. … ►The principal branch corresponds to the principal branches of $J_{\pm \nu }\left(z\right)$ in (10.2.3) and (10.2.4), with a cut in the $z$-plane along the interval $(-\mathrm{\infty },0]$. … ►Each solution has a branch point at $z=0$ for all $\nu \in ℂ$. The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the $z$-plane along the interval $(-\mathrm{\infty },0]$. …

27: 10.21 Zeros

… ►In Figures 10.21.1, 10.21.3, and 10.21.5 the two continuous curves that join the points $\pm 1$ are the boundaries of $𝐊$, that is, the eye-shaped domain depicted in Figure 10.20.3. … ►The first set of zeros of the principal value of $Y_n\left(nz\right)$ are the points $z=y_{n,m}/n$, $m=1,2,\mathrm{\dots }$, on the positive real axis (§10.21(i)). …Lastly, there are two conjugate sets, with $n$ zeros in each set, that are asymptotically close to the boundary of $𝐊$ as $n\to \mathrm{\infty }$. … ►In Figures 10.21.2, 10.21.4, and 10.21.6 the continuous curve that joins the points $\pm 1$ is the lower boundary of $𝐊$. … ►The only other set comprises $n$ zeros that are asymptotically close to the lower boundary of $𝐊$ as $n\to \mathrm{\infty }$. …

28: 3.5 Quadrature

… ►where $h=b-a$, $f\in C^2[a,b]$, and . … ►Let $h=\frac{1}{2}(b-a)$ and $f\in C^4[a,b]$. … ►If $f\in C^{2m+2}[a,b]$, then the remainder $E_n(f)$ in (3.5.2) can be expanded in the form … ►If the extreme members of the set of nodes $x_1,x_2,\mathrm{\dots },x_n$ are the endpoints $a$ and $b$, then the quadrature rule is said to be closed. … ►If we add $-1$ and $1$ to this set of $x_k$, then the resulting closed formula is the frequently-used Clenshaw–Curtis formula, whose weights are positive and given by …

29: 14.15 Uniform Asymptotic Approximations

… ►uniformly for $\theta \in (0,\pi -\delta ]$. … ►uniformly with respect to $x\in [0,1)$ and $\mu \in [0,(1-\delta )(\nu +\frac{1}{2})]$. … ►The points $x={\left(1-{\alpha }^2\right)}^{1/2}$, $x=1$, and $x=\mathrm{\infty }$ are mapped to $y={\alpha }^2$, $y=0$, and $y=-\mathrm{\infty }$, respectively. … ►uniformly with respect to $x\in [0,1)$ and $\mu \in [\delta (\nu +\frac{1}{2}),\nu +\frac{1}{2}]$. … ►uniformly with respect to $x\in (-1,1)$ and $\mu \in [\nu +\frac{1}{2},(1/\delta )(\nu +\frac{1}{2})]$. …

30: 14.20 Conical (or Mehler) Functions

… ►uniformly for $\theta \in (0,\pi -\delta ]$, where $I$ and $K$ are the modified Bessel functions (§10.25(ii)) and $\delta$ is an arbitrary constant such that . … ►uniformly for $x\in [-1+\delta ,1)$ and $\mu \in [0,A\tau ]$. … ►The interval is mapped one-to-one to the interval , with the points $x=-1$ and $x=1$ corresponding to $\eta =\mathrm{\infty }$ and $\eta =0$, respectively. … ►uniformly for $x\in (-1,1)$ and $\tau \in [0,A\mu ]$. … ►The interval is mapped one-to-one to the interval , with the points $x=-1$, $x=0$, and $x=1$ corresponding to $\rho =-\mathrm{\infty }$, $\rho =0$, and $\rho =\mathrm{\infty }$, respectively. …