西武芸術科学大学商科文凭证书【仿证 微kaa77788】】uppere

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11: Mathematical Introduction

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$ℂ$	complex plane (excluding infinity).
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$sup$	least upper bound (supremum).
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12: 30.8 Expansions in Series of Ferrers Functions

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30.8.9 $${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)=\sum _{k=-\mathrm{\infty }}^{-N-1}{(-1)}^{k}a_{}^{\prime }{}_{n,k}{}^m({\gamma }^2){𝖯}_{n+2k}^m\left(x\right)+\sum _{k=-N}^{\mathrm{\infty }}{(-1)}^k{a}_{n,k}^m({\gamma }^2){𝖰}_{n+2k}^m\left(x\right),$$

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Symbols:

: Ferrers function of the first kind, : Ferrers function of the second kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $a_{n,k}^m({\gamma }^2)$: coefficients and $N$: upper limit

A&S Ref:

21.7.21 (in different form)

Permalink:

http://dlmf.nist.gov/30.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §30.8(ii), §30.8 and Ch.30

►where ${𝖯}_n^m$ and ${𝖰}_n^m$ are again the Ferrers functions and $N=\lfloor \frac{1}{2}(n+m)\rfloor$. … ►

30.8.10 $$A_{-N-1}a_{}^{\prime }{}_{n,-N-2}{}^m({\gamma }^2)+\left(B_{-N-1}-{\lambda }_n^m\left({\gamma }^2\right)\right)a_{}^{\prime }{}_{n,-N-1}{}^m({\gamma }^2)+C^{\prime }{a}_{n,-N}^m({\gamma }^2)=0,$$

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Symbols:

${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $a_{n,k}^m({\gamma }^2)$: coefficients, $A_k$ , $B_k$ , $N$: upper limit and $C^{\prime }$

Permalink:

http://dlmf.nist.gov/30.8.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §30.8(ii), §30.8 and Ch.30

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13: Guide to Searching the DLMF

… ►DLMF search is generally case-insensitive except when it is important to be case-sensitive, as when two different special functions have the same standard names but one name has a lower-case initial and the other is has an upper-case initial, such as si and Si, gamma and Gamma. …

14: 10.44 Sums

… ►If $𝒵=I$ and the upper signs are taken, then the restriction on $\lambda$ is unnecessary. …

15: 14.24 Analytic Continuation

… ►The behavior of $P_{\nu }^{-\mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ as $z\to -1$ from the left on the upper or lower side of the cut from $-\mathrm{\infty }$ to $1$ can be deduced from (14.8.7)–(14.8.11), combined with (14.24.1) and (14.24.2) with $s=\pm 1$.

16: 19.34 Mutual Inductance of Coaxial Circles

… ►is the square of the maximum (upper signs) or minimum (lower signs) distance between the circles. …

17: 8.17 Incomplete Beta Functions

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8.17.24 $$I_x(m,n)={(1-x)}^n\sum _{j=m}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{n+j-1}{j}\right)x^j,$$ $m,n$ positive integers; .

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Symbols:

$I_x(a,b)$: incomplete beta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer and $x$: variable

A&S Ref:

26.5.26 (The upper limit of summation has been corrected.)

Referenced by:

§8.17(i), Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/8.17.E24

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation is the same as Equation (26.5.26) of Abramowitz and Stegun (1964), except that the upper limit of the summation has been corrected to $\mathrm{\infty }$.

Suggested 2011-03-23 by Stephen Bourn

See also:

Annotations for §8.17(vii), §8.17 and Ch.8

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18: Bibliography E

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An upper bound for the zeros of the derivative of Bessel functions. Rend. Circ. Mat. Palermo (2) 46 (1), pp. 123–130.

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External Links:

ISSN 0009-725X, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(vii).

Encodings:

BibTeX

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19: 4.23 Inverse Trigonometric Functions

… ►upper signs being taken on upper sides, and lower signs on lower sides. … ►the upper/lower signs corresponding to the upper/lower sides. … ►the upper/lower sign corresponding to the right/left side. …

20: 7.19 Voigt Functions

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