# arc(s)

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## 9 matching pages

##### 2: 19.25 Relations to Other Functions
19.25.32 $\operatorname{arcps}\left(x,k\right)=R_{F}\left(x^{2},x^{2}+\Delta(\mathrm{q,p% }),x^{2}+\Delta(\mathrm{r,p})\right),$
19.25.33 $\operatorname{arcsp}\left(x,k\right)=xR_{F}\left(1,1+\Delta(\mathrm{q,p})x^{2}% ,1+\Delta(\mathrm{r,p})x^{2}\right),$
##### 4: 7.20 Mathematical Applications
###### §7.20(ii) Cornu’s Spiral
Let the set $\{x(t),y(t),t\}$ be defined by $x(t)=C\left(t\right)$, $y(t)=S\left(t\right)$, $t\geq 0$. Then the set $\{x(t),y(t)\}$ is called Cornu’s spiral: it is the projection of the corkscrew on the $\{x,y\}$-plane. …Then the arc length between the origin and $P(t)$ equals $t$, and is directly proportional to the curvature at $P(t)$, which equals $\pi t$. …
##### 5: Sidebar 9.SB1: Supernumerary Rainbows
The faint line below the main colored arc is a ‘supernumerary rainbow’, produced by the interference of different sun-rays traversing a raindrop and emerging in the same direction. …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. The house in the picture is Newton’s birthplace. …
##### 6: 4.42 Solution of Triangles
where $s=\tfrac{1}{2}(a+b+c)$ (the semiperimeter). …
4.42.10 $\sin a\cos B=\cos b\sin c-\sin b\cos c\cos A,$
4.42.11 $\cos a\cos C=\sin a\cot b-\sin C\cot B,$
##### 7: Bibliography B
• R. W. Barnard, K. Pearce, and K. C. Richards (2000) A monotonicity property involving ${}_{3}F_{2}$ and comparisons of the classical approximations of elliptical arc length. SIAM J. Math. Anal. 32 (2), pp. 403–419.
• B. C. Berndt (1989) Ramanujan’s Notebooks. Part II. Springer-Verlag, New York.
• B. C. Berndt (1991) Ramanujan’s Notebooks. Part III. Springer-Verlag, Berlin-New York.
• M. V. Berry (1976) Waves and Thom’s theorem. Advances in Physics 25 (1), pp. 1–26.
• M. V. Berry (1989) Uniform asymptotic smoothing of Stokes’s discontinuities. Proc. Roy. Soc. London Ser. A 422, pp. 7–21.
• ##### 8: 1.10 Functions of a Complex Variable
If $f_{2}(z)$, analytic in $D_{2}$, equals $f_{1}(z)$ on an arc in $D=D_{1}\cap D_{2}$, or on just an infinite number of points with a limit point in $D$, then they are equal throughout $D$ and $f_{2}(z)$ is called an analytic continuation of $f_{1}(z)$. … Suppose $z(t)=x(t)+\mathrm{i}y(t)$, $a\leq t\leq b$, is an arc and $a=t_{0}. …
##### 9: 28.33 Physical Applications
We shall derive solutions to the uniform, homogeneous, loss-free, and stretched elliptical ring membrane with mass $\rho$ per unit area, and radial tension $\tau$ per unit arc length. …
28.33.4 $w^{\prime\prime}(t)+\left(b-f\cos\left(2\omega t\right)\right)w(t)=0,$
Substituting $z=\omega t$, $a=\ifrac{b}{\omega^{2}}$, and $2q=\ifrac{f}{\omega^{2}}$, we obtain Mathieu’s standard form (28.2.1). …