北达科他大学文凭毕业证怎么制作【言正 微aptao168】74b

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1: 5.12 Beta Function

… ►In (5.12.1)–(5.12.4) it is assumed $\mathrm{\Re }a>0$ and $\mathrm{\Re }b>0$. … ►

5.12.1 $$\mathrm{B}(a,b)={\int }_0^1{t}^{a-1}{(1-t)}^{b-1}dt=\frac{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{\mathrm{\Gamma }\left(a+b\right)}.$$

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Defines:

$\mathrm{B}(a,b)$: beta function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $a$: real or complex variable and $b$: real or complex variable

A&S Ref:

6.2.1 and 6.2.2

Referenced by:

§10.22(ii), §10.43(iii), §19.20(i), §19.20(iv), §2.6(iii), §5.12, §5.12, §7.7(ii)

Permalink:

http://dlmf.nist.gov/5.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.12, §5.12 and Ch.5

… ►

5.12.3 $${\int }_0^{\mathrm{\infty }}\frac{t^{a-1}dt}{{(1+t)}^{a+b}}=\mathrm{B}(a,b).$$

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Symbols:

$\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\int$: integral, $a$: real or complex variable and $b$: real or complex variable

A&S Ref:

6.2.1 and 6.2.2

Referenced by:

§5.12

Permalink:

http://dlmf.nist.gov/5.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.12, §5.12 and Ch.5

… ►when $\mathrm{\Re }b>0$, $a$ is not an integer and the contour cuts the real axis between $-1$ and the origin. … ►When $a,b\in ℂ$ …

2: 22.18 Mathematical Applications

… ►with $a\ge b>0$, is parametrized by … ►

$y=b\mathrm{cn}(u,k),$

►where $k=\sqrt{1-(b^2/a^2)}$ is the eccentricity, and $0\le u\le 4K\left(k\right)$. … ►in which $a,b,c,d,e,f$ are real constants, can be achieved in terms of single-valued functions. … ►With the identification $x=\mathrm{sn}(z,k)$, $y=d(\mathrm{sn}(z,k))/dz$, the addition law (22.18.8) is transformed into the addition theorem (22.8.1); see Akhiezer (1990, pp. 42, 45, 73–74) and McKean and Moll (1999, §§2.14, 2.16). …

3: Bibliography V

… ►

Calculation of spheroidal wave functions. J. Acoust. Soc. Amer. 51, pp. 414–416.

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External Links:

Document, ZentralBlatt

Referenced by:

§30.16(ii), §30.18(iii).

Encodings:

BibTeX

… ►

On the method of saddle points. Appl. Sci. Research B. 2, pp. 33–45.

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External Links:

Document, MathReview (E. T. Copson), ZentralBlatt

Referenced by:

§2.4(vi), §7.20(i).

Encodings:

BibTeX

… ►

Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.

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External Links:

Document

Referenced by:

§14.31(i).

Encodings:

BibTeX

… ►

Representation of Lie Groups and Special Functions. Volume 2: Class I Representations, Special Functions, and Integral Transforms. Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht.

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Notes:

Translated from the Russian by V. A. Groza and A. A. Groza

External Links:

ISBN 0-7923-1492-1, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

… ►

Generalized Associated Legendre Functions and their Applications. World Scientific Publishing Co. Inc., Singapore.

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External Links:

ISBN 981-02-4353-7, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§14.29.

Encodings:

BibTeX

…

4: Bibliography O

… ►

Complete elliptic integrals resulting from infinite integrals of Bessel functions. J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.

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External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

Encodings:

BibTeX

►

Complete elliptic integrals resulting from infinite integrals of Bessel functions. II. J. Res. Nat. Bur. Standards Sect. B 79B (3-4), pp. 137–170.

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External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

Encodings:

BibTeX

… ►

Hyperterminants. I. J. Comput. Appl. Math. 76 (1-2), pp. 255–264.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

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

Encodings:

BibTeX

… ►

Hyperterminants. II. J. Comput. Appl. Math. 89 (1), pp. 87–95.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

… ►

Error analysis of Miller’s recurrence algorithm. Math. Comp. 18 (85), pp. 65–74.

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External Links:

FullText, MathReview (J.C.P. Miller), ZentralBlatt

Referenced by:

§3.6(iii).

Encodings:

BibTeX

…

5: 10.6 Recurrence Relations and Derivatives

… ►

$p_{\nu }=J_{\nu }\left(a\right)Y_{\nu }\left(b\right)-J_{\nu }\left(b\right)Y_{\nu }\left(a\right),$

►

$q_{\nu }=J_{\nu }\left(a\right)Y_{\nu }^{\prime }\left(b\right)-J_{\nu }^{\prime }\left(b\right)Y_{\nu }\left(a\right),$

►

$r_{\nu }=J_{\nu }^{\prime }\left(a\right)Y_{\nu }\left(b\right)-J_{\nu }\left(b\right)Y_{\nu }^{\prime }\left(a\right),$

►

$s_{\nu }=J_{\nu }^{\prime }\left(a\right)Y_{\nu }^{\prime }\left(b\right)-J_{\nu }^{\prime }\left(b\right)Y_{\nu }^{\prime }\left(a\right),$

►where $a$ and $b$ are independent of $\nu$. …

6: Bibliography

… ►

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

… ►

On the intensity of light in the neighbourhood of a caustic. Trans. Camb. Phil. Soc. 6, pp. 379–402.

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Referenced by:

§9.16, §9.16.

Encodings:

BibTeX

… ►

Sharp bounds for the Bernoulli numbers. Arch. Math. (Basel) 74 (3), pp. 207–211.

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External Links:

ISSN 0003-889X, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.9.

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for exponential integrals $E_n(x)$ and Bickley functions ${\mathrm{Ki}}_n(x)$ . ACM Trans. Math. Software 9 (4), pp. 467–479.

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External Links:

ISSN 0098-3500, Document, MathReview (Marietta J. Tretter), ZentralBlatt

Referenced by:

§10.43(iii).

Encodings:

BibTeX

… ►

Mathematical Methods for Physicists. 6th edition, Elsevier, Oxford.

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External Links:

ISBN 0-12-059876-0;0-12-088584-0, MathReview, ZentralBlatt

Referenced by:

§1.17(ii), §1.17(iii).

Encodings:

BibTeX

…

7: Bibliography B

Bibliography B

… ►

A note on integrals involving parabolic cylinder functions. SIAM J. Appl. Math. 16 (1), pp. 71–74.

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External Links:

Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§12.12.

Encodings:

BibTeX

… ►

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.

Encodings:

BibTeX

… ►

The asymptotic analysis of canonical problems in high-frequency scattering theory. II. The circular and parabolic cylinders. Proc. Cambridge Philos. Soc. 74, pp. 313–332.

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External Links:

Document, MathReview (Y. M. Chen)

Referenced by:

§12.17.

Encodings:

BibTeX

… ►

The hypergeometric identities of Cayley, Orr, and Bailey. Proc. London Math. Soc. (2) 50, pp. 56–74.

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External Links:

Document, MathReview (N. A. Hall), ZentralBlatt

Referenced by:

§15.16.

Encodings:

BibTeX

…

8: 19.29 Reduction of General Elliptic Integrals

… ►and assume that the line segment with endpoints $a_{\alpha }+b_{\alpha }x$ and $a_{\alpha }+b_{\alpha }y$ lies in $ℂ\setminus (-\mathrm{\infty },0)$ for $1\le \alpha \le 4$. … ►where the arguments of the $R_D$ function are, in order, $(a-b)(u-c)$, $(b-c)(a-u)$, $(a-b)(b-c)$. … ►It can be expressed in terms of symmetric integrals by setting $a_5=1$ and $b_5=0$ in (19.29.8). … ►If both square roots in (19.29.22) are 0, then the indeterminacy in the two preceding equations can be removed by using (19.27.8) to evaluate the integral as $R_G(a_1{b}_2,a_2{b}_1,0)$ multiplied either by $-2/(b_1{b}_2)$ or by $-2/(a_1{a}_2)$ in the cases of (19.29.20) or (19.29.21), respectively. … ►In the cubic case, in which $a_2=1$, $b_2=0$, (19.29.26) reduces further to …

9: Bibliography G

… ►

On the computation of generalized Fermi-Dirac and Bose-Einstein integrals. Comput. Phys. Comm. 74 (2), pp. 233–238.

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External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§25.12(iii), §25.18(i), 5th item.

Encodings:

BibTeX

… ►

A table of integrals of the error function. II. Additions and corrections. J. Res. Nat. Bur. Standards Sect. B 75B, pp. 149–163.

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Notes:

See for the first part same journal, Sect. B 73, 1-20 (1969)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii).

Encodings:

BibTeX

… ►

Recurrence relations for cross-products of Bessel functions. Quart. J. Mech. Appl. Math. 2 (1), pp. 72–74.

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External Links:

ISSN 0033-5614, Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§10.6(iii).

Encodings:

BibTeX

… ►

Constructing wavefunctions for nonlocal potentials. J. Chem. Phys. 52, pp. 6211–6217.

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External Links:

Document, MathReview (D. B. Fairlie)

Referenced by:

(9.12.22), (9.12.23), §9.12(vii), §9.17(iii).

Encodings:

BibTeX

… ►

Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.

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External Links:

FullText, MathReview (B. M. Stewart), ZentralBlatt

Referenced by:

§24.15(iii), §24.6.

Encodings:

BibTeX

…

10: 12.7 Relations to Other Functions

… ►

12.7.12 $$u_1(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}M(\frac{1}{2}a+\frac{1}{4},\frac{1}{2},\frac{1}{2}{z}^2)={\mathrm{e}}^{\frac{1}{4}{z}^2}M(-\frac{1}{2}a+\frac{1}{4},\frac{1}{2},-\frac{1}{2}{z}^2),$$

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Defines:

$u_1(a,z)$: solution (locally)

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ 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 $$u_2(a,z)=z{\mathrm{e}}^{-\frac{1}{4}{z}^2}M(\frac{1}{2}a+\frac{3}{4},\frac{3}{2},\frac{1}{2}{z}^2)=z{\mathrm{e}}^{\frac{1}{4}{z}^2}M(-\frac{1}{2}a+\frac{3}{4},\frac{3}{2},-\frac{1}{2}{z}^2).$$

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Defines:

$u_2(a,z)$: solution (locally)

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ 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

… ►

12.7.14 $$U(a,z)=2^{-\frac{1}{4}-\frac{1}{2}a}{\mathrm{e}}^{-\frac{1}{4}{z}^2}U(\frac{1}{2}a+\frac{1}{4},\frac{1}{2},\frac{1}{2}{z}^2)=2^{-\frac{3}{4}-\frac{1}{2}a}z{\mathrm{e}}^{-\frac{1}{4}{z}^2}U(\frac{1}{2}a+\frac{3}{4},\frac{3}{2},\frac{1}{2}{z}^2)=2^{-\frac{1}{2}a}{z}^{-\frac{1}{2}}{W}_{-\frac{1}{2}a,\pm \frac{1}{4}}\left(\frac{1}{2}{z}^2\right).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.12.2 (modification of)

Referenced by:

§12.12, §12.20, §12.7(iv), §12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.7.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iv), §12.7 and Ch.12

…