B�cklund transformations

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1: 21.5 Modular Transformations

§21.5 Modular Transformations

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§21.5(i) Riemann Theta Functions

… ►Equation (21.5.4) is the modular transformation property for Riemann theta functions. … ►($𝐁$ symmetric with integer elements and even diagonal elements.) …($𝐁$ symmetric with integer elements.) …

2: 24.7 Integral Representations

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24.7.11 $$B_n\left(x\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{-c-\mathrm{i}\mathrm{\infty }}^{-c+\mathrm{i}\mathrm{\infty }}{(x+t)}^n{\left(\frac{\pi }{\sin \left(\pi t\right)}\right)}^{2}dt,$$ .

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: 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

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3: 37.15 Orthogonal Polynomials on the Ball

… ►In particular, the various explicit bases of orthogonal or biorthogonal polynomials on ${\mathrm{△}}^d$ are related to similar explicit bases on ${𝔹}^d$ by quadratic transformations. … ►Second, the biorthogonal bases (37.14.9) and (37.14.8) on ${\mathrm{△}}^d$ are related to the biorthogonal bases (37.15.10) and (37.15.11) on ${𝔹}^d$ by the quadratic transformations …

4: 1.3 Determinants, Linear Operators, and Spectral Expansions

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1.3.7 $$det(𝐀𝐁)=det(𝐀)det(𝐁).$$

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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 $𝐀$.

See also:

Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

… ►Let the columns of matrix $𝐒$ be these eigenvectors ${𝐚}_1,\mathrm{\dots },{𝐚}_n$, then ${𝐒}^{-1}={𝐒}^{\mathrm{H}}$, and the similarity transformation (1.2.73) is now of the form ${𝐒}^{\mathrm{H}}𝐀𝐒={\lambda }_i{\delta }_{i,j}$. For Hermitian matrices $𝐒$ is unitary, and for real symmetric matrices $𝐒$ is an orthogonal transformation. ►For self-adjoint $𝐀$ and $𝐁$, if $[𝐀,𝐁]=𝟎$, see (1.2.66), simultaneous eigenvectors of $𝐀$ and $𝐁$ always exist. …

5: null

error generating summary

6: 18.17 Integrals

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§18.17(v) Fourier Transforms

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Jacobi

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Ultraspherical

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§18.17(vi) Laplace Transforms

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§18.17(vii) Mellin Transforms

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7: 31.2 Differential Equations

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$F$-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

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Composite Transformations

►There are $8\cdot 24=192$ automorphisms of equation (31.2.1) by compositions of $F$-homotopic and homographic transformations. …

8: 32.7 Bäcklund Transformations

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32.7.43 $$(A,B,C,D)=(\frac{1}{2}{({\mathrm{\Theta }}_{\mathrm{\infty }}-1)}^2,-\frac{1}{2}{\mathrm{\Theta }}_0^2,\frac{1}{2}{\mathrm{\Theta }}_1^2,\frac{1}{2}(1-{\mathrm{\Theta }}_2^2)),$$

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Symbols:

${\mathrm{\Theta }}_j$: 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

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9: 24.13 Integrals

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24.13.1 $$\int B_n\left(t\right)dt=\frac{B_{n+1}\left(t\right)}{n+1}+\text{const.},$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $dx$: 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

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24.13.6 $${\int }_0^1{B}_n\left(t\right)B_m\left(t\right)dt=\frac{{(-1)}^{n-1}m!n!}{(m+n)!}{B}_{m+n}.$$

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $dx$: 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 ${\int }_0^x{B}_n\left(t\right)B_m\left(t\right)dt$ and ${\int }_0^x{B}_n\left(t\right)B_m\left(t\right)B_k\left(t\right)dt$ 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). …

10: Bibliography P

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A Kummer-type transformation for a ${}_2{}^{}F_2^{}$ hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

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External Links:

ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

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$A=B$ . A K Peters Ltd., Wellesley, MA.

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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

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

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External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.37.

Encodings:

BibTeX

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Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-59209-7; 0-521-59771-4, MathReview, ZentralBlatt

Referenced by:

§1.14(vii).

Encodings:

BibTeX

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Integrals and Series: Direct Laplace Transforms, Vol. 4. Gordon and Breach Science Publishers, New York.

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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

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