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41: 3.7 Ordinary Differential Equations

… ►The path is partitioned at

P+1

points labeled successively

z_{0},z_{1},\dots,z_{P}

, with

z_{0}=a

z_{P}=b

. … ►Write

\tau_{j}=z_{j+1}-z_{j}

j=0,1,\dots,P

, expand

w(z)

and

w^{\prime}(z)

in Taylor series (§1.10(i)) centered at

z=z_{j}

, and apply (3.7.2). … ►If, for example,

\beta_{0}=\beta_{1}=0

, then on moving the contributions of

w(z_{0})

and

w(z_{P})

to the right-hand side of (3.7.13) the resulting system of equations is not tridiagonal, but can readily be made tridiagonal by annihilating the elements of

\mathbf{A}_{P}

that lie below the main diagonal and its two adjacent diagonals. … ►The values

\lambda_{k}

are the eigenvalues and the corresponding solutions

w_{k}

of the differential equation are the eigenfunctions. … ►where

h=z_{n+1}-z_{n}

and …

42: 3.6 Linear Difference Equations

… ►Given numerical values of

w_{0}

and

w_{1}

, the solution

w_{n}

of the equation …These errors have the effect of perturbing the solution by unwanted small multiples of

w_{n}

and of an independent solution

g_{n}

, say. … ►The unwanted multiples of

g_{n}

now decay in comparison with

w_{n}

, hence are of little consequence. … ►The latter method is usually superior when the true value of

w_{0}

is zero or pathologically small. … ►beginning with

e_{0}=w_{0}

. …

43: 18.8 Differential Equations

… ►

Table 18.8.1: Classical OP’s: differential equations

A(x)f^{\prime\prime}(x)+B(x)f^{\prime}(x)+C(x)f(x)+\lambda_{n}f(x)=0

► ►►►►►

#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	$\lambda_{n}$
…
11	${\mathrm{e}}^{-n^{-1}x}x^{\ell+1}L^{(2\ell+1)}_{n-\ell-1}\left(2n^{-1}x\right)$	$1$	$0$	$\frac{2}{x}-\frac{\ell(\ell+1)}{x^{2}}$	$-\frac{1}{n^{2}}$
12	$H_{n}\left(x\right)$	$1$	$-2x$	$0$	$2n$
…
14	$\mathit{He}_{n}\left(x\right)$	$1$	$-x$	$0$	$n$

► ►Item 11 of Table 18.8.1 yields (18.39.36) for

Z=1

44: 3.2 Linear Algebra

… ►where

u_{j}=c_{j}

j=1,2,\dots,n-1

d_{1}=b_{1}

, and …Forward elimination for solving

\mathbf{A}\mathbf{x}=\mathbf{f}

then becomes

y_{1}=f_{1}

, …and back substitution is

x_{n}=y_{n}/d_{n}

, followed by … ►Define the Lanczos vectors

\mathbf{v}_{j}

and coefficients

\alpha_{j}

and

\beta_{j}

\mathbf{v}_{0}=\boldsymbol{{0}}

, a normalized vector

\mathbf{v}_{1}

(perhaps chosen randomly),

\alpha_{1}=\mathbf{v}_{1}^{\rm T}\mathbf{A}\mathbf{v}_{1}

\beta_{1}=0

, and for

j=1,2,\ldots,n-1

by the recursive scheme … ►Start with

\mathbf{v}_{0}=\boldsymbol{{0}}

, vector

\mathbf{v}_{1}

such that

\mathbf{v}_{1}^{\rm T}\mathbf{S}\mathbf{v}_{1}=1

\alpha_{1}=\mathbf{v}_{1}^{\rm T}\mathbf{A}\mathbf{v}_{1}

\beta_{1}=0

. …

45: 24.19 Methods of Computation

… ►Equations (24.5.3) and (24.5.4) enable

B_{n}

and

E_{n}

to be computed by recurrence. …For example, the tangent numbers

T_{n}

can be generated by simple recurrence relations obtained from (24.15.3), then (24.15.4) is applied. … ►For other information see Chellali (1988) and Zhang and Jin (1996, pp. 1–11). For algorithms for computing

B_{n}

E_{n}

B_{n}\left(x\right)

, and

E_{n}\left(x\right)

see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180). ►

§24.19(ii) Values of $B_{n}$ Modulo $p$

…

46: Bibliography O

… ►

K. Okamoto (1986) Studies on the Painlevé equations. III. Second and fourth Painlevé equations,

P_{{\rm II}}

and

P_{\mathrm{IV}}

. Math. Ann. 275 (2), pp. 221–255.

ⓘ

External Links:: ISSN 0025-5831, Document, MathReview (Hélène Airault), ZentralBlatt
Referenced by:: §32.10(ii), §32.10(iv), §32.6(iii), §32.6(v), §32.7(viii).
Encodings:: BibTeX

… ►

A. B. Olde Daalhuis (1998a) Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. Proc. Roy. Soc. London Ser. A 454, pp. 1–29.

ⓘ

External Links:: ISSN 1364-5021, Document, MathReview (Donald A. Lutz), ZentralBlatt
Referenced by:: §2.11(v), §2.7(ii).
Encodings:: BibTeX

… ►

F. W. J. Olver (1974) Error bounds for stationary phase approximations. SIAM J. Math. Anal. 5 (1), pp. 19–29.

ⓘ

Notes:: See also Olver (1997b)
External Links:: Document, MathReview (L. Berg), ZentralBlatt
Referenced by:: §2.3(iv).
Encodings:: BibTeX

… ►

S. Olver (2011) Numerical solution of Riemann-Hilbert problems: Painlevé II. Found. Comput. Math. 11 (2), pp. 153–179.

ⓘ

External Links:: ISSN 1615-3375, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §32.17, §32.17.
Encodings:: BibTeX

… ►

H. Oser (1960) Algorithm 22: Riccati-Bessel functions of first and second kind. Comm. ACM 3 (11), pp. 600–601.

ⓘ

Notes:: Includes Algol code, minimum accuracy 9S. For certification see same journal 13(1970), p. 448.
External Links:: ISSN 0001-0782, Document
Referenced by:: §10.77(ii).
Encodings:: BibTeX

…

47: 17.5 ${{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}$ Functions

§17.5 ${{}_{0}\phi_{0}},{{}_{1}\phi_{0}},{{}_{1}\phi_{1}}$ Functions

… ►

17.5.1

{{}_{0}\phi_{0}}\left(-;-;q,z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{% \genfrac{(}{)}{0.0pt}{}{n}{2}}z^{n}}{\left(q;q\right)_{n}}=\left(z;q\right)_{% \infty};

ⓘ

Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §17.2(iii), Erratum (V1.1.11) for Equation (17.5.1)
Permalink:: http://dlmf.nist.gov/17.5.E1
Encodings:: pMML, png, TeX
Correction (effective with 1.1.11):: The constraint $|z|<1$ is not necessary and was removed.
See also:: Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.2

{{}_{1}\phi_{0}}\left(a;-;q,z\right)=\frac{\left(az;q\right)_{\infty}}{\left(z% ;q\right)_{\infty}},

|z|<1

;

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.3

{{}_{1}\phi_{0}}\left(q^{-n};-;q,z\right)=\left(zq^{-n};q\right)_{n}.

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/17.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §17.5, §17.5 and Ch.17

… ►

17.5.4

{{}_{1}\phi_{0}}\left(0;-;q,z\right)=\sum_{n=0}^{\infty}\frac{z^{n}}{\left(q;q% \right)_{n}}=\frac{1}{\left(z;q\right)_{\infty}},

|z|<1

;

ⓘ

Symbols:: $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$ : $q$ -Pochhammer symbol (or $q$ -shifted factorial), ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$ : basic hypergeometric (or $q$ -hypergeometric) function, $q$ : complex base, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §17.2(iii)
Permalink:: http://dlmf.nist.gov/17.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §17.5, §17.5 and Ch.17

…

48: 20.7 Identities

… ►

20.7.6

{\theta_{4}}^{2}\left(0,q\right)\theta_{1}\left(w+z,q\right)\theta_{1}\left(w-% z,q\right)={\theta_{3}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right)-{% \theta_{2}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Referenced by:: §20.7(ii)
Permalink:: http://dlmf.nist.gov/20.7.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(ii), §20.7 and Ch.20

►

20.7.7

{\theta_{4}}^{2}\left(0,q\right)\theta_{2}\left(w+z,q\right)\theta_{2}\left(w-% z,q\right)={\theta_{4}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right)-{% \theta_{1}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.7.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(ii), §20.7 and Ch.20

►

20.7.8

{\theta_{4}}^{2}\left(0,q\right)\theta_{3}\left(w+z,q\right)\theta_{3}\left(w-% z,q\right)={\theta_{4}}^{2}\left(w,q\right){\theta_{3}}^{2}\left(z,q\right)-{% \theta_{1}}^{2}\left(w,q\right){\theta_{2}}^{2}\left(z,q\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(ii), §20.7 and Ch.20

… ►

20.7.10

\theta_{1}\left(2z,q\right)=2\frac{\theta_{1}\left(z,q\right)\theta_{2}\left(z% ,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left(z,q\right)}{\theta_{2}\left% (0,q\right)\theta_{3}\left(0,q\right)\theta_{4}\left(0,q\right)}.

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $z$ : complex and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.7.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §20.7(iii), §20.7 and Ch.20

… ►

20.7.34

\frac{\theta_{1}\left(z,q^{2}\right)\theta_{3}\left(z,q^{2}\right)}{\theta_{1}% \left(z,iq\right)}=\frac{\theta_{2}\left(z,q^{2}\right)\theta_{4}\left(z,q^{2}% \right)}{\theta_{2}\left(z,iq\right)}=i^{-1/4}\sqrt{\frac{\theta_{2}\left(0,q^% {2}\right)\theta_{4}\left(0,q^{2}\right)}{2}}.

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\mathrm{i}$ : imaginary unit, $z$ : complex and $q$ : nome
Referenced by:: §20.7(iv), Erratum (V1.0.5) for Chapters 8, 20, 36
Permalink:: http://dlmf.nist.gov/20.7.E34
Encodings:: pMML, png, TeX
Addition (effective with 1.0.5):: This equation has been added. See Walker (2012).
See also:: Annotations for §20.7(ix), §20.7 and Ch.20

…

49: 11.14 Tables

… ►

•

Abramowitz and Stegun (1964, Chapter 12) tabulates $\mathbf{H}_{n}\left(x\right)$ , $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$ , and $I_{n}\left(x\right)-\mathbf{L}_{n}\left(x\right)$ for $n=0,1$ and $x=0(.1)5$ , $x^{-1}=0(.01)0.2$ to 6D or 7D.

►

•

Agrest et al. (1982) tabulates $\mathbf{H}_{n}\left(x\right)$ and $e^{-x}\mathbf{L}_{n}\left(x\right)$ for $n=0,1$ and $x=0(.001)5(.005)15(.01)100$ to 11D.

… ►

•

Abramowitz and Stegun (1964, Chapter 12) tabulates $\int_{0}^{x}(I_{0}\left(t\right)-\mathbf{L}_{0}\left(t\right))\,\mathrm{d}t$ and $(2/\pi)\int_{x}^{\infty}t^{-1}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ for $x=0(.1)5$ to 5D or 7D; $\int_{0}^{x}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,\mathrm{d}t-(2% /\pi)\ln x$ , $\int_{0}^{x}(I_{0}\left(t\right)-\mathbf{L}_{0}\left(t\right))\,\mathrm{d}t-(2% /\pi)\ln x$ , and $\int_{x}^{\infty}t^{-1}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,% \mathrm{d}t$ for $x^{-1}=0(.01)0.2$ to 6D.

►

•

Agrest et al. (1982) tabulates $\int_{0}^{x}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ and $e^{-x}\int_{0}^{x}\mathbf{L}_{0}\left(t\right)\,\mathrm{d}t$ for $x=0(.001)5(.005)15(.01)100$ to 11D.

… ►

•

Agrest and Maksimov (1971, Chapter 11) defines incomplete Struve, Anger, and Weber functions and includes tables of an incomplete Struve function $\mathbf{H}_{n}\left(x,\alpha\right)$ for $n=0,1$ , $x=0(.2)10$ , and $\alpha=0(.2)1.4,\tfrac{1}{2}\pi$ , together with surface plots.

50: 3.11 Approximation Techniques

… ►Beginning with

u_{n+1}=0

u_{n}=c_{n}

, we apply … ►With

b_{0}=1

, the last

q

equations give

b_{1},\dots,b_{q}

as the solution of a system of linear equations. … ►(3.11.29) is a system of

n+1

linear equations for the coefficients

a_{0},a_{1},\dots,a_{n}

. … ►With this choice of

a_{k}

and

f_{j}=f(x_{j})

, the corresponding sum (3.11.32) vanishes. … ►Two are endpoints:

(x_{0},y_{0})

and

(x_{3},y_{3})

; the other points

(x_{1},y_{1})

and

(x_{2},y_{2})

are control points. …