SL%282%2CZ%29 bilinear transformation

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11: Bibliography B

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Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.12.

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Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

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Quasinormal ringing of Kerr black holes: The excitation factors. Phys. Rev. D 74 (104020), pp. 1–27.

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External Links:

ISSN 0556-2821, Document, MathReview

Referenced by:

6th item.

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Über Sturm-Liouvillesche Polynomsysteme. Math. Z. 29 (1), pp. 730–736.

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External Links:

ISSN 0025-5874, Link, MathReview Entry, ZentralBlatt

Referenced by:

item 1..

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The Borel transform and its use in the summation of asymptotic expansions. Stud. Appl. Math. 105 (2), pp. 83–113.

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External Links:

ISSN 0022-2526, Document, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§2.11(v).

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12: Bibliography

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Complex Analysis: An Introduction of the Theory of Analytic Functions of One Complex Variable. 2nd edition, McGraw-Hill Book Co., New York.

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Notes:

A third edition was published in 1978 by McGraw-Hill, New York.

External Links:

MathReview (P. Lelong), ZentralBlatt

Referenced by:

§1.9(iii), §23.18.

Encodings:

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On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.

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External Links:

ISSN 0024-1318, MathReview (James M. Horner)

Referenced by:

§13.9(i).

Encodings:

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Transformations $R{S}_4^2(3)$ of the ranks $\le 4$ and algebraic solutions of the sixth Painlevé equation. Comm. Math. Phys. 228 (1), pp. 151–176.

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External Links:

ISSN 0010-3616, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.9(iii).

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Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.

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External Links:

MathReview (R. N. Jain), ZentralBlatt

Referenced by:

§17.11.

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Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).

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Notes:

Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. In Russian. English translation: Russian Math. Surveys, 29(1974), no. 2, pp. 10–50.

External Links:

Document, MathReview (G. R. Belickii), ZentralBlatt

Referenced by:

§36.2(i).

Encodings:

BibTeX

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13: 6.14 Integrals

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§6.14(i) Laplace Transforms

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6.14.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{Ci}\left(t\right)dt=-\frac{1}{2a}\ln \left(1+a^2\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{Ci}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{\Re }$: real part

A&S Ref:

5.2.28

Referenced by:

§6.14(i)

Permalink:

http://dlmf.nist.gov/6.14.E2

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See also:

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

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6.14.4 $${\int }_0^{\mathrm{\infty }}{E}_1^2\left(t\right)dt=2\ln 2,$$

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Symbols:

$dx$: differential of $x$ , $E_1\left(z\right)$: exponential integral, $\int$: integral and $\ln z$: principal branch of logarithm function

A&S Ref:

5.1.33

Permalink:

http://dlmf.nist.gov/6.14.E4

Encodings:

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See also:

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

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6.14.6 $${\int }_0^{\mathrm{\infty }}{\mathrm{Ci}}^2\left(t\right)dt={\int }_0^{\mathrm{\infty }}{\mathrm{si}}^2\left(t\right)dt=\frac{1}{2}\pi ,$$

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Symbols:

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

14: 32.2 Differential Equations

… ►They are distinct modulo Möbius (bilinear) transformations … ►In ${\text{P}}_{\text{III}}$, if $w(z)={\zeta }^{-1/2}u(\zeta )$ with $\zeta =z^2$, then … ►In ${\text{P}}_{\text{IV}}$, if $w(z)=2\sqrt{2}{(u(\zeta ))}^2$ with $\zeta =\sqrt{2}z$ and $\alpha =2\nu +1$, then … ►where ${\mu }_1$, ${\mu }_2$, ${\mu }_3$ are constants, $f_1$, $f_2$, $f_3$ are functions of $z$, with … ►where ${\mu }_1$, ${\mu }_2$, ${\mu }_3$, ${\mu }_4$ are constants, $f_1$, $f_2$, $f_3$, $f_4$ are functions of $z$, with …

15: 29 Lamé Functions

Chapter 29 Lamé Functions

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16: 4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

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$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
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$\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$2-\sqrt{3}$	$\sqrt{2}(\sqrt{3}+1)$	$\sqrt{2}(\sqrt{3}-1)$	$2+\sqrt{3}$
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$\pi /4$	$\frac{1}{2}\sqrt{2}$	$\frac{1}{2}\sqrt{2}$	$1$	$\sqrt{2}$	$\sqrt{2}$	$1$
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$2\pi /3$	$\frac{1}{2}\sqrt{3}$	$-\frac{1}{2}$	$-\sqrt{3}$	$\frac{2}{3}\sqrt{3}$	$-2$	$-\frac{1}{3}\sqrt{3}$
$3\pi /4$	$\frac{1}{2}\sqrt{2}$	$-\frac{1}{2}\sqrt{2}$	$-1$	$\sqrt{2}$	$-\sqrt{2}$	$-1$
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4.17.3 $$\underset{z\to 0}{lim}\frac{1-\cos z}{z^2}=\frac{1}{2}.$$

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Symbols:

$\cos z$: cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

17: Bibliography O

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Hyperasymptotic solutions of second-order linear differential equations. I. Methods Appl. Anal. 2 (2), pp. 173–197.

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External Links:

ISSN 1073-2772, FullText, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§10.17(v), §10.40(iv), §13.29(i), §13.7(iii), §2.11(v), §9.7(v).

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Hyperasymptotic solutions of second-order linear differential equations. II. Methods Appl. Anal. 2 (2), pp. 198–211.

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External Links:

ISSN 1073-2772, FullText, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§10.17(v), §10.40(iv), §2.11(v), §9.7(v).

Encodings:

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

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External Links:

ISSN 1364-5021, Document, MathReview (Donald A. Lutz), ZentralBlatt

Referenced by:

§2.11(v), §2.7(ii).

Encodings:

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Inverse factorial-series solutions of difference equations. Proc. Edinb. Math. Soc. (2) 47 (2), pp. 421–448.

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External Links:

ISSN 0013-0915, Document, MathReview (Niro Yanagihara), ZentralBlatt

Referenced by:

§2.9(i).

Encodings:

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

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Notes:

See also Olver (1997b)

External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§2.3(iv).

Encodings:

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18: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ►For $k=2$ … ► $M_2$ is the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with $a_1$ cycles of length 1, $a_2$ cycles of length 2, $\mathrm{\dots }$, and $a_n$ cycles of length $n$: …$M_3$ is the number of set partitions of $\{1,2,\mathrm{\dots },n\}$ with $a_1$ subsets of size 1, $a_2$ subsets of size 2, $\mathrm{\dots }$, and $a_n$ subsets of size $n$: …For each $n$ all possible values of $a_1,a_2,\mathrm{\dots },a_n$ are covered. … ►where the summation is over all nonnegative integers $n_1,n_2,\mathrm{\dots },n_k$ such that $n_1+n_2+\mathrm{\cdots }+n_k=n$. …

19: 1.9 Calculus of a Complex Variable

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Bilinear Transformation

… ►The cross ratio of $z_1,z_2,z_3,z_4\in ℂ\cup \{\mathrm{\infty }\}$ is defined by …or its limiting form, and is invariant under bilinear transformations. ►Other names for the bilinear transformation are fractional linear transformation, homographic transformation, and Möbius transformation. …

20: 28.15 Expansions for Small $q$

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28.15.1 $${\lambda }_{\nu }\left(q\right)={\nu }^2+\frac{1}{2({\nu }^2-1)}{q}^2+\frac{5{\nu }^2+7}{32{({\nu }^2-1)}^3({\nu }^2-4)}{q}^4+\frac{9{\nu }^4+58{\nu }^2+29}{64{({\nu }^2-1)}^5({\nu }^2-4)({\nu }^2-9)}{q}^6+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $\nu$: complex parameter

A&S Ref:

20.3.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.15.E1

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See also:

Annotations for §28.15(i), §28.15 and Ch.28

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28.15.2 $$a-{\nu }^2-\frac{q^2}{a-{(\nu +2)}^2-}\frac{q^2}{a-{(\nu +4)}^2-}\mathrm{\cdots }=\frac{q^2}{a-{(\nu -2)}^2-}\frac{q^2}{a-{(\nu -4)}^2-}\mathrm{\cdots }.$$

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Symbols:

$q=h^2$: parameter, $\nu$: complex parameter and $a$: parameter

Referenced by:

item (e)

Permalink:

http://dlmf.nist.gov/28.15.E2

Encodings:

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See also:

Annotations for §28.15(i), §28.15 and Ch.28

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28.15.3 $${\mathrm{me}}_{\nu }(z,q)={\mathrm{e}}^{\mathrm{i}\nu z}-\frac{q}{4}\left(\frac{1}{\nu +1}{\mathrm{e}}^{\mathrm{i}(\nu +2)z}-\frac{1}{\nu -1}{\mathrm{e}}^{\mathrm{i}(\nu -2)z}\right)+\frac{q^2}{32}\left(\frac{1}{(\nu +1)(\nu +2)}{\mathrm{e}}^{\mathrm{i}(\nu +4)z}+\frac{1}{(\nu -1)(\nu -2)}{\mathrm{e}}^{\mathrm{i}(\nu -4)z}-\frac{2({\nu }^2+1)}{{({\nu }^2-1)}^2}{\mathrm{e}}^{\mathrm{i}\nu z}\right)+\mathrm{\cdots };$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

20.3.17 (only two terms and without normalization)

Permalink:

http://dlmf.nist.gov/28.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(ii), §28.15 and Ch.28

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