B�cklund transformations

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1: 21.5 Modular Transformations

§21.5 Modular Transformations

►

§21.5(i) Riemann Theta Functions

… ►Equation (21.5.4) is the modular transformation property for Riemann theta functions. … ►(

\mathbf{B}

symmetric with integer elements and even diagonal elements.) …(

\mathbf{B}

symmetric with integer elements.) …

2: 24.7 Integral Representations

… ►

24.7.11

B_{n}\left(x\right)=\frac{1}{2\pi i}\int_{-c-i\infty}^{-c+i\infty}(x+t)^{n}% \left(\frac{\pi}{\sin\left(\pi t\right)}\right)^{2}\,\mathrm{d}t,

0<c<1

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $n$ : integer, $t$ : real or complex and $x$ : real or complex
Keywords:: Mellin transform
Referenced by:: §24.7(ii)
Permalink:: http://dlmf.nist.gov/24.7.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §24.7(ii), §24.7(ii), §24.7 and Ch.24

…

3: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►

1.3.7

\det(\mathbf{A}\mathbf{B})=\det(\mathbf{A})\det(\mathbf{B}).

ⓘ

Symbols:: $\det$ : determinant, $j$ : integer and $k$ : integer
Referenced by:: §1.2(vi), §1.3(i), Erratum (V1.2.0) for Equations (1.3.5), (1.3.6), (1.3.7), Erratum (V1.2.0) §1.3
Permalink:: http://dlmf.nist.gov/1.3.E7
Encodings:: pMML, png, TeX
Modification (effective with 1.2.0):: Previously we did use the notation $[a_{jk}]$ for $\mathbf{A}$ .
See also:: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

… ►Let the columns of matrix

\mathbf{S}

be these eigenvectors

\mathbf{a}_{1},\dots,\mathbf{a}_{n}

, then

{\mathbf{S}}^{-1}={\mathbf{S}}^{{\rm H}}

, and the similarity transformation (1.2.73) is now of the form

{\mathbf{S}}^{{\rm H}}\mathbf{A}\mathbf{S}=\lambda_{i}\delta_{i,j}

. For Hermitian matrices

\mathbf{S}

is unitary, and for real symmetric matrices

\mathbf{S}

is an orthogonal transformation. ►For self-adjoint

\mathbf{A}

and

\mathbf{B}

, if

[{\mathbf{A}},{\mathbf{B}}]=\boldsymbol{{0}}

, see (1.2.66), simultaneous eigenvectors of

\mathbf{A}

and

\mathbf{B}

always exist. …

4: 18.17 Integrals

… ►

§18.17(v) Fourier Transforms

… ►

Jacobi

… ►

Ultraspherical

… ►

§18.17(vi) Laplace Transforms

… ►

§18.17(vii) Mellin Transforms

…

5: 31.2 Differential Equations

… ►

$F$ -Homotopic Transformations

… ►By composing these three steps, there result

2^{3}=8

possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1). ►

Homographic Transformations

… ►

Composite Transformations

►There are

8\cdot 24=192

automorphisms of equation (31.2.1) by compositions of

F

-homotopic and homographic transformations. …

6: 32.7 Bäcklund Transformations

… ►

32.7.43

(A,B,C,D)=\left(\tfrac{1}{2}(\Theta_{\infty}-1)^{2},-\tfrac{1}{2}\Theta_{0}^{2% },\tfrac{1}{2}\Theta_{1}^{2},\tfrac{1}{2}(1-\Theta_{2}^{2})\right),

ⓘ

Symbols:: $\Theta_{j}$ : transformation
Permalink:: http://dlmf.nist.gov/32.7.E43
Encodings:: pMML, png, TeX
See also:: Annotations for §32.7(vii), §32.7 and Ch.32

…

7: 24.13 Integrals

… ►

24.13.1

\int B_{n}\left(t\right)\,\mathrm{d}t=\frac{B_{n+1}\left(t\right)}{n+1}+\text{% const.},

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer and $t$ : real or complex
A&S Ref:: 23.1.11
Referenced by:: §24.13(i)
Permalink:: http://dlmf.nist.gov/24.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §24.13(i), §24.13 and Ch.24

… ►

24.13.6

\int_{0}^{1}B_{n}\left(t\right)B_{m}\left(t\right)\,\mathrm{d}t=\frac{(-1)^{n-% 1}m!n!}{(m+n)!}B_{m+n}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $m$ : integer, $n$ : integer and $t$ : real or complex
A&S Ref:: 23.1.12
Referenced by:: §24.13(i)
Permalink:: http://dlmf.nist.gov/24.13.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §24.13(i), §24.13 and Ch.24

►For integrals of the form

\int_{0}^{x}B_{n}\left(t\right)B_{m}\left(t\right)\,\mathrm{d}t

and

\int_{0}^{x}B_{n}\left(t\right)B_{m}\left(t\right)B_{k}\left(t\right)\,\mathrm% {d}t

see Agoh and Dilcher (2011). … ►

§24.13(iii) Compendia

►For Laplace and inverse Laplace transforms see Prudnikov et al. (1992a, §§3.28.1–3.28.2) and Prudnikov et al. (1992b, §§3.26.1–3.26.2). …

8: Bibliography P

… ►

R. B. Paris (2005a) A Kummer-type transformation for a

{}_{2}F_{2}

hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

… ►

M. Petkovšek, H. S. Wilf, and D. Zeilberger (1996)

A=B

. A K Peters Ltd., Wellesley, MA.

ⓘ

Notes:: With a separately available computer disk
External Links:: ISBN 1-56881-063-6, MathReview (Peter Paule), ZentralBlatt
Referenced by:: §16.23(iv).
Encodings:: BibTeX

►

E. Petropoulou (2000) Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

ⓘ

External Links:: ISSN 1065-2469, Document, MathReview, ZentralBlatt
Referenced by:: §10.37.
Encodings:: BibTeX

… ►

A. Pinkus and S. Zafrany (1997) Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.

ⓘ

External Links:: ISBN 0-521-59209-7; 0-521-59771-4, MathReview, ZentralBlatt
Referenced by:: §1.14(vii).
Encodings:: BibTeX

… ►

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992a) Integrals and Series: Direct Laplace Transforms, Vol. 4. Gordon and Breach Science Publishers, New York.

ⓘ

Notes:: Table erratum: Math. Comp. v. 66 (1997), no. 220, p. 1766.
External Links:: ISBN 2-88124-837-3, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §1.14(viii), §1.4(iv), §10.22(vi), §10.43(vi), §10.71(iii), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(ii), §13.10(vi), §13.23(i), §14.17(v), §15.14, §16.20, §16.5, §18.17(ix), §19.13(iii), §20.10(iii), §24.13(iii), §25.11(ix), §25.12(ii), §25.7, §5.13, §6.14(iii), §7.14(iii), §8.14, §9.10(ix).
Encodings:: BibTeX

…

9: 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 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 $p$

►For number-theoretic applications it is important to compute

B_{2n}\pmod{p}

for

2n\leq p-3

; in particular to find the irregular pairs

(2n,p)

for which

B_{2n}\equiv 0\pmod{p}

. … ►

•

Buhler et al. (1992) uses the expansion

24.19.3

\frac{t^{2}}{\cosh t-1}=-2\sum_{n=0}^{\infty}(2n-1)B_{2n}\frac{t^{2n}}{(2n)!},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $!$ : factorial (as in $n!$ ), $\cosh\NVar{z}$ : hyperbolic cosine function, $n$ : integer and $t$ : real or complex
Referenced by:: §24.19(ii)
Permalink:: http://dlmf.nist.gov/24.19.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.19(ii), §24.19 and Ch.24

and computes inverses modulo $p$ of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).

…

10: Bibliography U

… ►

H. Umemura (2000) On the transformation group of the second Painlevé equation. Nagoya Math. J. 157, pp. 15–46.

ⓘ

External Links:: ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.7(viii).
Encodings:: BibTeX

… ►

J. Urbanowicz (1988) On the equation

f(1)1^{k}+f(2)2^{k}+\cdots+f(x)x^{k}+{R}(x)={B}y^{2}

. Acta Arith. 51 (4), pp. 349–368.

ⓘ

External Links:: ISSN 0065-1036, Pdf, MathReview (István Gaál), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

…