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11: 3.8 Nonlinear Equations

… ►for all $n$ sufficiently large, where $A$ and $p$ are independent of $n$, then the sequence is said to have convergence of the $p$th order. … … ►This is useful when $f(z)$ satisfies a second-order linear differential equation because of the ease of computing $f^{\prime \prime }(z_n)$. … ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. …

12: Bibliography I

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The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of $J_0(z)-i{J}_1(z)$ and of Bessel functions $J_m(z)$ of any real order $m$ . Linear Algebra Appl. 194, pp. 35–70.

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External Links:

ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

BibTeX

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Computing zeros and orders of Bessel functions. J. Comput. Appl. Math. 38 (1-3), pp. 169–184.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vi), §3.8(iii).

Encodings:

BibTeX

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

Encodings:

BibTeX

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Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.

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External Links:

ISSN 1073-2772, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(v).

Encodings:

BibTeX

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13: 3.4 Differentiation

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First-Order

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Second-Order

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Fourth-Order

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3.4.33 $${\nabla }^4{u}_{0,0}=\frac{1}{h^4}(20u_{0,0}-8(u_{1,0}+u_{0,1}+u_{-1,0}+u_{0,-1})+2(u_{1,1}+u_{1,-1}+u_{-1,1}+u_{-1,-1})+(u_{0,2}+u_{2,0}+u_{-2,0}+u_{0,-2}))+O\left(h^2\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding and $u$: function

A&S Ref:

25.3.32

Permalink:

http://dlmf.nist.gov/3.4.E33

Encodings:

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See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

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3.4.34 $${\nabla }^4{u}_{0,0}=\frac{1}{6h^4}(184u_{0,0}-(u_{0,3}+u_{0,-3}+u_{3,0}+u_{-3,0})+14(u_{0,2}+u_{0,-2}+u_{2,0}+u_{-2,0})-77(u_{0,1}+u_{0,-1}+u_{1,0}+u_{-1,0})+20(u_{1,1}+u_{1,-1}+u_{-1,1}+u_{-1,-1})-(u_{1,2}+u_{2,1}+u_{1,-2}+u_{2,-1}+u_{-1,2}+u_{-2,1}+u_{-1,-2}+u_{-2,-1}))+O\left(h^4\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding and $u$: function

A&S Ref:

25.3.33

Permalink:

http://dlmf.nist.gov/3.4.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

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14: Bibliography P

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Smoothing of the Stokes phenomenon for high-order differential equations. Proc. Roy. Soc. London Ser. A 436, pp. 165–186.

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External Links:

ISSN 0962-8444, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

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Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.

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External Links:

ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§10.20(ii).

Encodings:

BibTeX

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Zonal Polynomials of Order $1$ Through $12$ . In Selected Tables in Mathematical Statistics, H. L. Harter and D. B. Owen (Eds.), Vol. 2, pp. 199–388.

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External Links:

MathReview

Referenced by:

§35.11.

Encodings:

BibTeX

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Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:

Double-precision Fortran, maximum accuracy 20S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

6th item.

Encodings:

BibTeX

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Fast analytic formulas for the modified Bessel functions of imaginary order for spectral line broadening calculations. J. Quantit. Spec. and Rad. Trans. 62 (4), pp. 389–395.

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External Links:

Document

Referenced by:

§10.76(ii).

Encodings:

BibTeX

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15: 9.9 Zeros

… ►They are denoted by $a_k$, $a_k^{\prime }$, $b_k$, $b_k^{\prime }$, respectively, arranged in ascending order of absolute value for $k=1,2,\mathrm{\dots }.$ … ►They lie in the sectors and , and are denoted by ${\beta }_k$, ${\beta }_k^{\prime }$, respectively, in the former sector, and by $\overline{{\beta }_k}$, $\overline{{\beta }_k^{\prime }}$, in the conjugate sector, again arranged in ascending order of absolute value (modulus) for $k=1,2,\mathrm{\dots }.$ See §9.3(ii) for visualizations. … ►

9.9.6 $$a_k=-T\left(\frac{3}{8}\pi (4k-1)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $a_k$: $k$th zero of Airy $\mathrm{Ai}$ , $k$: nonnegative integer and $T$: expansion

Source:

Olver (1954, (A 20))

Permalink:

http://dlmf.nist.gov/9.9.E6

Encodings:

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See also:

Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.7 $${\mathrm{Ai}}^{\prime }\left(a_k\right)={(-1)}^{k-1}V\left(\frac{3}{8}\pi (4k-1)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $a_k$: $k$th zero of Airy $\mathrm{Ai}$ , $k$: nonnegative integer and $V$: expansion

Source:

Olver (1954, (A 20))

Permalink:

http://dlmf.nist.gov/9.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.8 $$a_k^{\prime }=-U\left(\frac{3}{8}\pi (4k-3)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $a_k^{\prime }$: $k$th zero of Airy ${\mathrm{Ai}}^{\prime }$ , $k$: nonnegative integer and $U$: expansion

Source:

Olver (1954, (A 20))

Permalink:

http://dlmf.nist.gov/9.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

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