# B�zier curves

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## 8 matching pages

##### 1: 10.20 Uniform Asymptotic Expansions for Large Order
The curves $BP_{1}E_{1}$ and $BP_{2}E_{2}$ in the $z$-plane are the inverse maps of the line segments …
##### 2: 10.41 Asymptotic Expansions for Large Order
The curve $E_{1}BE_{2}$ in the $z$-plane is the upper boundary of the domain $\mathbf{K}$ depicted in Figure 10.20.3 and rotated through an angle $-\tfrac{1}{2}\pi$. …
##### 3: 36.5 Stokes Sets
The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set: … For $z=0$, the set consists of the two curves
$B_{-}=-1.69916,$
$B_{+}=0.33912.$
In Figures 36.5.136.5.6 the plane is divided into regions by the dashed curves (Stokes sets) and the continuous curves (bifurcation sets). …
##### 4: 21.7 Riemann Surfaces
###### §21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces
Belokolos et al. (1994, §2.1)), they are obtainable from plane algebraic curves (Springer (1957), or Riemann (1851)). …Equation (21.7.1) determines a plane algebraic curve in ${\mathbb{C}}^{2}$, which is made compact by adding its points at infinity. …
###### §21.7(iii) Frobenius’ Identity
These are Riemann surfaces that may be obtained from algebraic curves of the form …
##### 5: 28.32 Mathematical Applications
Also let $\mathcal{L}$ be a curve (possibly improper) such that the quantity …
28.32.6 $w(z)=\int_{\mathcal{L}}K(z,\zeta)u(\zeta)\,\mathrm{d}\zeta$
defines a solution of Mathieu’s equation, provided that (in the case of an improper curve) the integral converges with respect to $z$ uniformly on compact subsets of $\mathbb{C}$. … is separated in this system, each of the separated equations can be reduced to the Whittaker–Hill equation (28.31.1), in which $A,B$ are separation constants. Two conditions are used to determine $A,B$. …
##### 6: 25.11 Hurwitz Zeta Function
25.11.6 $\zeta\left(s,a\right)=\frac{1}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)-% \frac{s(s+1)}{2}\int_{0}^{\infty}\frac{\widetilde{B}_{2}\left(x\right)-B_{2}}{% (x+a)^{s+2}}\,\mathrm{d}x,$ $s\neq 1$, $\Re s>-1$, $a>0$.
For $\widetilde{B}_{n}\left(x\right)$ see §24.2(iii). …
25.11.19 $\zeta'\left(s,a\right)=-\frac{\ln a}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}% \right)-\frac{a^{1-s}}{(s-1)^{2}}+\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{(% \widetilde{B}_{2}\left(x\right)-B_{2})\ln\left(x+a\right)}{(x+a)^{s+2}}\,% \mathrm{d}x-\frac{(2s+1)}{2}\int_{0}^{\infty}\frac{\widetilde{B}_{2}\left(x% \right)-B_{2}}{(x+a)^{s+2}}\,\mathrm{d}x,$ $\Re s>-1$, $s\neq 1$, $a>0$.
where $H_{n}$ are the harmonic numbers: …
##### 7: 10.21 Zeros
$B_{0}(\zeta)$ and $C_{0}(\zeta)$ are defined by (10.20.11) and (10.20.12) with $k=0$. … Each curve that represents an infinite string of nonreal zeros should be located on the opposite side of its straight line asymptote. … 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 $\mathbf{K}$, that is, the eye-shaped domain depicted in Figure 10.20.3. These curves therefore intersect the imaginary axis at the points $z=\pm\mathrm{i}c$, where $c=0.66274\dotsc$. … 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 $\mathbf{K}$. …
##### 8: 18.39 Applications in the Physical Sciences
18.39.54 $\Psi_{x,l}(r)=\\ B_{l}(x)\sum_{n=0}^{\infty}\frac{n!}{\Gamma(n+2l+2)}P^{(l+1)}_{n}\left(x;\frac% {2Z}{s},-\frac{2Z}{s}\right)\phi_{n,l}(sr),$ $x=\frac{8\epsilon-s^{2}}{8\epsilon+s^{2}}$,
Full expressions for both $A_{x_{i},l}$ and $B_{l}(x)$ are given in Yamani and Reinhardt (1975) and it is seen that $|\ifrac{A_{x_{i},l}}{B_{l}(x_{i})}|^{2}$ = $\ifrac{w_{i}^{N}}{w^{\mathrm{CP}}(x_{i})}$ where $w^{N}_{i}$ is the Gaussian-Pollaczek quadrature weight at $x=x_{i}$, and $w^{\mathrm{CP}}(x_{i})$ is the Gaussian-Pollaczek weight function at the same quadrature abscissa. …