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11: 19.33 Triaxial Ellipsoids

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§19.33(iii) Depolarization Factors

… ►The external field and the induced magnetization together produce a uniform field inside the ellipsoid with strength $H/(1+L_c\chi )$, where $L_c$ is the demagnetizing factor, given in cgs units by ►

19.33.7 $$L_c=2\pi abc{\int }_0^{\mathrm{\infty }}\frac{d\lambda }{\sqrt{(a^2+\lambda )(b^2+\lambda ){(c^2+\lambda )}^3}}=V{R}_D(a^2,b^2,c^2).$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $L_a$, $L_b$: factor and $V$: volume

Referenced by:

§19.15, §19.33(iii)

Permalink:

http://dlmf.nist.gov/19.33.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.33(iii), §19.33 and Ch.19

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19.33.8 $$L_a+L_b+L_c=4\pi ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter and $L_a$, $L_b$: factor

Permalink:

http://dlmf.nist.gov/19.33.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.33(iii), §19.33 and Ch.19

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12: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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28.5.1 $${\mathrm{fe}}_n(z,q)=C_n(q)\left(z{\mathrm{ce}}_n(z,q)+f_n(z,q)\right),$$

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Defines:

${\mathrm{fe}}_n(z,q)$: second solution, Mathieu’s equation

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $f_n(z,q)$: functions and $C_n(q)$: factor

A&S Ref:

20.3.6 (in different form)

Referenced by:

§28.5(i), §28.5(i)

Permalink:

http://dlmf.nist.gov/28.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

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28.5.2 $${\mathrm{ge}}_n(z,q)=S_n(q)\left(z{\mathrm{se}}_n(z,q)+g_n(z,q)\right),$$

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Defines:

${\mathrm{ge}}_n(z,q)$: second solution, Mathieu’s equation

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $g_n(z,q)$: functions and $S_n(q)$: factor

A&S Ref:

20.3.7 (in different form)

Referenced by:

§28.5(i), §28.5(i)

Permalink:

http://dlmf.nist.gov/28.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

… ►The factors $C_n(q)$ and $S_n(q)$ in (28.5.1) and (28.5.2) are normalized so that ►

28.5.5 $${(C_n(q))}^2{\int }_0^{2\pi }{(f_n(x,q))}^{2}dx={(S_n(q))}^2{\int }_0^{2\pi }{(g_n(x,q))}^{2}dx=\pi .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $q=h^2$: parameter, $n$: integer, $x$: real variable, $f_n(z,q)$: functions, $g_n(z,q)$: functions, $C_n(q)$: factor and $S_n(q)$: factor

Permalink:

http://dlmf.nist.gov/28.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

… ►As a consequence of the factor $z$ on the right-hand sides of (28.5.1), (28.5.2), all solutions of Mathieu’s equation that are linearly independent of the periodic solutions are unbounded as $z\to \pm \mathrm{\infty }$ on $ℝ$. …

13: 3.8 Nonlinear Equations

… ►After a zero $\zeta$ has been computed, the factor $z-\zeta$ is factored out of $p(z)$ as a by-product of Horner’s scheme (§1.11(i)) for the computation of $p(\zeta )$. … ►Let $z^2-sz-t$ be an approximation to the real quadratic factor of $p(z)$ that corresponds to a pair of conjugate complex zeros or to a pair of real zeros. … ►The method converges locally and quadratically, except when the wanted quadratic factor is a multiple factor of $q(z)$. … ►This example illustrates the fact that the method succeeds even if the two zeros of the wanted quadratic factor are real and the same. … ►The perturbation factor (3.8.14) is given by …

14: 33.8 Continued Fractions

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33.8.1 $$\frac{F_{\mathrm{\ell }}^{\prime }}{F_{\mathrm{\ell }}}=S_{\mathrm{\ell }+1}-\frac{R_{\mathrm{\ell }+1}^2}{T_{\mathrm{\ell }+1}-}\frac{R_{\mathrm{\ell }+2}^2}{T_{\mathrm{\ell }+2}-}\mathrm{\cdots }.$$

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Symbols:

$F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $S_{\mathrm{\ell }}$: factor and $T_{\mathrm{\ell }}$: factor

Referenced by:

§33.8, §33.8

Permalink:

http://dlmf.nist.gov/33.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.8 and Ch.33

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15: 27.18 Methods of Computation: Primes

… ►Two simple algorithms for proving primality require a knowledge of all or part of the factorization of $n-1,n+1$, or both; see Crandall and Pomerance (2005, §§4.1–4.2). …

16: 28.1 Special Notation

… ►The notation for the joining factors is … ►

$f_{o,n}(h).$

…

17: 18.10 Integral Representations

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18.10.8 $$p_n(x)=\frac{g_0(x)}{2\pi \mathrm{i}}{\int }_C{\left(g_1(z,x)\right)}^n{g}_2(z,x){(z-c)}^{-1}dz$$

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Defines:

$g_0(x)$: factors (locally), $g_1(z,x)$: factors (locally), $g_2(z,x)$: factor (locally) and $c$: center (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $p_n(x)$: polynomial of degree $n$ , $z$: complex variable, $n$: nonnegative integer, $C$: closed contour and $x$: real variable

Referenced by:

Table 18.10.1, (18.17.21_1)

Permalink:

http://dlmf.nist.gov/18.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(iii), §18.10 and Ch.18

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18: 19.15 Advantages of Symmetry

… ►For example, the computation of depolarization factors for solid ellipsoids is simplified considerably; compare (19.33.7) with Cronemeyer (1991). …

19: 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 ${\mathrm{e}}^{2\nu \pi \mathrm{i}}$. …

20: 33.13 Complex Variable and Parameters

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33.13.2 $$R_{\mathrm{\ell }}=(2\mathrm{\ell }+1)C_{\mathrm{\ell }}\left(\eta \right)/C_{\mathrm{\ell }-1}\left(\eta \right).$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\eta$: real parameter and $R_{\mathrm{\ell }}$: factor

Permalink:

http://dlmf.nist.gov/33.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.13 and Ch.33

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