.%E5%8E%86%E5%B1%8A%E4%B8%96%E7%95%8C%E6%9D%AF%E9%87%91%E7%90%83%E5%A5%96%E6%95%B0%E9%87%8F%E3%80%8E%E7%BD%91%E5%9D%80%3Amxsty.cc%E3%80%8F.2017%E8%B6%B3%E7%90%83%E4%B8%96%E7%95%8C%E6%9D%AF%E6%AF%94%E8%B5%9B-m6q3s2-2022%E5%B9%B411%E6%9C%8829%E6%97%A56%E6%97%B617%E5%88%8650%E7%A7%92.nzpvd1p5z

SearchAdvancedHelp

(0.017 seconds)

21—30 of 743 matching pages

21: 12.3 Graphics

… ►

►►

Figure 12.3.1: $U(a,x)$, $a$ = 0. 5, 2, 3. 5, 5, 8. Magnify

►

►►

Figure 12.3.2: $V(a,x)$, $a$ = 0. …5, 5, 8. Magnify

… ►

►►

Figure 12.3.5: $U(-8,x)$, $\overline{U}(-8,x)$, $F(-8,x)$, $-4\sqrt{2}\le x\le 4\sqrt{2}$. Magnify

►

►►

Figure 12.3.6: $U^{\prime }(-8,x)$, ${\overline{U}}^{\prime }(-8,x)$, $G(-8,x)$, $-4\sqrt{2}\le x\le 4\sqrt{2}$. Magnify

…

22: 17.16 Mathematical Applications

… ►More recent applications are given in Gasper and Rahman (2004, Chapter 8) and Fine (1988, Chapters 1 and 2).

23: Bibliography

… ►

Tables of $\ln \mathrm{\Gamma }(z)$ for Complex Argument. Pergamon Press, New York.

ⓘ

External Links:

ZentralBlatt

Referenced by:

§5.22(iii).

Encodings:

BibTeX

… ►

Supplement to a paper “On the intensity of light in the neighbourhood of a caustic”. Trans. Camb. Phil. Soc. 8, pp. 595–599.

ⓘ

Referenced by:

§9.16.

Encodings:

BibTeX

… ►

Frequency response characteristics of the multiport planar elliptic patch. IEEE Trans. Microwave Theory Tech. 40 (8), pp. 1726–1730.

ⓘ

External Links:

Document

Referenced by:

6th item.

Encodings:

BibTeX

… ►

Algorithm 588. Fast Hankel transforms using related and lagged convolutions. ACM Trans. Math. Software 8 (4), pp. 369–370.

ⓘ

Notes:

Single-precision Fortran, maximum accuracy 8S.

External Links:

ISSN 0098-3500, Document, GAMS (module 588; package TOMS)

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

Chiral-symmetry breaking in QCD. I. The infrared domain. Phys. Rev. D (3) 37 (8), pp. 2290–2295.

ⓘ

External Links:

ISSN 0556-2821, Document, MathReview

Referenced by:

§15.18.

Encodings:

BibTeX

…

24: 24.21 Software

… ►

§24.21(ii) $B_n$, $B_n\left(x\right)$, $E_n$, and $E_n\left(x\right)$

…

25: 24.8 Series Expansions

… ►If $n=1$ with , or $n=2,3,\mathrm{\dots }$ with $0\le x\le 1$, then … ►

24.8.4 $$E_{2n}\left(x\right)={(-1)}^n\frac{4(2n)!}{{\pi }^{2n+1}}\sum _{k=0}^{\mathrm{\infty }}\frac{\sin \left((2k+1)\pi x\right)}{{(2k+1)}^{2n+1}},$$

ⓘ

Symbols:

$E_n\left(x\right)$: Euler polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\sin z$: sine function, $k$: integer, $n$: integer and $x$: real or complex

A&S Ref:

23.1.18

Referenced by:

§24.8(i), §24.9

Permalink:

http://dlmf.nist.gov/24.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.8(i), §24.8 and Ch.24

►

24.8.5 $$E_{2n-1}\left(x\right)={(-1)}^n\frac{4(2n-1)!}{{\pi }^{2n}}\sum _{k=0}^{\mathrm{\infty }}\frac{\cos \left((2k+1)\pi x\right)}{{(2k+1)}^{2n}}.$$

ⓘ

Symbols:

$E_n\left(x\right)$: Euler polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $k$: integer, $n$: integer and $x$: real or complex

A&S Ref:

23.1.17

Referenced by:

§24.8(i)

Permalink:

http://dlmf.nist.gov/24.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.8(i), §24.8 and Ch.24

… ►

24.8.6 $$B_{4n+2}=(8n+4)\sum _{k=1}^{\mathrm{\infty }}\frac{k^{4n+1}}{{\mathrm{e}}^{2\pi k}-1},$$ $n=1,2,\mathrm{\dots }$,

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $k$: integer and $n$: integer

Referenced by:

§24.8(ii)

Permalink:

http://dlmf.nist.gov/24.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.8(ii), §24.8 and Ch.24

… ►

24.8.9 $$E_{2n}={(-1)}^n\sum _{k=1}^{\mathrm{\infty }}\frac{k^{2n}}{\cosh \left(\frac{1}{2}\pi k\right)}-4\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{(2k+1)}^{2n}}{{\mathrm{e}}^{2\pi (2k+1)}-1},$$ $n=1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $k$: integer and $n$: integer

Keywords:

Euler polynomials, infinite series expansions, other

Permalink:

http://dlmf.nist.gov/24.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.8(ii), §24.8 and Ch.24

26: 9.4 Maclaurin Series

… ►

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

ⓘ

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

►

9.4.2 $${\mathrm{Ai}}^{\prime }\left(z\right)={\mathrm{Ai}}^{\prime }\left(0\right)\left(1+\frac{2}{3!}{z}^3+\frac{2\cdot 5}{6!}{z}^6+\frac{2\cdot 5\cdot 8}{9!}{z}^9+\mathrm{\cdots }\right)+\mathrm{Ai}\left(0\right)\left(\frac{1}{2!}{z}^2+\frac{1\cdot 4}{5!}{z}^5+\frac{1\cdot 4\cdot 7}{8!}{z}^8+\mathrm{\cdots }\right),$$

ⓘ

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

►

9.4.3 $$\mathrm{Bi}\left(z\right)=\mathrm{Bi}\left(0\right)\left(1+\frac{1}{3!}{z}^3+\frac{1\cdot 4}{6!}{z}^6+\frac{1\cdot 4\cdot 7}{9!}{z}^9+\mathrm{\cdots }\right)+{\mathrm{Bi}}^{\prime }\left(0\right)\left(z+\frac{2}{4!}{z}^4+\frac{2\cdot 5}{7!}{z}^7+\frac{2\cdot 5\cdot 8}{10!}{z}^{10}+\mathrm{\cdots }\right),$$

ⓘ

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

►

9.4.4 $${\mathrm{Bi}}^{\prime }\left(z\right)={\mathrm{Bi}}^{\prime }\left(0\right)\left(1+\frac{2}{3!}{z}^3+\frac{2\cdot 5}{6!}{z}^6+\frac{2\cdot 5\cdot 8}{9!}{z}^9+\mathrm{\cdots }\right)+\mathrm{Bi}\left(0\right)\left(\frac{1}{2!}{z}^2+\frac{1\cdot 4}{5!}{z}^5+\frac{1\cdot 4\cdot 7}{8!}{z}^8+\mathrm{\cdots }\right).$$

ⓘ

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

27: 10.62 Graphs

… ►

►►

Figure 10.62.1: $\mathrm{ber}x$, $\mathrm{bei}x$, ${\mathrm{ber}}^{\prime }x$, ${\mathrm{bei}}^{\prime }x$, $0\le x\le 8$. Magnify

►

►►

Figure 10.62.2: $\ker x$, $\mathrm{kei}x$, ${\ker }^{\prime }x$, ${\mathrm{kei}}^{\prime }x$, $0\le x\le 8$. Magnify

►

►►

Figure 10.62.3: ${\mathrm{e}}^{-x/\sqrt{2}}\mathrm{ber}x$, ${\mathrm{e}}^{-x/\sqrt{2}}\mathrm{bei}x$, ${\mathrm{e}}^{-x/\sqrt{2}}M\left(x\right)$, $0\le x\le 8$. Magnify

►

►►

Figure 10.62.4: ${\mathrm{e}}^{x/\sqrt{2}}\ker x$, ${\mathrm{e}}^{x/\sqrt{2}}\mathrm{kei}x$, ${\mathrm{e}}^{x/\sqrt{2}}N\left(x\right)$, $0\le x\le 8$. Magnify

28: Software Index

… ► ►►►►►►►

	Open Source													With Book						Commercial
…
8 Incomplete Gamma and Related Functions
…
8.28(vi) $E_p\left(x\right)$, $x\in ℝ$, $p\in \mathbb{Z}$	✓		✓		✓				✓		✓	✓	✓	✓	✓	✓			✓	✓	✓	✓	✓
8.28(vii) $E_p\left(z\right)$, $z,p\in ℂ$	✓											✓	✓								✓	✓	✓
…
11.16(v) ${𝐉}_{\nu }\left(z\right)$, ${𝐄}_{\nu }\left(z\right)$, ${𝐀}_{\nu }\left(z\right)$												a	✓				✓				✓	✓	a
…
24.21(ii) $B_n$, $B_n\left(x\right)$, $E_n$, $E_n\left(x\right)$	✓	✓						✓		✓		a	✓	✓			✓		✓		✓	✓	✓		Derive, MuPAD
…

…

29: 19.2 Definitions

… ►The integral for $E(\varphi ,k)$ is well defined if $k^2={\sin }^2\varphi =1$, and the Cauchy principal value (§1.4(v)) of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ is taken if $1-{\alpha }^2{\sin }^2\varphi$ vanishes at an interior point of the integration path. … ►The principal branch of $K\left(k\right)$ and $E\left(k\right)$ is $|\mathrm{ph}\left(1-k^2\right)|\le \pi$, that is, the branch-cuts are $(-\mathrm{\infty },-1]\cup [1,+\mathrm{\infty })$. The principal values of $K\left(k\right)$ and $E\left(k\right)$ are even functions. … ►

§19.2(iv) A Related Function: $R_C(x,y)$

… ►For the special cases of $R_C(x,x)$ and $R_C(0,y)$ see (19.6.15). …

30: 6.8 Inequalities

… ►

6.8.1

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

5.1.20

Referenced by:

§6.8

Permalink:

http://dlmf.nist.gov/6.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.8 and Ch.6

►

6.8.2

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $x$: real variable

Referenced by:

§6.8

Permalink:

http://dlmf.nist.gov/6.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.8 and Ch.6

►

6.8.3

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $x$: real variable

Referenced by:

§6.8

Permalink:

http://dlmf.nist.gov/6.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.8 and Ch.6