# points

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##### 1: 36.4 Bifurcation Sets
###### Critical Points for Umbilics
This is the codimension-one surface in $\mathbf{x}$ space where critical points coalesce, satisfying (36.4.1) and … This is the codimension-one surface in $\mathbf{x}$ space where critical points coalesce, satisfying (36.4.2) and …
##### 2: 9.15 Mathematical Applications
Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …
##### 3: 3.1 Arithmetics and Error Measures
A nonzero normalized binary floating-point machine number $x$ is represented as … …
##### 4: 36.12 Uniform Approximation of Integrals
###### §36.12(i) General Theory for Cuspoids
Correspondence between the $u_{j}(\mathbf{y})$ and the $t_{j}(\mathbf{x})$ is established by the order of critical points along the real axis when $\mathbf{y}$ and $\mathbf{x}$ are such that these critical points are all real, and by continuation when some or all of the critical points are complex. …In (36.12.10), both second derivatives vanish when critical points coalesce, but their ratio remains finite. … For further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).
##### 5: 12.16 Mathematical Applications
PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). …
##### 6: 15.11 Riemann’s Differential Equation
The most general form is given by … Here $\{a_{1},a_{2}\}$, $\{b_{1},b_{2}\}$, $\{c_{1},c_{2}\}$ are the exponent pairs at the points $\alpha$, $\beta$, $\gamma$, respectively. …
15.11.3 $w=P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.$
These constants can be chosen to map any two sets of three distinct points $\{\alpha,\beta,\gamma\}$ and $\{\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\}$ onto each other. …
15.11.6 $P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}\widetilde{\alpha}&\widetilde{% \beta}&\widetilde{\gamma}&\\ a_{1}&b_{1}&c_{1}&t\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.$
##### 7: 28.7 Analytic Continuation of Eigenvalues
The only singularities are algebraic branch points, with $a_{n}\left(q\right)$ and $b_{n}\left(q\right)$ finite at these points. The number of branch points is infinite, but countable, and there are no finite limit points. …The branch points are called the exceptional values, and the other points normal values. … For a visualization of the first branch point of $a_{0}\left(\mathrm{i}\hat{q}\right)$ and $a_{2}\left(\mathrm{i}\hat{q}\right)$ see Figure 28.7.1. …
##### 8: Sidebar 9.SB1: Supernumerary Rainbows
Airy invented his function in 1838 precisely to describe this phenomenon more accurately than Young had done in 1800 when pointing out that supernumerary rainbows require the wave theory of light and are impossible to explain with Newton’s picture of light as a stream of independent corpuscles. …
##### 9: 10.72 Mathematical Applications
###### Simple Turning Points
These expansions are uniform with respect to $z$, including the turning point $z_{0}$ and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities. …
##### 10: 33.23 Methods of Computation
Inside the turning points, that is, when $\rho<\rho_{\operatorname{tp}}\left(\eta,\ell\right)$, there can be a loss of precision by a factor of approximately $|G_{\ell}|^{2}$. … WKBJ approximations (§2.7(iii)) for $\rho>\rho_{\operatorname{tp}}\left(\eta,\ell\right)$ are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq. … Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for $F_{0}$ and $G_{0}$ in the region inside the turning point: $\rho<\rho_{\operatorname{tp}}\left(\eta,\ell\right)$.