SL(2,Z) bilinear transformation

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42: 7.14 Integrals

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Fourier Transform

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Laplace Transforms

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Laplace Transforms

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7.14.7

\int_{0}^{\infty}e^{-at}C\left(\sqrt{\frac{2t}{\pi}}\right)\,\mathrm{d}t=\frac% {(\sqrt{a^{2}+1}+a)^{\frac{1}{2}}}{2a\sqrt{a^{2}+1}},

\Re a>0

ⓘ

Symbols:: $C\left(\NVar{z}\right)$ : Fresnel integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $\Re$ : real part
Keywords:: Laplace transform
A&S Ref:: 7.4.29 in modified form
Referenced by:: §7.14(ii)
Permalink:: http://dlmf.nist.gov/7.14.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

… ►For collections of integrals see Apelblat (1983, pp. 131–146), Erdélyi et al. (1954a, vol. 1, pp. 40, 96, 176–177), Geller and Ng (1971), Gradshteyn and Ryzhik (2000, §§5.4 and 6.28–6.32), Marichev (1983, pp. 184–189), Ng and Geller (1969), Oberhettinger (1974, pp. 138–139, 142–143), Oberhettinger (1990, pp. 48–52, 155–158), Oberhettinger and Badii (1973, pp. 171–172, 179–181), Prudnikov et al. (1986b, vol. 2, pp. 30–36, 93–143), Prudnikov et al. (1992a, §§3.7–3.8), and Prudnikov et al. (1992b, §§3.7–3.8). …

43: 6.14 Integrals

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§6.14(i) Laplace Transforms

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6.14.2

\int_{0}^{\infty}e^{-at}\operatorname{Ci}\left(t\right)\,\mathrm{d}t=-\frac{1}% {2a}\ln\left(1+a^{2}\right),

\Re a>0

ⓘ

Symbols:: $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $\Re$ : real part
A&S Ref:: 5.2.28
Referenced by:: §6.14(i)
Permalink:: http://dlmf.nist.gov/6.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §6.14(i), §6.14 and Ch.6

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6.14.4

\int_{0}^{\infty}{E_{1}}^{2}\left(t\right)\,\mathrm{d}t=2\ln 2,

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\int$ : integral and $\ln\NVar{z}$ : principal branch of logarithm function
A&S Ref:: 5.1.33
Permalink:: http://dlmf.nist.gov/6.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §6.14(ii), §6.14 and Ch.6

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6.14.6

\int_{0}^{\infty}{\operatorname{Ci}}^{2}\left(t\right)\,\mathrm{d}t=\int_{0}^{% \infty}{\operatorname{si}}^{2}\left(t\right)\,\mathrm{d}t=\tfrac{1}{2}\pi,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\operatorname{si}\left(\NVar{z}\right)$ : sine integral
A&S Ref:: 5.2.31
Permalink:: http://dlmf.nist.gov/6.14.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §6.14(ii), §6.14 and Ch.6

… ►For collections of integrals, see Apelblat (1983, pp. 110–123), Bierens de Haan (1939, pp. 373–374, 409, 479, 571–572, 637, 664–673, 680–682, 685–697), Erdélyi et al. (1954a, vol. 1, pp. 40–42, 96–98, 177–178, 325), Geller and Ng (1969), Gradshteyn and Ryzhik (2000, §§5.2–5.3 and 6.2–6.27), Marichev (1983, pp. 182–184), Nielsen (1906b), Oberhettinger (1974, pp. 139–141), Oberhettinger (1990, pp. 53–55 and 158–160), Oberhettinger and Badii (1973, pp. 172–179), Prudnikov et al. (1986b, vol. 2, pp. 24–29 and 64–92), Prudnikov et al. (1992a, §§3.4–3.6), Prudnikov et al. (1992b, §§3.4–3.6), and Watrasiewicz (1967).

44: 21.6 Products

… ►Also, let

\mathbf{Z}

be an arbitrary

g\times h

matrix. … ►

21.6.3

\prod_{j=1}^{h}\theta\left(\sum_{k=1}^{h}T_{jk}\mathbf{z}_{k}\middle|% \boldsymbol{{\Omega}}\right)=\frac{1}{\mathcal{D}^{g}}\sum_{\mathbf{A}\in% \mathcal{K}}\sum_{\mathbf{B}\in\mathcal{K}}e^{2\pi i\operatorname{tr}\left[% \frac{1}{2}\mathbf{A}^{\mathrm{T}}\boldsymbol{{\Omega}}\mathbf{A}+\mathbf{A}^{% \mathrm{T}}[\mathbf{Z}+\mathbf{B}]\right]}\*\prod_{j=1}^{h}\theta\left(\mathbf% {z}_{j}+\boldsymbol{{\Omega}}\mathbf{a}_{j}+\mathbf{b}_{j}\middle|\boldsymbol{% {\Omega}}\right),

►where

\mathbf{z}_{j}

\mathbf{a}_{j}

\mathbf{b}_{j}

denote respectively the

j

th columns of

\mathbf{Z}

\mathbf{A}

\mathbf{B}

. … ►

21.6.5

\mathbf{T}=\frac{1}{2}\begin{bmatrix}1&1&1&1\\ 1&1&-1&-1\\ 1&-1&1&-1\\ 1&-1&-1&1\end{bmatrix}.

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Referenced by:: Erratum (V1.0.17) for Equation (21.6.5)
Permalink:: http://dlmf.nist.gov/21.6.E5
Encodings:: pMML, png, TeX
Errata (effective with 1.0.17):: Originally the prefactor $\frac{1}{2}$ on the right-hand side was missing.
Reported 2017-08-12 by Wolfgang Bauhardt
See also:: Annotations for §21.6(i), §21.6(i), §21.6 and Ch.21

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21.6.7

\theta\genfrac{[}{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}+\mathbf{c}_{2}+% \mathbf{c}_{3}+\mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}+\mathbf{d}_{2}+% \mathbf{d}_{3}+\mathbf{d}_{4}]}\left(\frac{\mathbf{x}+\mathbf{y}+\mathbf{u}+% \mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{% }{\tfrac{1}{2}[\mathbf{c}_{1}+\mathbf{c}_{2}-\mathbf{c}_{3}-\mathbf{c}_{4}]}{% \tfrac{1}{2}[\mathbf{d}_{1}+\mathbf{d}_{2}-\mathbf{d}_{3}-\mathbf{d}_{4}]}% \left(\frac{\mathbf{x}+\mathbf{y}-\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol% {{\Omega}}\right)\*\theta\genfrac{[}{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}-% \mathbf{c}_{2}+\mathbf{c}_{3}-\mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}-% \mathbf{d}_{2}+\mathbf{d}_{3}-\mathbf{d}_{4}]}\left(\frac{\mathbf{x}-\mathbf{y% }+\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{% [}{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}-\mathbf{c}_{2}-\mathbf{c}_{3}+% \mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}-\mathbf{d}_{2}-\mathbf{d}_{3}+% \mathbf{d}_{4}]}\left(\frac{\mathbf{x}-\mathbf{y}-\mathbf{u}+\mathbf{v}}{2}% \middle|\boldsymbol{{\Omega}}\right)\\ =\frac{1}{2^{g}}\sum_{\boldsymbol{{\alpha}}\in\frac{1}{2}{\mathbb{Z}}^{g}/{% \mathbb{Z}}^{g}}\,\sum_{\boldsymbol{{\beta}}\in\frac{1}{2}{\mathbb{Z}}^{g}/{% \mathbb{Z}}^{g}}e^{-2\pi i\boldsymbol{{\beta}}\cdot[\mathbf{c}_{1}+\mathbf{c}_% {2}+\mathbf{c}_{3}+\mathbf{c}_{4}]}\*\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_% {1}+\boldsymbol{{\alpha}}}{\mathbf{d}_{1}+\boldsymbol{{\beta}}}\left(\mathbf{x% }\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_% {2}+\boldsymbol{{\alpha}}}{\mathbf{d}_{2}+\boldsymbol{{\beta}}}\left(\mathbf{y% }\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_% {3}+\boldsymbol{{\alpha}}}{\mathbf{d}_{3}+\boldsymbol{{\beta}}}\left(\mathbf{u% }\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_% {4}+\boldsymbol{{\alpha}}}{\mathbf{d}_{4}+\boldsymbol{{\beta}}}\left(\mathbf{v% }\middle|\boldsymbol{{\Omega}}\right).

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathbb{Z}$ : set of all integers, $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.6.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §21.6(i), §21.6(i), §21.6 and Ch.21

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45: 18.8 Differential Equations

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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

► ►►►►►►

#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	$\lambda_{n}$
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2	$\begin{gathered}\textstyle\left(\sin\frac{1}{2}x\right)^{\alpha+\frac{1}{2}}% \left(\cos\frac{1}{2}x\right)^{\beta+\frac{1}{2}}\\ \times P^{(\alpha,\beta)}_{n}\left(\cos x\right)\end{gathered}$	$1$	$0$	$\dfrac{\tfrac{1}{4}-\alpha^{2}}{4{\sin}^{2}\frac{1}{2}x}+\dfrac{\frac{1}{4}-% \beta^{2}}{4{\cos}^{2}\frac{1}{2}x}$	$\left(n+\frac{1}{2}(\alpha+\beta+1)\right)^{2}$
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9	${\mathrm{e}}^{-\frac{1}{2}x^{2}}x^{\alpha+\frac{1}{2}}L^{(\alpha)}_{n}\left(x^% {2}\right)$	$1$	$0$	$-x^{2}+(\frac{1}{4}-\alpha^{2})x^{-2}$	$4n+2\alpha+2$
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12	$H_{n}\left(x\right)$	$1$	$-2x$	$0$	$2n$
13	${\mathrm{e}}^{-\frac{1}{2}x^{2}}H_{n}\left(x\right)$	$1$	$0$	$-x^{2}$	$2n+1$
…

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

Z=1

46: 29.21 Tables

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•

Ince (1940a) tabulates the eigenvalues $a^{m}_{\nu}\left(k^{2}\right)$ , $b^{m+1}_{\nu}\left(k^{2}\right)$ (with $a^{2m+1}_{\nu}$ and $b^{2m+1}_{\nu}$ interchanged) for $k^{2}=0.1,0.5,0.9$ , $\nu=-\frac{1}{2},0(1)25$ , and $m=0,1,2,3$ . Precision is 4D.

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•

Arscott and Khabaza (1962) tabulates the coefficients of the polynomials $P$ in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues $h$ for $k^{2}=0.1(.1)0.9$ , $n=1(1)30$ . Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.

47: 1.16 Distributions

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§1.16(vii) Fourier Transforms of Tempered Distributions

… ►Then its Fourier transform is … ►The Fourier transform

\mathscr{F}\left(u\right)

of a tempered distribution is again a tempered distribution, and … ►

§1.16(viii) Fourier Transforms of Special Distributions

… ►Since

\sqrt{2\pi}\mathscr{F}\left(\delta\right)=1

, we have …

48: 13.10 Integrals

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§13.10(ii) Laplace Transforms

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§13.10(iii) Mellin Transforms

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§13.10(iv) Fourier Transforms

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§13.10(v) Hankel Transforms

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49: 2.6 Distributional Methods

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§2.6(ii) Stieltjes Transform

… ►The Stieltjes transform of

f(t)

is defined by … ►

\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)

being the Mellin transform of

f(t)

or its analytic continuation (§2.5(ii)). … ►Corresponding results for the generalized Stieltjes transform … ►where

\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)

is the Mellin transform of

f

or its analytic continuation. …

50: 19.22 Quadratic Transformations

§19.22 Quadratic Transformations

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Bartky’s Transformation

… ►Descending Gauss transformations include, as special cases, transformations of complete integrals into complete integrals; ascending Landen transformations do not. … ►

SL(2,Z) bilinear transformation

41—50 of 814 matching pages

41: 13.23 Integrals

§13.23(i) Laplace and Mellin Transforms

§13.23(ii) Fourier Transforms

§13.23(iii) Hankel Transforms

§13.23(iv) Integral Transforms in terms of Whittaker Functions

42: 7.14 Integrals

Fourier Transform

Laplace Transforms

Laplace Transforms

43: 6.14 Integrals

§6.14(i) Laplace Transforms

44: 21.6 Products

45: 18.8 Differential Equations

46: 29.21 Tables

47: 1.16 Distributions

§1.16(vii) Fourier Transforms of Tempered Distributions

§1.16(viii) Fourier Transforms of Special Distributions

48: 13.10 Integrals

§13.10(ii) Laplace Transforms

§13.10(iii) Mellin Transforms

§13.10(iv) Fourier Transforms

§13.10(v) Hankel Transforms

49: 2.6 Distributional Methods

§2.6(ii) Stieltjes Transform

50: 19.22 Quadratic Transformations

§19.22 Quadratic Transformations

Bartky’s Transformation