# multipliers

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## 1—10 of 26 matching pages

##### 1: Bonita V. Saunders

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►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.
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##### 2: 20.12 Mathematical Applications

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►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)).
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##### 3: 31.6 Path-Multiplicative Solutions

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►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 ${\mathrm{e}}^{2\nu \pi \mathrm{i}}$.
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##### 4: 3.2 Linear Algebra

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►During this reduction process we store the

*multipliers*${\mathrm{\ell}}_{jk}$ 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
$$\mathbf{L}=\left[\begin{array}{cccc}1& 0& \mathrm{\cdots}& 0\\ {\mathrm{\ell}}_{21}& 1& \mathrm{\cdots}& 0\\ \mathrm{\vdots}& \mathrm{\ddots}& \mathrm{\ddots}& \mathrm{\vdots}\\ {\mathrm{\ell}}_{n1}& \mathrm{\cdots}& {\mathrm{\ell}}_{n,n-1}& 1\end{array}\right].$$

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►In practice, if any of the multipliers
${\mathrm{\ell}}_{jk}$ are unduly large in magnitude compared with unity, then Gaussian elimination is unstable.
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##### 5: 19.10 Relations to Other Functions

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►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.
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##### 6: 2.7 Differential Equations

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►

${w}_{2}(z)={\mathrm{e}}^{-2\pi \mathrm{i}{\mu}_{2}}{w}_{2}(z{\mathrm{e}}^{2\pi \mathrm{i}})+{C}_{2}{w}_{1}(z),$

►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

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►Sign changes of $Z(t)$ are determined by multiplying (25.9.3) by $\mathrm{exp}\left(\mathrm{i}\vartheta (t)\right)$ to obtain the

*Riemann–Siegel formula*: …##### 8: 9.7 Asymptotic Expansions

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►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 $n\ge 0$ in the first case and $n\ge 1$ in the second case.
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►The $n$th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by
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##### 9: Bibliography O

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On the calculation of Stokes multipliers for linear differential equations of the second order.
Methods Appl. Anal. 2 (3), pp. 348–367.
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##### 10: 19.25 Relations to Other Functions

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19.25.35
$$z+2\omega =\pm {R}_{F}(\mathrm{\wp}\left(z\right)-{e}_{1},\mathrm{\wp}\left(z\right)-{e}_{2},\mathrm{\wp}\left(z\right)-{e}_{3}),$$

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19.25.37
$$\zeta \left(z+2\omega \right)+(z+2\omega )\mathrm{\wp}\left(z\right)=\pm 2{R}_{G}(\mathrm{\wp}\left(z\right)-{e}_{1},\mathrm{\wp}\left(z\right)-{e}_{2},\mathrm{\wp}\left(z\right)-{e}_{3}),$$

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19.25.40
$$z+2\omega =\pm \sigma \left(z\right){R}_{F}({\sigma}_{1}^{2}(z),{\sigma}_{2}^{2}(z),{\sigma}_{3}^{2}(z)),$$

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