multipliers

(0.001 seconds)

1—10 of 26 matching pages

1: Bonita V. Saunders

… ►As the principal developer of graphics for the DLMF, she has collaborated with other NIST mathematicians, computer scientists, and student interns to produce informative graphs and dynamic interactive visualizations of elementary and higher mathematical functions over both simply and multiply connected domains. …

2: 20.12 Mathematical Applications

… ►This ability to uniformize multiply-connected spaces (manifolds), or multi-sheeted functions of a complex variable (Riemann (1899), Rauch and Lebowitz (1973), Siegel (1988)) has led to applications in string theory (Green et al. (1988a, b), Krichever and Novikov (1989)), and also in statistical mechanics (Baxter (1982)). …

3: 31.6 Path-Multiplicative Solutions

… ►This denotes a set of solutions of (31.2.1) with the property that if we pass around a simple closed contour in the

z

-plane that encircles

s_{1}

and

s_{2}

once in the positive sense, but not the remaining finite singularity, then the solution is multiplied by a constant factor

{\mathrm{e}}^{2\nu\pi i}

. …

4: 3.2 Linear Algebra

… ►During this reduction process we store the multipliers

\ell_{jk}

that are used in each column to eliminate other elements in that column. This yields a lower triangular matrix of the form ►

3.2.4

\mathbf{L}=\begin{bmatrix}1&0&\cdots&0\\ \ell_{21}&1&\cdots&0\\ \vdots&\ddots&\ddots&\vdots\\ \ell_{n1}&\cdots&\ell_{n,n-1}&1\end{bmatrix}.

ⓘ

Symbols:: $\ell_{jk}$ : multipliers
Referenced by:: §1.2(vi)
Permalink:: http://dlmf.nist.gov/3.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §3.2(i), §3.2 and Ch.3

… ►In practice, if any of the multipliers

\ell_{jk}

are unduly large in magnitude compared with unity, then Gaussian elimination is unstable. … …

5: 19.10 Relations to Other Functions

… ►In each case when

y=1

, the quantity multiplying

R_{C}

supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0. …

6: 2.7 Differential Equations

… ►

w_{2}(z)=e^{-2\pi i\mu_{2}}w_{2}(ze^{2\pi i})+C_{2}w_{1}(z),

►in which

C_{1}

C_{2}

are constants, the so-called Stokes multipliers. … ►For the calculation of Stokes multipliers see Olde Daalhuis and Olver (1995b). …

7: 25.10 Zeros

… ►Sign changes of

Z(t)

are determined by multiplying (25.9.3) by

\exp\left(i\vartheta(t)\right)

to obtain the Riemann–Siegel formula: …

8: 9.7 Asymptotic Expansions

… ►In (9.7.7) and (9.7.8) the

n

th error term is bounded in magnitude by the first neglected term multiplied by

\chi(n+\sigma)+1

where

\sigma=\frac{1}{6}

for (9.7.7) and

\sigma=0

for (9.7.8), provided that

n\geq 0

in the first case and

n\geq 1

in the second case. … ►The

n

th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by …

9: Bibliography O

… ►

A. B. Olde Daalhuis and F. W. J. Olver (1995b) On the calculation of Stokes multipliers for linear differential equations of the second order. Methods Appl. Anal. 2 (3), pp. 348–367.

ⓘ

External Links:: ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt
Referenced by:: §2.7(ii).
Encodings:: BibTeX

…

10: 19.25 Relations to Other Functions

… ►

19.25.35

z+2\omega=\pm R_{F}\left(\wp\left(z\right)-e_{1},\wp\left(z\right)-e_{2},\wp% \left(z\right)-e_{3}\right),

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots and $z$ : complex number
Proof sketch:: Use (23.6.36) with $z=\wp\left(\omega\right)$ as prescribed in the text that follows (23.6.36), substitute $u=t+\wp\left(\omega\right)$ 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

… ►

19.25.37

\zeta\left(z+2\omega\right)+(z+2\omega)\wp\left(z\right)=\pm 2R_{G}\left(\wp% \left(z\right)-e_{1},\wp\left(z\right)-e_{2},\wp\left(z\right)-e_{3}\right),

ⓘ

Symbols:: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $e_{\NVar{j}}$ : Weierstrass lattice roots, $\zeta\left(\NVar{z}\right)$ (= $\zeta\left(z|\mathbb{L}\right)$ = $\zeta\left(z;g_{2},g_{3}\right)$ ): 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 $\zeta\left(z\right)+z\wp\left(z\right)$ , has been replaced by $\zeta\left(z+2\omega\right)+(z+2\omega)\wp\left(z\right)$ , 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}\left(\sigma_{1}^{2}(z),\sigma_{2}^{2}(z% ),\sigma_{3}^{2}(z)\right),

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): 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

…