Sears’ balanced 4ϕ3 transformation

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11: 32.7 Bäcklund Transformations

… ►Then the transformations … ►Next, let $W_j=W(z;{\alpha }_j,{\beta }_j,1,-1)$, $j=0,1,2,3,4$, be solutions of ${\text{P}}_{\text{III}}$ with … ►The quadratic transformation …transforms ${\text{P}}_{\text{VI}}$ with $\alpha =-\beta$ and $\gamma =\frac{1}{2}-\delta$ to ${\text{P}}_{\text{VI}}$ with $({\alpha }_1,{\beta }_1,{\gamma }_1,{\delta }_1)=(4\alpha ,-4\gamma ,0,\frac{1}{2})$. … ►

12: 3.9 Acceleration of Convergence

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§3.9(i) Sequence Transformations

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§3.9(ii) Euler’s Transformation of Series

… ►Euler’s transformation is usually applied to alternating series. … ►

§3.9(iv) Shanks’ Transformation

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§3.9(vi) Applications and Further Transformations

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13: 16.16 Transformations of Variables

§16.16 Transformations of Variables

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§16.16(i) Reduction Formulas

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§16.16(ii) Other Transformations

… ►For quadratic transformations of Appell functions see Carlson (1976).

14: 9.10 Integrals

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§9.10(v) Laplace Transforms

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§9.10(vi) Mellin Transform

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§9.10(vii) Stieltjes Transforms

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§9.10(ix) Compendia

►For further integrals, including the Airy transform, see §9.11(iv), Widder (1979), Prudnikov et al. (1990, §1.8.1), Prudnikov et al. (1992a, pp. 405–413), Prudnikov et al. (1992b, §4.3.25), Vallée and Soares (2010, Chapters 3, 4).

15: 7.14 Integrals

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Fourier Transform

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Laplace Transforms

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7.14.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{erf}\left(bt\right)dt=\frac{1}{a}{\mathrm{e}}^{a^2/(4b^2)}\mathrm{erfc}\left(\frac{a}{2b}\right),$$ $\mathrm{\Re }a>0$, ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.17

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

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7.14.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{erf}\sqrt{bt}dt=\frac{1}{a}\sqrt{\frac{b}{a+b}},$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }b>0$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.19

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

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Laplace Transforms

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16: Bibliography W

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The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.

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External Links:

ZentralBlatt

Referenced by:

§15.8(v), §15.8(v).

Encodings:

BibTeX

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The Airy transform. Amer. Math. Monthly 86 (4), pp. 271–277.

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External Links:

ISSN 0002-9890, FullText, Document, MathReview (A. G. Ramm), ZentralBlatt

Referenced by:

(9.10.13), §9.10(ix), §9.10(v).

Encodings:

BibTeX

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Asymptotics for the ${}_4{}^{}F_3^{}$ polynomials. J. Approx. Theory 66 (1), pp. 58–71.

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External Links:

ISSN 0021-9045, Document, MathReview (R. G. Buschman), ZentralBlatt

Referenced by:

§18.26(v).

Encodings:

BibTeX

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Algorithm 44: Bessel functions computed recursively. Comm. ACM 4 (4), pp. 177–178.

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Notes:

Includes Algol code, maximum accuracy 8S.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

Encodings:

BibTeX

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Asymptotic expansions of Hankel transforms of functions with logarithmic singularities. Comput. Math. Appl. 3 (4), pp. 271–286.

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External Links:

ISSN 0886-9561, Document, MathReview, ZentralBlatt

Referenced by:

§10.22(v).

Encodings:

BibTeX

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17: Guide to Searching the DLMF

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Table 1: Query Examples

► ►►►►►

Query	Matching records contain
"Fourier transform" and series	both the phrase “Fourier transform” and the word “series”.
…
Fourier (transform or series)	at least one of “Fourier transform” or “Fourier series”.
1/(2pi) and "Fourier transform"	both $1/(2\pi )$ and the phrase “Fourier transform”.
…

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Table 4: Font and Accent Examples

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Query	Matches
…

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18: 6.14 Integrals

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§6.14(i) Laplace Transforms

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6.14.5 $${\int }_0^{\mathrm{\infty }}\cos t\mathrm{Ci}\left(t\right)dt={\int }_0^{\mathrm{\infty }}\sin t\mathrm{si}\left(t\right)dt=-\frac{1}{4}\pi ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\int$: integral, $\mathrm{si}\left(z\right)$: sine integral and $\sin z$: sine function

A&S Ref:

5.2.30

Permalink:

http://dlmf.nist.gov/6.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(ii), §6.14 and Ch.6

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19: 20.10 Integrals

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§20.10(i) Mellin Transforms with respect to the Lattice Parameter

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20.10.1 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}{\theta }_2\left(0\right|\mathrm{i}{x}^2)dx=2^s(1-2^{-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E1

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_2\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_4\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.31) and (20.4.5).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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20.10.3 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}(1-{\theta }_4\left(0\right|\mathrm{i}{x}^2))dx=(1-2^{1-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E3

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>0$ because $1-{\theta }_4\left(0\right|\mathrm{i}{x}^2)=1-x^{-1}{\theta }_2\left(0\right|\mathrm{i}{x}^{-2})\sim 1$ as $x\to 0+$, which follows from (20.7.33) and (20.4.3).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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§20.10(ii) Laplace Transforms with respect to the Lattice Parameter

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20.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_3\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_4\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=\frac{\mathrm{\ell }}{\sqrt{s}}\cosh \left(\beta \sqrt{s}\right)\mathrm{csch}\left(\mathrm{\ell }\sqrt{s}\right).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

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20: 23.15 Definitions

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$k^{\prime }={\displaystyle \frac{{\theta }_4^2(0,q)}{{\theta }_3^2(0,q)}}.$

►Also $𝒜$ denotes a bilinear transformation on $\tau$, given by …The set of all bilinear transformations of this form is denoted by SL$(2,\mathbb{Z})$ (Serre (1973, p. 77)). … ►

23.15.6 $$\lambda \left(\tau \right)=\frac{{\theta }_2^4(0,q)}{{\theta }_3^4(0,q)};$$

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Defines:

$\lambda \left(\tau \right)$: elliptic modular function

Symbols:

${\theta }_j(z,q)$: theta function, $q$: nome and $\tau$: complex variable

Referenced by:

§20.9(ii), §23.17(iii)

Permalink:

http://dlmf.nist.gov/23.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

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