Buy quesn - www.barnett - Price hydrologe slatere 200mg. For lucas, artists. Get online - barnett

SearchAdvancedHelp

(0.003 seconds)

11—20 of 68 matching pages

11: 17 q-Hypergeometric and Related Functions

…

12: 1.11 Zeros of Polynomials

… ►Add $-\frac{1}{4}a$ to the roots of $g(w)=0$ to get those of $f(z)=0$. …

13: 15 Hypergeometric Function

…

14: 16 Generalized Hypergeometric Functions & Meijer G-Function

…

15: 10.73 Physical Applications

… ►See Jackson (1999, Chapter 9, §9.6), Jones (1986, Chapters 7, 8), and Lord Rayleigh (1945, Vol. I, Chapter IX, §§200–211, 218, 219, 221a; Vol. II, Chapter XIII, §272a; Chapter XV, §302; Chapter XVIII; Chapter XIX, §350; Chapter XX, §357; Chapter XXI, §369). …See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). …

16: Errata

… ►

In the paragraph immediately below (25.10.4), it was originally stated that “more than one-third of all zeros in the critical strip lie on the critical line.” which referred to Levinson (1974). This sentence has been updated with “one-third” being replaced with “41%” now referring to Bui et al. (2011) (suggested by Gergő Nemes on 2021-08-23).

…

17: Bibliography T

… ►

Numerical evaluation of exponential integral: Theis well function approximation. Journal of Hydrology 205 (1-2), pp. 38–51.

ⓘ

External Links:

Document

Referenced by:

§6.18(i).

Encodings:

BibTeX

…

18: 19.25 Relations to Other Functions

… ►

19.25.35 $$z+2\omega =\pm R_F(\mathrm{\wp }\left(z\right)-e_1,\mathrm{\wp }\left(z\right)-e_2,\mathrm{\wp }\left(z\right)-e_3),$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots and $z$: complex number

Proof sketch:

Use (23.6.36) with $z=\mathrm{\wp }\left(\omega \right)$ as prescribed in the text that follows (23.6.36), substitute $u=t+\mathrm{\wp }\left(\omega \right)$ and compare with (19.16.1). The undetermined $\pm$ is a consequence of the multivaluedness of the square-roots in (19.16.4).

Referenced by:

(19.25.38), (19.25.40), §19.25(v), §19.25(vi), §19.25(vi), §20.9(i), Erratum (V1.1.0) for Subsection 19.25(vi)

Permalink:

http://dlmf.nist.gov/19.25.E35

Encodings:

pMML, png, TeX

Correction (effective with 1.1.0):

This equation has been corrected, namely the left-hand side which was originally $z$, has been replaced by $z+2\omega$ and the right-hand side has been multiplied by $\pm 1$.

See also:

Annotations for §19.25(vi), §19.25 and Ch.19

… ►

19.25.36 $$\mathrm{\wp }\left(z\right)-e_j\in ℂ\setminus (-\mathrm{\infty },0],$$ $j=1,2,3.$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $ℂ$: complex plane, $\in$: element of, $(a,b]$: half-closed interval, $\setminus$: set subtraction, $j$: $(=1,2,3)$ and $z$: complex number

Referenced by:

(19.25.39)

Permalink:

http://dlmf.nist.gov/19.25.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(vi), §19.25 and Ch.19

… ►

19.25.37 $$\zeta \left(z+2\omega \right)+(z+2\omega )\mathrm{\wp }\left(z\right)=\pm 2R_G(\mathrm{\wp }\left(z\right)-e_1,\mathrm{\wp }\left(z\right)-e_2,\mathrm{\wp }\left(z\right)-e_3),$$

ⓘ

Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function and $z$: complex number

Referenced by:

§19.25(vi), Erratum (V1.0.18) for Equation (19.25.37), Erratum (V1.1.0) for Subsection 19.25(vi)

Permalink:

http://dlmf.nist.gov/19.25.E37

Encodings:

pMML, png, TeX

Correction (effective with 1.1.0):

This equation has been corrected, namely the left-hand side which was originally $\zeta \left(z\right)+z\mathrm{\wp }\left(z\right)$, has been replaced by $\zeta \left(z+2\omega \right)+(z+2\omega )\mathrm{\wp }\left(z\right)$, and the right-hand side has been multiplied by $\pm 1$.

Clarification (effective with 1.0.18):

The Weierstrass zeta function in this equation incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the functions appeared correct.

See also:

Annotations for §19.25(vi), §19.25 and Ch.19

… ►

19.25.40 $$z+2\omega =\pm \sigma \left(z\right)R_F({\sigma }_1^2(z),{\sigma }_2^2(z),{\sigma }_3^2(z)),$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $z$: complex number and ${\sigma }_j(z)$: function

Proof sketch:

Combine Erdélyi et al. (1953b, §§13.12(22), 13.13(22)), (19.25.35) and (19.16.1).

Referenced by:

§19.25(vi), §19.25(vi), Erratum (V1.1.0) for Subsection 19.25(vi)

Permalink:

http://dlmf.nist.gov/19.25.E40

Encodings:

pMML, png, TeX

Correction (effective with 1.1.0):

This equation has been corrected, namely the left-hand side which was originally $z$, has been replaced by $z+2\omega$ and the right-hand side has been multiplied by $\pm 1$.

See also:

Annotations for §19.25(vi), §19.25 and Ch.19

…

19: 8.24 Physical Applications

… ►The function $\gamma (a,x)$ appears in: discussions of power-law relaxation times in complex physical systems (Sornette (1998)); logarithmic oscillations in relaxation times for proteins (Metzler et al. (1999)); Gaussian orbitals and exponential (Slater) orbitals in quantum chemistry (Shavitt (1963), Shavitt and Karplus (1965)); population biology and ecological systems (Camacho et al. (2002)). …

20: 13.30 Tables

… ►

Slater (1960) tabulates $M(a,b,x)$ for $a=-1(.1)1$, $b=0.1(.1)1$, and $x=0.1(.1)10$, 7–9S; $M(a,b,1)$ for $a=-11(.2)2$ and $b=-4(.2)1$, 7D; the smallest positive $x$-zero of $M(a,b,x)$ for $a=-4(.1)-0.1$ and $b=0.1(.1)2.5$, 7D.

…