differentiation of asymptotic approximations

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1: 2.1 Definitions and Elementary Properties

… ►This result also holds with both $O$’s replaced by $o$’s. … ►means that for each $n$, the difference between $f(x)$ and the $n$th partial sum on the right-hand side is $O\left({(x-c)}^n\right)$ as $x\to c$ in $𝐗$. …

2: 10.57 Uniform Asymptotic Expansions for Large Order

§10.57 Uniform Asymptotic Expansions for Large Order

►Asymptotic expansions for ${𝗃}_n\left((n+\frac{1}{2})z\right)$, ${𝗒}_n\left((n+\frac{1}{2})z\right)$, ${𝗁}_n^{(1)}\left((n+\frac{1}{2})z\right)$, ${𝗁}_n^{(2)}\left((n+\frac{1}{2})z\right)$, ${𝗂}_n^{(1)}\left((n+\frac{1}{2})z\right)$, and ${𝗄}_n\left((n+\frac{1}{2})z\right)$ as $n\to \mathrm{\infty }$ that are uniform with respect to $z$ can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9). …

3: 2.3 Integrals of a Real Variable

… ►(In other words, differentiation of (2.3.8) with respect to the parameter $\lambda$ (or $\mu$) is legitimate.) …

4: 9.10 Integrals

… ►

§9.10(ii) Asymptotic Approximations

… ►

9.10.8 $$\int zw(z)dz=w^{\prime }(z),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $w$: ODE solution

Proof sketch:

To verify, differentiate and refer to (9.2.1).

Permalink:

http://dlmf.nist.gov/9.10.E8

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See also:

Annotations for §9.10(iii), §9.10 and Ch.9

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9.10.9 $$\int z^{2}w(z)dz=z{w}^{\prime }(z)-w(z),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $w$: ODE solution

Proof sketch:

To verify, differentiate and refer to (9.2.1).

Permalink:

http://dlmf.nist.gov/9.10.E9

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See also:

Annotations for §9.10(iii), §9.10 and Ch.9

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9.10.10 $$\int z^{n+3}w(z)dz=z^{n+2}{w}^{\prime }(z)-(n+2)z^{n+1}w(z)+(n+1)(n+2)\int z^{n}w(z)dz,$$ $n=0,1,2,\mathrm{\dots }.$

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Defines:

$n$: index (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $w$: ODE solution

Proof sketch:

To verify, differentiate and refer to (9.2.1).

Permalink:

http://dlmf.nist.gov/9.10.E10

Encodings:

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See also:

Annotations for §9.10(iii), §9.10 and Ch.9

… ►

9.10.20 $${\int }_0^x{\int }_0^v\mathrm{Ai}\left(t\right)dtdv=x{\int }_0^x\mathrm{Ai}\left(t\right)dt-{\mathrm{Ai}}^{\prime }\left(x\right)+{\mathrm{Ai}}^{\prime }\left(0\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $dx$: differential of $x$ , $\int$: integral, $x$: real variable and $v$: parameter

Proof sketch:

To verify, differentiate and refer to (9.2.1)

Referenced by:

2nd item, 2nd item

Permalink:

http://dlmf.nist.gov/9.10.E20

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See also:

Annotations for §9.10(viii), §9.10 and Ch.9

…

5: 2.8 Differential Equations with a Parameter

… ►dots denoting differentiations with respect to $\xi$. … ►In both cases uniform asymptotic approximations are obtained in terms of Bessel functions of order $1/(\lambda +2)$. … ►For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24. ►For further examples of uniform asymptotic approximations in terms of Bessel functions or modified Bessel functions of variable order see §§13.21(ii), 14.15(ii), 14.15(iv), 14.20(viii), 30.9(i), 30.9(ii). ►For examples of uniform asymptotic approximations in terms of Whittaker functions with fixed second parameter see §18.15(i) and §28.8(iv). …

6: Bibliography M

… ►

Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

Encodings:

BibTeX

… ►

Approximation of Functions: Theory and Numerical Methods. Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag, New York.

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Notes:

Expanded translation from the German edition. Translated by Larry L. Schumaker

External Links:

MathReview, ZentralBlatt

Referenced by:

§3.11(i), §3.11(iii).

Encodings:

BibTeX

… ►

Asymptotic approximations for prolate spheroidal wave functions. Studies in Appl. Math. 54 (4), pp. 315–349.

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External Links:

MathReview (M. L. Mehta), ZentralBlatt

Referenced by:

§30.9(i).

Encodings:

BibTeX

… ►

A continued fraction approximation of the gamma function. J. Math. Anal. Appl. 402 (2), pp. 405–410.

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External Links:

ISSN 0022-247X, Document, FullText, MathReview (Paul Levrie), ZentralBlatt

Referenced by:

§5.10.

Encodings:

BibTeX

… ►

On asymptotic expansions of ellipsoidal wave functions. Math. Nachr. 32, pp. 157–172.

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External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.11, §29.7(ii).

Encodings:

BibTeX

…

7: 9.12 Scorer Functions

… ►

§9.12(viii) Asymptotic Expansions

… ►For other phase ranges combine these results with the connection formulas (9.12.11)–(9.12.14) and the asymptotic expansions given in §9.7. … ►

Integrals

… ►

9.12.31 $${\int }_0^z\mathrm{Hi}\left(-t\right)dt\sim \frac{1}{\pi }\ln z+\frac{2\gamma +\ln 3}{3\pi }+\frac{1}{\pi }\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}\frac{(3k-1)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $k$: nonnegative integer, $x$: real variable, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

Except for the constant term, this be verified by termwise integration of (9.12.27). To evaluate the constant term replace $z$ by $-x$ ($\le 0$) in (9.12.20) and integrate (§1.5(v)) to obtain $\pi {\int }_0^x\mathrm{Hi}\left(-t\right)dt={\int }_0^{\mathrm{\infty }}(1-{\mathrm{e}}^{-xt}){\mathrm{e}}^{-\frac{1}{3}{t}^3}{t}^{-1}dt$. Next, integrate the right-hand side of this equation by parts—integrating the factor $t^{-1}$ and differentiating the rest. As $x\to \mathrm{\infty }$ the asymptotic expansions of ${\int }_0^{\mathrm{\infty }}x{\mathrm{e}}^{-xt}{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ and ${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xt}{t}^2{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ follow from (2.3.9). Also, ${\int }_0^{\mathrm{\infty }}{t}^2{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ can be found by replacing $\frac{1}{3}{t}^3$ by $t$ and referring to the first of (5.9.18). (This equation first appeared in Rothman (1954a). As noted in this reference these results were derived by the author of the present DLMF chapter, but the proof was not included.)

Referenced by:

(9.12.30)

Permalink:

http://dlmf.nist.gov/9.12.E31

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See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

… ►For the above properties and further results, including the distribution of complex zeros, asymptotic approximations for the numerically large real or complex zeros, and numerical tables see Gil et al. (2003c). …

8: 18.40 Methods of Computation

… ►See Gautschi (1983) for examples of numerically stable and unstable use of the above recursion relations, and how one can then usefully differentiate between numerical results of low and high precision, as produced thereby. … ►

Stieltjes Inversion via (approximate) Analytic Continuation

… ►Interpolation of the midpoints of the jumps followed by differentiation with respect to $x$ yields a Stieltjes–Perron inversion to obtain $w^{\mathrm{RCP}}(x)$ to a precision of $\sim 4$ decimal digits for $N=120$. … ►Here $x(t,N)$ is an interpolation of the abscissas $x_{i,N},i=1,2,\mathrm{\dots },N$, that is, $x(i,N)=x_{i,N}$, allowing differentiation by $i$. … ►In Figure 18.40.2 the approximations were carried out with a precision of 50 decimal digits.

9: 9.8 Modulus and Phase

… ►Primes denote differentiations with respect to $x$, which is continued to be assumed real and nonpositive. … ►

§9.8(iv) Asymptotic Expansions

… ►

9.8.20 $$M^2\left(x\right)\sim \frac{1}{\pi {(-x)}^{1/2}}\sum _{k=0}^{\mathrm{\infty }}\frac{1\cdot 3\cdot 5\mathrm{\cdots }(6k-1)}{k!{(96)}^k}\frac{1}{x^{3k}},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $M\left(z\right)$: Airy modulus function and $x$: real variable

Proof sketch:

Combine (9.8.9)–(9.8.12) with §10.18(iii)

Referenced by:

§9.8(iv)

Permalink:

http://dlmf.nist.gov/9.8.E20

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See also:

Annotations for §9.8(iv), §9.8 and Ch.9

… ►Also, approximate values (25S) of the coefficients of the powers $x^{-15}$, $x^{-18}$, $\mathrm{\dots }$, $x^{-56}$ are available in Sherry (1959).