# trigonometric form

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##### 11: 7.20 Mathematical Applications Figure 7.20.1: Cornu’s spiral, formed from Fresnel integrals, is defined parametrically by x = C ⁡ ( t ) , y = S ⁡ ( t ) , t ∈ [ 0 , ∞ ) . Magnify
Similar results hold for the trapezoidal rule in the formIf $f\in C^{2m+2}[a,b]$, then the remainder $E_{n}(f)$ in (3.5.2) can be expanded in the formThe $p_{n}(x)$ are the monic Legendre polynomials, that is, the polynomials $P_{n}\left(x\right)$18.3) scaled so that the coefficient of the highest power of $x$ in their explicit forms is unity. … Integrals of the formThe integral (3.5.39) has the form (3.5.35) if we set $\zeta=tp$, $c=t\sigma$, and $f(\zeta)=t^{-1}\zeta^{s}G(\zeta/t)$. …
##### 13: 28.8 Asymptotic Expansions for Large $q$
28.8.9 $W_{m}^{\pm}(x)=\frac{e^{\pm 2h\sin x}}{(\cos x)^{m+1}}\begin{cases}\left(\cos% \left(\frac{1}{2}x+\frac{1}{4}\pi\right)\right)^{2m+1},\\ \left(\sin\left(\frac{1}{2}x+\frac{1}{4}\pi\right)\right)^{2m+1},\end{cases}$
28.8.11 $P_{m}(x)\sim 1+\dfrac{s}{2^{3}h{\cos^{2}}x}+\dfrac{1}{h^{2}}\left(\dfrac{s^{4}% +86s^{2}+105}{2^{11}{\cos^{4}}x}-\dfrac{s^{4}+22s^{2}+57}{2^{11}{\cos^{2}}x}% \right)+\cdots,$
28.8.12 $Q_{m}(x)\sim\dfrac{\sin x}{{\cos^{2}}x}\left(\dfrac{1}{2^{5}h}(s^{2}+3)+\dfrac% {1}{2^{9}h^{2}}\left(s^{3}+3s+\dfrac{4s^{3}+44s}{{\cos^{2}}x}\right)\right)+\cdots.$
##### 15: 36.7 Zeros
There are also three sets of zero lines in the plane $z=0$ related by $2\pi/3$ rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates $(x=r\cos\theta,\;y=r\sin\theta)$ is given by …
##### 17: 4.37 Inverse Hyperbolic Functions
For the corresponding results for $\operatorname{arccsch}z$, $\operatorname{arcsech}z$, and $\operatorname{arccoth}z$, use (4.37.7)–(4.37.9); compare §4.23(iv). …
##### 18: 14.5 Special Values
14.5.20 $\mathsf{P}_{\frac{1}{2}}\left(\cos\theta\right)=\frac{2}{\pi}\left(2E\left(% \sin\left(\tfrac{1}{2}\theta\right)\right)-K\left(\sin\left(\tfrac{1}{2}\theta% \right)\right)\right),$
14.5.26 $\boldsymbol{Q}_{\frac{1}{2}}\left(\cosh\xi\right)=2\pi^{-1/2}\cosh\xi% \operatorname{sech}\left(\tfrac{1}{2}\xi\right)K\left(\operatorname{sech}\left% (\tfrac{1}{2}\xi\right)\right)-4\pi^{-1/2}\cosh\left(\tfrac{1}{2}\xi\right)E% \left(\operatorname{sech}\left(\tfrac{1}{2}\xi\right)\right),$
##### 20: 9.5 Integral Representations
9.5.2 $\mathrm{Ai}\left(-x\right)=\frac{x^{\ifrac{1}{2}}}{\pi}\int_{-1}^{\infty}\cos% \left(x^{\ifrac{3}{2}}(\tfrac{1}{3}t^{3}+t^{2}-\tfrac{2}{3})\right)\mathrm{d}t,$ $x>0$.