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18 Orthogonal Polynomials Classical Orthogonal Polynomials 18.3 Definitions 18.5 Explicit Representations

§18.4 Graphics

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Permalink:: http://dlmf.nist.gov/18.4
See also:: Annotations for Ch.18

Contents

§18.4(i) Graphs
§18.4(ii) Surfaces

§18.4(i) Graphs

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Notes:: These graphs were produced at NIST.
Permalink:: http://dlmf.nist.gov/18.4.i
See also:: Annotations for §18.4 and Ch.18

See accompanying text

Figure 18.4.1: Jacobi polynomials $P^{(1.5,-0.5)}_{n}\left(x\right)$ ,

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Symbols:: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, : nonnegative integer and : real variable
Keywords:: Jacobi polynomials, graphs
Referenced by:: §18.2(vi)
Permalink:: http://dlmf.nist.gov/18.4.F1
Encodings:: pdf, png
See also:: Annotations for §18.4(i), §18.4 and Ch.18

See accompanying text

Figure 18.4.2: Jacobi polynomials $P^{(1.25,0.75)}_{n}\left(x\right)$ ,

. This illustrates inequalities for extrema of a Jacobi polynomial; see (18.14.16). See also Askey (1990). Magnify

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Symbols:: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, : nonnegative integer and : real variable
Keywords:: Jacobi polynomials, graphs
Permalink:: http://dlmf.nist.gov/18.4.F2
Encodings:: pdf, png
See also:: Annotations for §18.4(i), §18.4 and Ch.18

See accompanying text

Figure 18.4.3: Chebyshev polynomials $T_{n}\left(x\right)$ ,

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Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, : nonnegative integer and : real variable
Keywords:: Chebyshev polynomials, graphs
Referenced by:: §3.11(ii)
Permalink:: http://dlmf.nist.gov/18.4.F3
Encodings:: pdf, png
See also:: Annotations for §18.4(i), §18.4 and Ch.18

See accompanying text

Figure 18.4.4: Legendre polynomials $P_{n}\left(x\right)$ ,

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Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, : nonnegative integer and : real variable
Keywords:: Legendre polynomials, graphs
Permalink:: http://dlmf.nist.gov/18.4.F4
Encodings:: pdf, png
See also:: Annotations for §18.4(i), §18.4 and Ch.18

See accompanying text

Figure 18.4.5: Laguerre polynomials $L_{n}\left(x\right)$ ,

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Symbols:: $L_{\NVar{n}}\left(\NVar{x}\right)=L^{(0)}_{n}\left(x\right)$ : Laguerre polynomial, : nonnegative integer and : real variable
Keywords:: Laguerre polynomials, graphics
Permalink:: http://dlmf.nist.gov/18.4.F5
Encodings:: pdf, png
See also:: Annotations for §18.4(i), §18.4 and Ch.18

See accompanying text

Figure 18.4.6: Laguerre polynomials $L^{(\alpha)}_{3}\left(x\right)$ , $\alpha=0,1,2,3,4$ . Magnify

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Symbols:: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial and : real variable
Keywords:: Laguerre polynomials, graphics
Permalink:: http://dlmf.nist.gov/18.4.F6
Encodings:: pdf, png
See also:: Annotations for §18.4(i), §18.4 and Ch.18

See accompanying text

Figure 18.4.7: Monic Hermite polynomials $h_{n}(x)=2^{-n}H_{n}\left(x\right)$ ,

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Symbols:: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, : nonnegative integer and : real variable
Keywords:: Hermite polynomials, graphs, monic
Referenced by:: §18.2(vi)
Permalink:: http://dlmf.nist.gov/18.4.F7
Encodings:: pdf, png
See also:: Annotations for §18.4(i), §18.4 and Ch.18

§18.4(ii) Surfaces

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Notes:: These surfaces were produced at NIST.
Permalink:: http://dlmf.nist.gov/18.4.ii
See also:: Annotations for §18.4 and Ch.18

See accompanying text

Figure 18.4.8: Laguerre polynomials $L^{(\alpha)}_{3}\left(x\right)$ , $0\leq\alpha\leq 3$ , $0\leq x\leq 10$ . Magnify 3D Help

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Symbols:: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial and : real variable
Permalink:: http://dlmf.nist.gov/18.4.F8
Encodings:: Vizualization, pdf, png
See also:: Annotations for §18.4(ii), §18.4 and Ch.18

See accompanying text

Figure 18.4.9: Laguerre polynomials $L^{(\alpha)}_{4}\left(x\right)$ , $0\leq\alpha\leq 3$ , $0\leq x\leq 10$ . Magnify 3D Help

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Symbols:: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial and : real variable
Permalink:: http://dlmf.nist.gov/18.4.F9
Encodings:: Vizualization, pdf, png
See also:: Annotations for §18.4(ii), §18.4 and Ch.18

18.3 Definitions 18.5 Explicit Representations

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