愛知東邦大学学士成绩单【购证 微kaa77788】54Z

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11: Bibliography P

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Complex zeros of the modified Bessel function $K_n(Z)$ . Math. Comp. 26 (120), pp. 949–953.

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External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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The exact distribution of the Wald statistic. Econometrica 54 (4), pp. 881–895.

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External Links:

ISSN 0012-9682, FullText, MathReview (Sadanori Konishi), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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12: 9.4 Maclaurin Series

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9.4.1 $$\mathrm{Ai}\left(z\right)=\mathrm{Ai}\left(0\right)\left(1+\frac{1}{3!}{z}^3+\frac{1\cdot 4}{6!}{z}^6+\frac{1\cdot 4\cdot 7}{9!}{z}^9+\mathrm{\cdots }\right)+{\mathrm{Ai}}^{\prime }\left(0\right)\left(z+\frac{2}{4!}{z}^4+\frac{2\cdot 5}{7!}{z}^7+\frac{2\cdot 5\cdot 8}{10!}{z}^{10}+\mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: 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 $${\mathrm{Ai}}^{\prime }\left(z\right)={\mathrm{Ai}}^{\prime }\left(0\right)\left(1+\frac{2}{3!}{z}^3+\frac{2\cdot 5}{6!}{z}^6+\frac{2\cdot 5\cdot 8}{9!}{z}^9+\mathrm{\cdots }\right)+\mathrm{Ai}\left(0\right)\left(\frac{1}{2!}{z}^2+\frac{1\cdot 4}{5!}{z}^5+\frac{1\cdot 4\cdot 7}{8!}{z}^8+\mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: 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 $$\mathrm{Bi}\left(z\right)=\mathrm{Bi}\left(0\right)\left(1+\frac{1}{3!}{z}^3+\frac{1\cdot 4}{6!}{z}^6+\frac{1\cdot 4\cdot 7}{9!}{z}^9+\mathrm{\cdots }\right)+{\mathrm{Bi}}^{\prime }\left(0\right)\left(z+\frac{2}{4!}{z}^4+\frac{2\cdot 5}{7!}{z}^7+\frac{2\cdot 5\cdot 8}{10!}{z}^{10}+\mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: 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 $${\mathrm{Bi}}^{\prime }\left(z\right)={\mathrm{Bi}}^{\prime }\left(0\right)\left(1+\frac{2}{3!}{z}^3+\frac{2\cdot 5}{6!}{z}^6+\frac{2\cdot 5\cdot 8}{9!}{z}^9+\mathrm{\cdots }\right)+\mathrm{Bi}\left(0\right)\left(\frac{1}{2!}{z}^2+\frac{1\cdot 4}{5!}{z}^5+\frac{1\cdot 4\cdot 7}{8!}{z}^8+\mathrm{\cdots }\right).$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: 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

13: 22.16 Related Functions

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22.16.32 $$\mathrm{Z}\left(x|k\right)=ℰ(x,k)-(E\left(k\right)/K\left(k\right))x.$$

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Defines:

$\mathrm{Z}\left(x|k\right)$: Jacobi’s zeta function

Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $x$: real and $k$: modulus

Referenced by:

§22.20(vi)

Permalink:

http://dlmf.nist.gov/22.16.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(iii), §22.16(iii), §22.16 and Ch.22

… ►(Sometimes in the literature $\mathrm{Z}\left(x|k\right)$ is denoted by $\mathrm{Z}(\mathrm{am}(x,k),k^2)$.) … ► $\mathrm{Z}\left(x|k\right)$ satisfies the same quasi-addition formula as the function $ℰ(x,k)$, given by (22.16.27). … ►

22.16.34 $$\mathrm{Z}\left(x+2K|k\right)=\mathrm{Z}\left(x|k\right).$$

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Symbols:

$\mathrm{Z}\left(x|k\right)$: Jacobi’s zeta function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $x$: real and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.16.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(iii), §22.16(iii), §22.16 and Ch.22

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Figure 22.16.3: Jacobi’s zeta function $\mathrm{Z}\left(x|k\right)$ for $0\le x\le 10\pi$ and $k=0.4,0.7,0.99,0.999999$. Magnify

14: 10.43 Integrals

… ►Let ${𝒵}_{\nu }\left(z\right)$ be defined as in §10.25(ii). … ►

$$\int z^{\nu +1}{𝒵}_{\nu }\left(z\right)dz=z^{\nu +1}{𝒵}_{\nu +1}\left(z\right),$$

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$$\int z^{-\nu +1}{𝒵}_{\nu }\left(z\right)dz=z^{-\nu +1}{𝒵}_{\nu -1}\left(z\right).$$

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$$\int {\mathrm{e}}^{\pm z}{z}^{\nu }{𝒵}_{\nu }\left(z\right)dz=\frac{{\mathrm{e}}^{\pm z}{z}^{\nu +1}}{2\nu +1}\left({𝒵}_{\nu }\left(z\right)\mp {𝒵}_{\nu +1}\left(z\right)\right),$$ $\nu \ne -\frac{1}{2}$,

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$$\int {\mathrm{e}}^{\pm z}{z}^{-\nu }{𝒵}_{\nu }\left(z\right)dz=\frac{{\mathrm{e}}^{\pm z}{z}^{-\nu +1}}{1-2\nu }\left({𝒵}_{\nu }\left(z\right)\mp {𝒵}_{\nu -1}\left(z\right)\right),$$ $\nu \ne \frac{1}{2}$.

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15: 10.25 Definitions

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Symbol ${𝒵}_{\nu }\left(z\right)$

►Corresponding to the symbol ${𝒞}_{\nu }$ introduced in §10.2(ii), we sometimes use ${𝒵}_{\nu }\left(z\right)$ to denote $I_{\nu }\left(z\right)$, ${\mathrm{e}}^{\nu \pi \mathrm{i}}{K}_{\nu }\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$. …

16: 22.21 Tables

… ►Lawden (1989, pp. 280–284 and 293–297) tabulates $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, $\mathrm{dn}(x,k)$, $ℰ(x,k)$, $\mathrm{Z}\left(x|k\right)$ to 5D for $k=0.1(.1)0.9$, $x=0(.1)X$, where $X$ ranges from 1. …

17: 27.16 Cryptography

… ►To code a message by this method, we replace each letter by two digits, say $A=01$, $B=02$, $\mathrm{\dots }$, $Z=26$, and divide the message into pieces of convenient length smaller than the public value $n=pq$. …

18: 35.1 Special Notation

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$a,b$	complex variables.
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$𝐙$	complex symmetric matrix.
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$Z_{\kappa }\left(𝐓\right)$	zonal polynomials.

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19: 10.66 Expansions in Series of Bessel Functions

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20: 22.1 Special Notation

… ►The functions treated in this chapter are the three principal Jacobian elliptic functions $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$; the nine subsidiary Jacobian elliptic functions $\mathrm{cd}(z,k)$, $\mathrm{sd}(z,k)$, $\mathrm{nd}(z,k)$, $\mathrm{dc}(z,k)$, $\mathrm{nc}(z,k)$, $\mathrm{sc}(z,k)$, $\mathrm{ns}(z,k)$, $\mathrm{ds}(z,k)$, $\mathrm{cs}(z,k)$; the amplitude function $\mathrm{am}(x,k)$; Jacobi’s epsilon and zeta functions $ℰ(x,k)$ and $\mathrm{Z}\left(x|k\right)$. …