logarithmic weight function

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1: 3.5 Quadrature

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Gauss Formula for a Logarithmic Weight Function

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Table 3.5.14: Nodes and weights for the 5-point Gauss formula for the logarithmic weight function.

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$x_k$		$w_k$
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Table 3.5.15: Nodes and weights for the 10-point Gauss formula for the logarithmic weight function.

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$x_k$		$w_k$
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Table 3.5.16: Nodes and weights for the 15-point Gauss formula for the logarithmic weight function.

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$x_k$		$w_k$
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Table 3.5.17: Nodes and weights for the 20-point Gauss formula for the logarithmic weight function.

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$x_k$		$w_k$
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2: 18.33 Polynomials Orthogonal on the Unit Circle

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18.33.31 $$\prod _{j=0}^{\mathrm{\infty }}\left(1-{\left|{\alpha }_j\right|}^2\right)=\exp \left(\frac{1}{2\pi \mathrm{i}}{\int }_{|z|=1}\ln \left(w(z)\right)\frac{dz}{z}\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\ln z$: principal branch of logarithm function, $w(x)$: weight function and $z$: complex variable

Proved:

Simon (2005a, (1.1.18))(proved)

Referenced by:

§18.33(vi)

Permalink:

http://dlmf.nist.gov/18.33.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

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18.33.32

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\ln z$: principal branch of logarithm function, $w(x)$: weight function and $z$: complex variable

Source:

Simon (2005a, (1.1.19))

Referenced by:

Erratum (V1.2.0) Section 18.33

Permalink:

http://dlmf.nist.gov/18.33.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

3: 18.19 Hahn Class: Definitions

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18.19.7 $$w^{(\lambda )}(z;\varphi )=\mathrm{\Gamma }\left(\lambda +iz\right)\mathrm{\Gamma }\left(\lambda -iz\right){\mathrm{e}}^{(2\varphi -\pi )z},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $w(x)$: weight function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/18.19.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.19, §18.19 and Ch.18

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18.19.8 $$w(x)=w^{(\lambda )}(x;\varphi )={\left|\mathrm{\Gamma }\left(\lambda +\mathrm{i}x\right)\right|}^2{\mathrm{e}}^{(2\varphi -\pi )x},$$ $\lambda >0$, ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $w(x)$: weight function and $x$: real variable

Referenced by:

§18.19, §18.19

Permalink:

http://dlmf.nist.gov/18.19.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.19, §18.19 and Ch.18

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4: 18.35 Pollaczek Polynomials

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18.35.6 $$w^{(\lambda )}(\cos \theta ;a,b)={\pi }^{-1}{\mathrm{e}}^{(2\theta -\pi ){\tau }_{a,b}(\theta )}{\left(2\sin \theta \right)}^{2\lambda -1}{\left|\mathrm{\Gamma }\left(\lambda +\mathrm{i}{\tau }_{a,b}(\theta )\right)\right|}^2,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $w(x)$: weight function and ${\tau }_{a,b}(\theta )$

Source:

Ismail (2009, (5.4.13))

Referenced by:

(18.35.5), §18.35(ii), §18.39(iv), §18.39(iv), Erratum (V1.2.0) for Equation (18.35.5)

Permalink:

http://dlmf.nist.gov/18.35.E6

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

The constraints in this equation have been moved to (18.35.5) except for .

See also:

Annotations for §18.35(ii), §18.35 and Ch.18

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18.35.6_1

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\ln z$: principal branch of logarithm function and $w(x)$: weight function

Referenced by:

§18.35(ii), Erratum (V1.2.0) Section 18.35

Permalink:

http://dlmf.nist.gov/18.35.E6_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.35(ii), §18.35 and Ch.18

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18.35.6_6 $$w^{(\lambda )}(\cos \theta ;a,b,c)=\frac{{\mathrm{e}}^{(2\theta -\pi ){\tau }_{a,b}(\theta )}{\left(2\sin \theta \right)}^{2\lambda -1}{\left|\mathrm{\Gamma }\left(c+\lambda +\mathrm{i}{\tau }_{a,b}(\theta )\right)\right|}^2}{\pi {\left|F(\genfrac{}{}{0pt}{}{1-\lambda +\mathrm{i}{\tau }_{a,b}(\theta ),c}{c+\lambda +\mathrm{i}{\tau }_{a,b}(\theta )};{\mathrm{e}}^{2\mathrm{i}\theta })\right|}^2},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $w(x)$: weight function and ${\tau }_{a,b}(\theta )$

Source:

Pollaczek (1950, (2))

Referenced by:

§18.35(ii), Erratum (V1.2.0) Section 18.35

Permalink:

http://dlmf.nist.gov/18.35.E6_6

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.35(ii), §18.35 and Ch.18

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5: 18.2 General Orthogonal Polynomials

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18.2.39

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $w(x)$: weight function and $x$: real variable

Source:

Szegő (1975, §12.1)

Referenced by:

§18.35(ii)

Permalink:

http://dlmf.nist.gov/18.2.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §18.2(xi), §18.2(xi), §18.2 and Ch.18

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6: 18.30 Associated OP’s

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18.30.11 $$w^{\lambda }(x,c)=\frac{x^{\lambda }{\mathrm{e}}^{-x}}{{\left|U(c,1-\lambda ,x{\mathrm{e}}^{-\mathrm{i}\pi })\right|}^2}.$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $w(x)$: weight function and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.30.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(iii), §18.30 and Ch.18

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18.30.18 $$w^{(\lambda )}(x,\varphi ,c)=\frac{{\mathrm{e}}^{(2\varphi -\pi )x}{\left(2\sin \varphi \right)}^{2\lambda }{\left|\mathrm{\Gamma }\left(c+\lambda +\mathrm{i}x\right)\right|}^2}{2\pi {\left|F(1-\lambda +\mathrm{i}x,c;c+\lambda +\mathrm{i}x;{\mathrm{e}}^{2\mathrm{i}\varphi })\right|}^2}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $w(x)$: weight function and $x$: real variable

Sources:

Askey and Wimp (1984, (1.10)); Pollaczek (1950, (16))

Permalink:

http://dlmf.nist.gov/18.30.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(v), §18.30 and Ch.18

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7: 18.36 Miscellaneous Polynomials

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18.36.10 $$w(x)=\frac{{\mathrm{e}}^{-x^2}}{{\left(4x^2+2\right)}^2},$$ $x\in (-\mathrm{\infty },\mathrm{\infty })$.

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Symbols:

$\in$: element of, $\mathrm{e}$: base of natural logarithm, $(a,b)$: open interval, $w(x)$: weight function and $x$: real variable

Referenced by:

§18.39(i), Erratum (V1.2.0) Section 18.36

Permalink:

http://dlmf.nist.gov/18.36.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

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8: 18.39 Applications in the Physical Sciences

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18.39.50 $$w^{\mathrm{CP}}(x)=\frac{(l+1+\frac{2Z}{s})}{\pi \mathrm{\Gamma }(2l+2)}{\mathrm{e}}^{(2\theta (x)-\pi )\tau (x)}{\left(4(1-x^2)\right)}^{l+\frac{1}{2}}{\left|\mathrm{\Gamma }\left(l+1+\mathrm{i}\tau (x)\right)\right|}^2,$$ $\theta (x)=\arccos (x)$, $\tau (x)=-\frac{2Z}{s}\sqrt{\frac{1-x}{1+x}}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\arccos z$: arccosine function, $w(x)$: weight function, $l$: nonnegative integer and $x$: real variable

Referenced by:

Figure 18.39.2, Figure 18.39.2, §18.39(iv), §18.40(ii)

Permalink:

http://dlmf.nist.gov/18.39.E50

Encodings:

pMML, png, TeX

See also:

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

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9: 18.5 Explicit Representations

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§18.5(i) Trigonometric Functions

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Chebyshev

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§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

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Laguerre

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Hermite

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10: 18.34 Bessel Polynomials

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§18.34(i) Definitions and Recurrence Relation

… ►Hence the full system of polynomials $y_n(x;a)$ cannot be orthogonal on the line with respect to a positive weight function, but this is possible for a finite system of such polynomials, the Romanovski–Bessel polynomials, if : …Explicit (but complicated) weight functions $w(x)$ taking both positive and negative values have been found such that (18.2.26) holds with $d\mu (x)=w(x)dx$; see Durán (1993), Evans et al. (1993), and Maroni (1995). ►Orthogonality of the full system on the unit circle can be given with a much simpler weight function: … ►With functions …