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11: 1.11 Zeros of Polynomials

… ►A polynomial of degree $n$ with real or complex coefficients has exactly $n$ real or complex zeros counting multiplicity. …

12: 14.28 Sums

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14.28.1 $$P_{\nu }\left(z_1{z}_2-{\left(z_1^2-1\right)}^{1/2}{\left(z_2^2-1\right)}^{1/2}\cos \varphi \right)=P_{\nu }\left(z_1\right)P_{\nu }\left(z_2\right)+2\sum _{m=1}^{\mathrm{\infty }}{(-1)}^m\frac{\mathrm{\Gamma }\left(\nu -m+1\right)}{\mathrm{\Gamma }\left(\nu +m+1\right)}{P}_{\nu }^m\left(z_1\right)P_{\nu }^m(z_2)\cos \left(m\varphi \right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\cos z$: cosine function, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $z$: complex variable, $m$: nonnegative integer and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.28(i), §14.28 and Ch.14

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13: 14.21 Definitions and Basic Properties

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14.21.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }^2}{1-z^2}\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.1.1

Referenced by:

§14.21(ii), §14.29, 2nd item

Permalink:

http://dlmf.nist.gov/14.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.21(i), §14.21 and Ch.14

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14: Bibliography V

… ►

The topological degree theory for the localization and computation of complex zeros of Bessel functions. Numer. Funct. Anal. Optim. 18 (1-2), pp. 227–234.

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External Links:

ISSN 0163-0563, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

Encodings:

BibTeX

15: 18.28 Askey–Wilson Class

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18.28.6 $${\int }_{-1}^1{p}_n(x)p_m(x)w(x)dx+\sum _{\mathrm{\ell }}{p}_n(x_{\mathrm{\ell }})p_m(x_{\mathrm{\ell }}){\omega }_{\mathrm{\ell }}=h_n{\delta }_{n,m},$$ $ab,ac,ad,bc,bd,cd\in \{z\in ℂ\mid |z|\le 1,z\ne 1\}$,

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $p_n(x)$: polynomial of degree $n$ , $w(x)$: weight function, $z$: complex variable, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $h_n$

Referenced by:

Erratum (V1.2.0) for Equation (18.28.6)

Permalink:

http://dlmf.nist.gov/18.28.E6

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

This constraint of this equation was updated to include $ab,ac,ad,bc,bd,cd\in \{z\in ℂ\mid |z|\le 1,z\ne 1\}$.

See also:

Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

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16: 18.22 Hahn Class: Recurrence Relations and Differences

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18.22.13 $$p_n(x)=p_n(x;a,b,\overline{a},\overline{b}),$$

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Symbols:

$\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(ii)

Permalink:

http://dlmf.nist.gov/18.22.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

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18.22.28 $${\delta }_x\left(w(x;a+\frac{1}{2},b+\frac{1}{2},\overline{a}+\frac{1}{2},\overline{b}+\frac{1}{2})p_n(x;a+\frac{1}{2},b+\frac{1}{2},\overline{a}+\frac{1}{2},\overline{b}+\frac{1}{2})\right)=-(n+1)w(x;a,b,\overline{a},\overline{b})p_{n+1}(x;a,b,\overline{a},\overline{b}).$$

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Symbols:

${\delta }_x$: central difference, $\overline{z}$: complex conjugate, $p_n(x)$: polynomial of degree $n$ , $w(x)$: weight function, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.20(i), §18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

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17: 18.39 Applications in the Physical Sciences

… ►Kuijlaars and Milson (2015, §1) refer to these, in this case complex zeros, as exceptional, as opposed to regular, zeros of the EOP’s, these latter belonging to the (real) orthogonality integration range. … ►

18.39.21 $$\mathscr{H}=-\frac{{\mathrm{\hslash }}^2}{2m}{\nabla }^2+V(𝐱),$$ $𝐱=(x,y,z)\in {ℝ}^3$,

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Symbols:

$\in$: element of, $ℝ$: real line, $y$: real variable, $z$: complex variable, $m$: nonnegative integer and $x$: real variable

Referenced by:

§18.39(ii)

Permalink:

http://dlmf.nist.gov/18.39.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(ii), §18.39 and Ch.18

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18: 18.33 Polynomials Orthogonal on the Unit Circle

… ►with complex coefficients $c_k$ and of a certain degree $n$ define the reversed polynomial $p^{\ast }(z)$ by …

19: 31.8 Solutions via Quadratures

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31.8.2 $$w_{\pm }(𝐦;\lambda ;z)=\sqrt{{\mathrm{\Psi }}_{g,N}(\lambda ,z)}\exp \left(\pm \frac{\mathrm{i}\nu (\lambda )}{2}{\int }_{z_0}^z\frac{t^{m_1}{(t-1)}^{m_2}{(t-a)}^{m_3}dt}{{\mathrm{\Psi }}_{g,N}(\lambda ,t)\sqrt{t(t-1)(t-a)}}\right)$$

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Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $\nu$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $\lambda =-4q$ , ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$: polynomial, $g$: degree of $\lambda$ and $N$: degree of $z$

Referenced by:

§31.8

Permalink:

http://dlmf.nist.gov/31.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.8 and Ch.31

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20: 3.5 Quadrature

… ►Complex orthogonal polynomials $p_n(1/\zeta )$ of degree $n=0,1,2,\mathrm{\dots }$, in $1/\zeta$ that satisfy the orthogonality condition …