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11: 19.20 Special Cases

… ►The general lemniscatic case is … ►The general lemniscatic case is …

12: 4.2 Definitions

… ►where $\mathrm{ph}z\in [-\pi ,\pi ]$ for the principal value of $z^a$, and is unrestricted in the general case. …

13: 15.12 Asymptotic Approximations

… ►For the more general case in which $a^2=o\left(c\right)$ and $b^2=o\left(c\right)$ see Wagner (1990). …

14: 32.11 Asymptotic Approximations for Real Variables

… ►In the generic case …

15: 20.7 Identities

… ►These are specific examples of modular transformations as discussed in §23.15; the corresponding results for the general case are given by Rademacher (1973, pp. 181–183). …

16: Bibliography S

… ►

Relations among the fundamental solutions of the generalized hypergeometric equation when $p=q+1$. II. Logarithmic cases. Bull. Amer. Math. Soc. 45 (12), pp. 927–935.

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External Links:

Document, MathReview (C. S. Meijer), ZentralBlatt

Referenced by:

§16.8(ii).

Encodings:

BibTeX

… ►

Error bounds for asymptotic solutions of differential equations. II. The general case. J. Res. Nat. Bur. Standards Sect. B 70B, pp. 187–210.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.7(ii).

Encodings:

BibTeX

…

17: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►

§1.18(vii) Continuous Spectra: More General Cases

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18: 8.21 Generalized Sine and Cosine Integrals

… ►When $\mathrm{ph}z=0$ (and when $a\ne -1,-3,-5,\mathrm{\dots }$, in the case of $\mathrm{Si}(a,z)$, or $a\ne 0,-2,-4,\mathrm{\dots }$, in the case of $\mathrm{Ci}(a,z)$) the principal values of $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). …

19: 25.11 Hurwitz Zeta Function

… ►For the more general case ${\zeta }^{\prime }(-m,a)$, $m=1,2,\mathrm{\dots }$, see Elizalde (1986). …

20: 14.29 Generalizations

§14.29 Generalizations

… ►

14.29.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }_1^2}{2(1-z)}-\frac{{\mu }_2^2}{2(1+z)}\right)w=0$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.29

Permalink:

http://dlmf.nist.gov/14.29.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.29 and Ch.14

►are called Generalized Associated Legendre Functions. As in the case of (14.21.1), the solutions are hypergeometric functions, and (14.29.1) reduces to (14.21.1) when ${\mu }_1={\mu }_2=\mu$. … …