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11—20 of 521 matching pages

11: Bibliography

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J. Abad and J. Sesma (1995) Computation of the regular confluent hypergeometric function. The Mathematica Journal 5 (4), pp. 74–76.

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External Links:: ISSN 1047-5974, FullText
Referenced by:: §13.32(iii).
Encodings:: BibTeX

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R. W. Abernathy and R. P. Smith (1993) Algorithm 724: Program to calculate F-percentiles. ACM Trans. Math. Software 19 (4), pp. 481–483.

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Notes:: Double-precision Fortran, maximum accuracy 13D.
External Links:: ISSN 0098-3500, Document, GAMS (module 724; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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V. È. Adler (1994) Nonlinear chains and Painlevé equations. Phys. D 73 (4), pp. 335–351.

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External Links:: ISSN 0167-2789, Document, MathReview (F. Pempinelli), ZentralBlatt
Referenced by:: §32.2(v).
Encodings:: BibTeX

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F. V. Andreev and A. V. Kitaev (2002) Transformations

RS^{2}_{4}(3)

of the ranks

\leq 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).
Encodings:: BibTeX

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G. E. Andrews (1972) 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.
Encodings:: BibTeX

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12: Bibliography N

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D. Naylor (1990) On an asymptotic expansion of the Kontorovich-Lebedev transform. Applicable Anal. 39 (4), pp. 249–263.

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External Links:: ISSN 0003-6811, Document, MathReview (Aloknath Chakrabarti), ZentralBlatt
Referenced by:: §10.43(v).
Encodings:: BibTeX

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G. Nemes (2014b) The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.

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External Links:: ISSN 1848-5979, Document, Link, MathReview (José Luis López), ZentralBlatt
Referenced by:: (11.11.10), (11.11.8), §11.11(iii).
Encodings:: BibTeX

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G. Nemes (2014c) The resurgence properties of the large order asymptotics of the Anger-Weber function II. J. Class. Anal. 4 (2), pp. 121–147.

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External Links:: ISSN 1848-5979, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: (11.11.10), (11.11.8), §11.11(iii).
Encodings:: BibTeX

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G. Nemes (2018) Error bounds for the large-argument asymptotic expansions of the Lommel and allied functions. Stud. Appl. Math. 140 (4), pp. 508–541.

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External Links:: ISSN 0022-2526, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt
Referenced by:: §11.11(i), §11.11(i).
Encodings:: BibTeX

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E. Neuman (2004) Inequalities involving Bessel functions of the first kind. JIPAM. J. Inequal. Pure Appl. Math. 5 (4), pp. Article 94, 4 pp. (electronic).

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External Links:: ISSN 1443-5756, FullText, MathReview, ZentralBlatt
Referenced by:: §10.14.
Encodings:: BibTeX

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13: 9.4 Maclaurin Series

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9.4.1

\operatorname{Ai}\left(z\right)=\operatorname{Ai}\left(0\right)\left(1+\frac{1% }{3!}z^{3}+\frac{1\cdot 4}{6!}z^{6}+\frac{1\cdot 4\cdot 7}{9!}z^{9}+\cdots% \right)+\operatorname{Ai}'\left(0\right)\left(z+\frac{2}{4!}z^{4}+\frac{2\cdot 5% }{7!}z^{7}+\frac{2\cdot 5\cdot 8}{10!}z^{10}+\cdots\right),

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $!$ : factorial (as in $n!$ ) and $z$ : complex variable
Source:: Olver (1997b, p. 54)
A&S Ref:: 10.4.2 (with 10.4.4 and 10.4.5)
Permalink:: http://dlmf.nist.gov/9.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §9.4 and Ch.9

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9.4.2

\operatorname{Ai}'\left(z\right)=\operatorname{Ai}'\left(0\right)\left(1+\frac% {2}{3!}z^{3}+\frac{2\cdot 5}{6!}z^{6}+\frac{2\cdot 5\cdot 8}{9!}z^{9}+\cdots% \right)+\operatorname{Ai}\left(0\right)\left(\frac{1}{2!}z^{2}+\frac{1\cdot 4}% {5!}z^{5}+\frac{1\cdot 4\cdot 7}{8!}z^{8}+\cdots\right),

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $!$ : factorial (as in $n!$ ) and $z$ : complex variable
Proof sketch:: Use methods of Olver (1997b, p. 54).
Permalink:: http://dlmf.nist.gov/9.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §9.4 and Ch.9

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9.4.3

\operatorname{Bi}\left(z\right)=\operatorname{Bi}\left(0\right)\left(1+\frac{1% }{3!}z^{3}+\frac{1\cdot 4}{6!}z^{6}+\frac{1\cdot 4\cdot 7}{9!}z^{9}+\cdots% \right)+\operatorname{Bi}'\left(0\right)\left(z+\frac{2}{4!}z^{4}+\frac{2\cdot 5% }{7!}z^{7}+\frac{2\cdot 5\cdot 8}{10!}z^{10}+\cdots\right),

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Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $!$ : factorial (as in $n!$ ) and $z$ : complex variable
Proof sketch:: Use methods of Olver (1997b, p. 54).
A&S Ref:: 10.4.3 (with 10.4.4 and 10.4.5)
Referenced by:: (9.12.15)
Permalink:: http://dlmf.nist.gov/9.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §9.4 and Ch.9

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9.4.4

\operatorname{Bi}'\left(z\right)=\operatorname{Bi}'\left(0\right)\left(1+\frac% {2}{3!}z^{3}+\frac{2\cdot 5}{6!}z^{6}+\frac{2\cdot 5\cdot 8}{9!}z^{9}+\cdots% \right)+\operatorname{Bi}\left(0\right)\left(\frac{1}{2!}z^{2}+\frac{1\cdot 4}% {5!}z^{5}+\frac{1\cdot 4\cdot 7}{8!}z^{8}+\cdots\right).

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Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $!$ : factorial (as in $n!$ ) and $z$ : complex variable
Proof sketch:: Use methods of Olver (1997b, p. 54).
Permalink:: http://dlmf.nist.gov/9.4.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §9.4 and Ch.9

14: 11.8 Analogs to Kelvin Functions

§11.8 Analogs to Kelvin Functions

►For properties of Struve functions of argument

xe^{\pm 3\pi i/4}

see McLachlan and Meyers (1936).

15: 22.10 Maclaurin Series

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22.10.1

\operatorname{sn}\left(z,k\right)=z-\left(1+k^{2}\right)\frac{z^{3}}{3!}+\left% (1+14k^{2}+k^{4}\right)\frac{z^{5}}{5!}-\left(1+135k^{2}+135k^{4}+k^{6}\right)% \frac{z^{7}}{7!}+O\left(z^{9}\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $!$ : factorial (as in $n!$ ), $z$ : complex and $k$ : modulus
A&S Ref:: 16.22.1
Permalink:: http://dlmf.nist.gov/22.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §22.10(i), §22.10 and Ch.22

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22.10.2

\operatorname{cn}\left(z,k\right)=1-\frac{z^{2}}{2!}+\left(1+4k^{2}\right)% \frac{z^{4}}{4!}-\left(1+44k^{2}+16k^{4}\right)\frac{z^{6}}{6!}+O\left(z^{8}% \right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $!$ : factorial (as in $n!$ ), $z$ : complex and $k$ : modulus
A&S Ref:: 16.22.2
Permalink:: http://dlmf.nist.gov/22.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.10(i), §22.10 and Ch.22

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22.10.3

\operatorname{dn}\left(z,k\right)=1-k^{2}\frac{z^{2}}{2!}+k^{2}\left(4+k^{2}% \right)\frac{z^{4}}{4!}-k^{2}\left(16+44k^{2}+k^{4}\right)\frac{z^{6}}{6!}+O% \left(z^{8}\right).

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $!$ : factorial (as in $n!$ ), $z$ : complex and $k$ : modulus
A&S Ref:: 16.22.3
Permalink:: http://dlmf.nist.gov/22.10.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.10(i), §22.10 and Ch.22

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22.10.4

\operatorname{sn}\left(z,k\right)=\sin z-\frac{k^{2}}{4}(z-\sin z\cos z)\cos z% +O\left(k^{4}\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.13.1
Referenced by:: §22.10(ii), §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.10.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.10(ii), §22.10 and Ch.22

►

22.10.5

\operatorname{cn}\left(z,k\right)=\cos z+\frac{k^{2}}{4}(z-\sin z\cos z)\sin z% +O\left(k^{4}\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.13.2
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.10.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §22.10(ii), §22.10 and Ch.22

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16: Bibliography H

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W. Hahn (1949) Über Orthogonalpolynome, die

q

-Differenzengleichungen genügen. Math. Nachr. 2, pp. 4–34 (German).

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External Links:: MathReview (N. J. Fine), ZentralBlatt
Referenced by:: §18.27(i).
Encodings:: BibTeX

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J. R. Herndon (1961a) Algorithm 55: Complete elliptic integral of the first kind. Comm. ACM 4 (4), pp. 180.

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Notes:: Includes Algol program, maximum accuracy 8D. For certification see same journal 6 (1966), pp. 166–167
External Links:: ISSN 0001-0782, Document
Referenced by:: §19.39(ii).
Encodings:: BibTeX

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J. R. Herndon (1961b) Algorithm 56: Complete elliptic integral of the second kind. Comm. ACM 4 (4), pp. 180–181.

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Notes:: Includes Algol program, maximum accuracy 8D. For certification see same journal 9 (1966), p. 12
External Links:: ISSN 0001-0782, Document
Referenced by:: §19.39(ii).
Encodings:: BibTeX

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E. Hille (1929) Note on some hypergeometric series of higher order. J. London Math. Soc. 4, pp. 50–54.

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External Links:: Document, ZentralBlatt
Referenced by:: §16.9.
Encodings:: BibTeX

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Y. P. Hsu (1993) Development of a Gaussian hypergeometric function code in complex domains. Internat. J. Modern Phys. C 4 (4), pp. 805–840.

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External Links:: ISSN 0129-1831, Document, MathReview (Walter Gautschi), ZentralBlatt
Referenced by:: §15.20(iii).
Encodings:: BibTeX

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17: 4.43 Cubic Equations

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A=\left(-\tfrac{4}{3}p\right)^{1/2},

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B=\left(\tfrac{4}{3}p\right)^{1/2}.

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(a)

$A\sin a$ , $A\sin\left(a+\frac{2}{3}\pi\right)$ , and $A\sin\left(a+\frac{4}{3}\pi\right)$ , with $\sin\left(3a\right)=4q/A^{3}$ , when $4p^{3}+27q^{2}\leq 0$ .

►

(b)

$A\cosh a$ , $A\cosh\left(a+\frac{2}{3}\pi i\right)$ , and $A\cosh\left(a+\frac{4}{3}\pi i\right)$ , with $\cosh\left(3a\right)=-4q/A^{3}$ , when $p<0$ , $q<0$ , and $4p^{3}+27q^{2}>0$ .

►

(c)

$B\sinh a$ , $B\sinh\left(a+\frac{2}{3}\pi i\right)$ , and $B\sinh\left(a+\frac{4}{3}\pi i\right)$ , with $\sinh\left(3a\right)=-4q/B^{3}$ , when $p>0$ .

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18: 24.13 Integrals

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24.13.5

\int_{1/4}^{3/4}B_{n}\left(t\right)\,\mathrm{d}t=\frac{E_{n}}{2^{2n+1}}.

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Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $E_{\NVar{n}}$ : Euler numbers, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer and $t$ : real or complex
Referenced by:: §24.13(i)
Permalink:: http://dlmf.nist.gov/24.13.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §24.13(i), §24.13 and Ch.24

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24.13.8

\int_{0}^{1}E_{n}\left(t\right)\,\mathrm{d}t=-2\frac{E_{n+1}\left(0\right)}{n+% 1}=\frac{4(2^{n+2}-1)}{(n+1)(n+2)}B_{n+2},

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : integer and $t$ : real or complex
Permalink:: http://dlmf.nist.gov/24.13.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §24.13(ii), §24.13 and Ch.24

… ►

24.13.11

\int_{0}^{1}E_{n}\left(t\right)E_{m}\left(t\right)\,\mathrm{d}t=(-1)^{n}4\frac% {(2^{m+n+2}-1)m!n!}{(m+n+2)!}B_{m+n+2}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler 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(ii)
Permalink:: http://dlmf.nist.gov/24.13.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §24.13(ii), §24.13 and Ch.24

… ►For other integrals see Prudnikov et al. (1990, pp. 55–57).

19: Bibliography G

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W. Gautschi (1979a) Algorithm 542: Incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 482–489.

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Notes:: Single-precision Fortran.
External Links:: ISSN 0098-3500, Document, GAMS (module 542; package TOMS), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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W. Gautschi (1979b) A computational procedure for incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 466–481.

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External Links:: ISSN 0098-3500, Document, ZentralBlatt
Referenced by:: §8.25(iv), §8.25(v).
Encodings:: BibTeX

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I. M. Gessel (2003) Applications of the classical umbral calculus. Algebra Universalis 49 (4), pp. 397–434.

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External Links:: ISSN 0002-5240, Document, MathReview (George E. Andrews), ZentralBlatt
Referenced by:: §24.4(viii).
Encodings:: BibTeX

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S. G. Gindikin (1964) Analysis in homogeneous domains. Uspehi Mat. Nauk 19 (4 (118)), pp. 3–92 (Russian).

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Notes:: In Russian. English translation: Russian Math. Surveys, 19(1964), no. 4, pp. 1–89
External Links:: Document, MathReview (A. Korányi), ZentralBlatt
Referenced by:: §35.3(i).
Encodings:: BibTeX

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V. I. Gromak (1976) The solutions of Painlevé’s fifth equation. Differ. Uravn. 12 (4), pp. 740–742 (Russian).

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Notes:: In Russian. English translation: Differential Equations 12(4), pp. 519–521 (1977)
External Links:: MathReview (Z. Nehari), ZentralBlatt
Referenced by:: §32.7(v).
Encodings:: BibTeX

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20: 12.7 Relations to Other Functions

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12.7.8

U\left(-2,z\right)=\frac{z^{5/2}}{4\sqrt{2\pi}}\left(2K_{\frac{1}{4}}\left(% \tfrac{1}{4}z^{2}\right)+3K_{\frac{3}{4}}\left(\tfrac{1}{4}z^{2}\right)-K_{% \frac{5}{4}}\left(\tfrac{1}{4}z^{2}\right)\right),

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $z$ : complex variable
A&S Ref:: 19.15.11 (modification of)
Permalink:: http://dlmf.nist.gov/12.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §12.7(iii), §12.7 and Ch.12

►

12.7.9

U\left(-1,z\right)=\frac{z^{3/2}}{2\sqrt{2\pi}}\left(K_{\frac{1}{4}}\left(% \tfrac{1}{4}z^{2}\right)+K_{\frac{3}{4}}\left(\tfrac{1}{4}z^{2}\right)\right),

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $z$ : complex variable
A&S Ref:: 19.15.10 (modification of)
Permalink:: http://dlmf.nist.gov/12.7.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §12.7(iii), §12.7 and Ch.12

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12.7.11

U\left(1,z\right)=\frac{z^{3/2}}{\sqrt{2\pi}}\left(K_{\frac{3}{4}}\left(\tfrac% {1}{4}z^{2}\right)-K_{\frac{1}{4}}\left(\tfrac{1}{4}z^{2}\right)\right).

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $z$ : complex variable
A&S Ref:: 19.15.3 (modification of)
Permalink:: http://dlmf.nist.gov/12.7.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §12.7(iii), §12.7 and Ch.12

… ►

12.7.12

u_{1}(a,z)=e^{-\tfrac{1}{4}z^{2}}M\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2% },\tfrac{1}{2}z^{2}\right)=e^{\tfrac{1}{4}z^{2}}M\left(-\tfrac{1}{2}a+\tfrac{1% }{4},\tfrac{1}{2},-\tfrac{1}{2}z^{2}\right),

ⓘ

Defines:: $u_{1}(a,z)$ : solution (locally)
Symbols:: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.2.2 (modification of)
Referenced by:: §12.20
Permalink:: http://dlmf.nist.gov/12.7.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §12.7(iv), §12.7 and Ch.12

►

12.7.13

u_{2}(a,z)=ze^{-\tfrac{1}{4}z^{2}}M\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{% 2},\tfrac{1}{2}z^{2}\right)=ze^{\tfrac{1}{4}z^{2}}M\left(-\tfrac{1}{2}a+\tfrac% {3}{4},\tfrac{3}{2},-\tfrac{1}{2}z^{2}\right).

ⓘ

Defines:: $u_{2}(a,z)$ : solution (locally)
Symbols:: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.2.4 (modification of)
Referenced by:: §12.20
Permalink:: http://dlmf.nist.gov/12.7.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §12.7(iv), §12.7 and Ch.12

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.体育推单 摩登4『世界杯佣金分红55%，咨询专员：@ky975』.wua.k2q1w9-eoiyaq2o2