# approximants

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## 1—10 of 16 matching pages

##### 1: 18.13 Continued Fractions
$T_{n}\left(x\right)$ is the denominator of the $n$th approximant to: …and $U_{n}\left(x\right)$ is the denominator of the $n$th approximant to: … $P_{n}\left(x\right)$ is the denominator of the $n$th approximant to: … $L_{n}\left(x\right)$ is the denominator of the $n$th approximant to: … $H_{n}\left(x\right)$ is the denominator of the $n$th approximant to: …
##### 2: 3.11 Approximation Techniques
is called a Padé approximant at zero of $f$ if …It is denoted by ${[p/q]_{f}}\left(z\right)$. Thus if $b_{0}\neq 0$, then the Maclaurin expansion of (3.11.21) agrees with (3.11.20) up to, and including, the term in $z^{p+q}$. … The array of Padé approximantsApproximants with the same denominator degree are located in the same column of the table. …
##### 3: 36.13 Kelvin’s Ship-Wave Pattern
36.13.1 $z(\phi,\rho)=\int_{-\pi/2}^{\pi/2}\cos\left(\rho\frac{\cos\left(\theta+\phi% \right)}{{\cos}^{2}\theta}\right)\,\mathrm{d}\theta,$
36.13.8 $z(\rho,\phi)=2\pi\left(\rho^{-1/3}u(\phi)\cos\left(\rho\widetilde{f}(\phi)% \right)\operatorname{Ai}\left(-\rho^{2/3}\Delta(\phi)\right)\*(1+O\left(1/\rho% \right))+\rho^{-2/3}v(\phi)\sin\left(\rho\widetilde{f}(\phi)\right)% \operatorname{Ai}'\left(-\rho^{2/3}\Delta(\phi)\right)\*(1+O\left(1/\rho\right% ))\right),$ $\rho\to\infty$.
##### 4: 3.10 Continued Fractions
3.10.2 $C_{n}=b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cdots}}\frac{a_{n}}{b_{n}% }=\frac{A_{n}}{B_{n}}.$
$C_{n}$ is the $n$th approximant or convergent to $C$. …
##### 5: 8.10 Inequalities
$C_{n}$ is called the $n$th approximant or convergent to $C$ . … …
1.12.27 $-\tfrac{1}{2}\pi+\delta<\operatorname{ph}C_{n}<\tfrac{1}{2}\pi-\delta,$ $n=1,2,3,\dots$,
• J. Wimp (1985) Some explicit Padé approximants for the function $\Phi^{\prime}/\Phi$ and a related quadrature formula involving Bessel functions. SIAM J. Math. Anal. 16 (4), pp. 887–895.