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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 $b_0\ne 0$, 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 approximants …Approximants with the same denominator degree are located in the same column of the table. …

3: 36.13 Kelvin’s Ship-Wave Pattern

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36.13.1 $$z(\varphi ,\rho )={\int }_{-\pi /2}^{\pi /2}\cos \left(\rho \frac{\cos \left(\theta +\varphi \right)}{{\cos }^2\theta }\right)d\theta ,$$

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Defines:

$z(\varphi ,\rho )$: wave height approximant (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential and $\int$: integral

Permalink:

http://dlmf.nist.gov/36.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.13 and Ch.36

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36.13.8 $$z(\rho ,\varphi )=2\pi \left({\rho }^{-1/3}u(\varphi )\cos \left(\rho \tilde{f}(\varphi )\right)\mathrm{Ai}\left(-{\rho }^{2/3}\mathrm{\Delta }(\varphi )\right)(1+O\left(1/\rho \right))+{\rho }^{-2/3}v(\varphi )\sin \left(\rho \tilde{f}(\varphi )\right){\mathrm{Ai}}^{\prime }\left(-{\rho }^{2/3}\mathrm{\Delta }(\varphi )\right)(1+O\left(1/\rho \right))\right),$$ $\rho \to \mathrm{\infty }$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z(\varphi ,\rho )$: wave height approximant, $u(\varphi )$: function and $v(\varphi )$: function

Referenced by:

Figure 36.13.1, Figure 36.13.1

Permalink:

http://dlmf.nist.gov/36.13.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.13 and Ch.36

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4: 3.10 Continued Fractions

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3.10.2 $$C_n=b_0+\frac{a_1}{b_1+}\frac{a_2}{b_2+}\mathrm{\cdots }\frac{a_n}{b_n}=\frac{A_n}{B_n}.$$

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Defines:

$A_n$: continued fraction numerator (locally), $B_n$: continued fraction denominator (locally) and $C_n$: continued fraction approximant (locally)

Referenced by:

§3.10(iii), §3.10(iii)

Permalink:

http://dlmf.nist.gov/3.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §3.10(i), §3.10 and Ch.3

► $C_n$ is the $n$th approximant or convergent to $C$. …

5: 8.10 Inequalities

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Padé Approximants

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6: 1.12 Continued Fractions

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1.12.4 $$C_n=b_0+\frac{a_1}{b_1+}\frac{a_2}{b_2+}\mathrm{\cdots }{\displaystyle \frac{a_n}{b_n}}=\frac{A_n}{B_n}.$$

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Symbols:

$n$: nonnegative integer, $A_n$: $n$th numerator, $B_n$: $n$th denominator and $C_n(w)$: continued fraction

A&S Ref:

3.10.1

Permalink:

http://dlmf.nist.gov/1.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §1.12(ii), §1.12 and Ch.1

► $C_n$ is called the $n$th approximant or convergent to $C$ . … … ►

1.12.27 $n=1,2,3,\mathrm{\dots }$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $n$: nonnegative integer and $C_n$: approximant

Permalink:

http://dlmf.nist.gov/1.12.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §1.12(v), §1.12(v), §1.12 and Ch.1

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7: 10.72 Mathematical Applications

… ►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. …

8: 13.31 Approximations

… ►For a discussion of the convergence of the Padé approximants that are related to the continued fraction (13.5.1) see Wimp (1985). …

9: Bibliography W

… ►

A rational approximant for the digamma function. Numer. Algorithms 33 (1-4), pp. 499–507.

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Notes:

International Conference on Numerical Algorithms, Vol. I (Marrakesh, 2001)

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§3.9(vi), §5.23(i).

Encodings:

BibTeX

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Some explicit Padé approximants for the function ${\mathrm{\Phi }}^{\prime }/\mathrm{\Phi }$ and a related quadrature formula involving Bessel functions. SIAM J. Math. Anal. 16 (4), pp. 887–895.

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External Links:

ISSN 0036-1410, Document, MathReview (Wyman Fair), ZentralBlatt

Referenced by:

§13.31(ii).

Encodings:

BibTeX

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10: 2.4 Contour Integrals

… ►For a coalescing saddle point and a pole see Wong (1989, Chapter 7) and van der Waerden (1951); in this case the uniform approximants are complementary error functions. For a coalescing saddle point and endpoint see Olver (1997b, Chapter 9) and Wong (1989, Chapter 7); if the endpoint is an algebraic singularity then the uniform approximants are parabolic cylinder functions with fixed parameter, and if the endpoint is not a singularity then the uniform approximants are complementary error functions. …