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14 Legendre and Related Functions Real Arguments 14.3 Definitions and Hypergeometric Representations 14.5 Special Values

§14.4 Graphics

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Permalink:: http://dlmf.nist.gov/14.4
See also:: Annotations for Ch.14

Contents

§14.4(i) Ferrers Functions: 2D Graphs
§14.4(ii) Ferrers Functions: 3D Surfaces
§14.4(iii) Associated Legendre Functions: 2D Graphs
§14.4(iv) Associated Legendre Functions: 3D Surfaces

§14.4(i) Ferrers Functions: 2D Graphs

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Keywords:: Ferrers functions, graphics
Notes:: These graphs were produced at NIST.
Permalink:: http://dlmf.nist.gov/14.4.i
See also:: Annotations for §14.4 and Ch.14

See accompanying text

Figure 14.4.1: $\mathsf{P}^{0}_{\nu}\left(x\right)$ , $\nu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F1
Encodings:: pdf, png
See also:: Annotations for §14.4(i), §14.4 and Ch.14

See accompanying text

Figure 14.4.2: $\mathsf{Q}^{0}_{\nu}\left(x\right)$ , $\nu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F2
Encodings:: pdf, png
See also:: Annotations for §14.4(i), §14.4 and Ch.14

See accompanying text

Figure 14.4.3: $\mathsf{P}^{-1/2}_{\nu}\left(x\right)$ , $\nu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F3
Encodings:: pdf, png
See also:: Annotations for §14.4(i), §14.4 and Ch.14

See accompanying text

Figure 14.4.4: $\mathsf{Q}^{1/2}_{\nu}\left(x\right)$ , $\nu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F4
Encodings:: pdf, png
See also:: Annotations for §14.4(i), §14.4 and Ch.14

See accompanying text

Figure 14.4.5: $\mathsf{P}^{-1}_{\nu}\left(x\right)$ , $\nu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F5
Encodings:: pdf, png
See also:: Annotations for §14.4(i), §14.4 and Ch.14

See accompanying text

Figure 14.4.6: $\mathsf{Q}^{1}_{\nu}\left(x\right)$ , $\nu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F6
Encodings:: pdf, png
See also:: Annotations for §14.4(i), §14.4 and Ch.14

See accompanying text

Figure 14.4.7: $\mathsf{P}^{-\mu}_{0}\left(x\right)$ , $\mu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F7
Encodings:: pdf, png
See also:: Annotations for §14.4(i), §14.4 and Ch.14

See accompanying text

Figure 14.4.8: $\mathsf{Q}^{\mu}_{0}\left(x\right)$ , $\mu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F8
Encodings:: pdf, png
See also:: Annotations for §14.4(i), §14.4 and Ch.14

See accompanying text

Figure 14.4.9: $\mathsf{P}^{-\mu}_{1/2}\left(x\right)$ , $\mu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F9
Encodings:: pdf, png
See also:: Annotations for §14.4(i), §14.4 and Ch.14

See accompanying text

Figure 14.4.10: $\mathsf{Q}^{\mu}_{1/2}\left(x\right)$ , $\mu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F10
Encodings:: pdf, png
See also:: Annotations for §14.4(i), §14.4 and Ch.14

See accompanying text

Figure 14.4.11: $\mathsf{P}^{-\mu}_{1}\left(x\right)$ , $\mu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F11
Encodings:: pdf, png
See also:: Annotations for §14.4(i), §14.4 and Ch.14

See accompanying text

Figure 14.4.12: $\mathsf{Q}^{\mu}_{1}\left(x\right)$ , $\mu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F12
Encodings:: pdf, png
See also:: Annotations for §14.4(i), §14.4 and Ch.14

§14.4(ii) Ferrers Functions: 3D Surfaces

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Keywords:: Ferrers functions, graphics
Notes:: These surfaces were produced at NIST.
Permalink:: http://dlmf.nist.gov/14.4.ii
See also:: Annotations for §14.4 and Ch.14

See accompanying text

Figure 14.4.13: $\mathsf{P}^{0}_{\nu}\left(x\right)$ , $0\leq\nu\leq 10,-1<x<1$ . Magnify 3D Help

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F13
Encodings:: Vizualization, pdf, png
See also:: Annotations for §14.4(ii), §14.4 and Ch.14

See accompanying text

Figure 14.4.14: $\mathsf{Q}^{0}_{\nu}\left(x\right)$ , $0\leq\nu\leq 10,-1<x<1$ . Magnify 3D Help

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Symbols:: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F14
Encodings:: Vizualization, pdf, png
See also:: Annotations for §14.4(ii), §14.4 and Ch.14

See accompanying text

Figure 14.4.15: $\mathsf{P}^{-\mu}_{0}\left(x\right)$ , $0\leq\mu\leq 10,-1<x<1$ . Magnify 3D Help

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F15
Encodings:: Vizualization, pdf, png
See also:: Annotations for §14.4(ii), §14.4 and Ch.14

See accompanying text

Figure 14.4.16: $\mathsf{Q}^{\mu}_{0}\left(x\right)$ , $0\leq\mu\leq 6.2,-1<x<1$ . Magnify 3D Help

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Symbols:: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, : real variable and $\mu$ : general order
Keywords:: Ferrers functions, graphics
Permalink:: http://dlmf.nist.gov/14.4.F16
Encodings:: Vizualization, pdf, png
See also:: Annotations for §14.4(ii), §14.4 and Ch.14

§14.4(iii) Associated Legendre Functions: 2D Graphs

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Keywords:: associated Legendre functions, graphics
Notes:: These graphs were produced at NIST.
Permalink:: http://dlmf.nist.gov/14.4.iii
See also:: Annotations for §14.4 and Ch.14

See accompanying text

Figure 14.4.17: $P^{0}_{\nu}\left(x\right)$ , $\nu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F17
Encodings:: pdf, png
See also:: Annotations for §14.4(iii), §14.4 and Ch.14

See accompanying text

Figure 14.4.18: $\boldsymbol{Q}^{0}_{\nu}\left(x\right)$ , $\nu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F18
Encodings:: pdf, png
See also:: Annotations for §14.4(iii), §14.4 and Ch.14

See accompanying text

Figure 14.4.19: $P^{-1/2}_{\nu}\left(x\right)$ , $\nu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F19
Encodings:: pdf, png
See also:: Annotations for §14.4(iii), §14.4 and Ch.14

See accompanying text

Figure 14.4.20: $\boldsymbol{Q}^{1/2}_{\nu}\left(x\right)$ , $\nu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F20
Encodings:: pdf, png
See also:: Annotations for §14.4(iii), §14.4 and Ch.14

See accompanying text

Figure 14.4.21: $P^{-1}_{\nu}\left(x\right)$ , $\nu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F21
Encodings:: pdf, png
See also:: Annotations for §14.4(iii), §14.4 and Ch.14

See accompanying text

Figure 14.4.22: $\boldsymbol{Q}^{1}_{\nu}\left(x\right)$ , $\nu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F22
Encodings:: pdf, png
See also:: Annotations for §14.4(iii), §14.4 and Ch.14

See accompanying text

Figure 14.4.23: $P^{-\mu}_{0}\left(x\right)$ , $\mu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F23
Encodings:: pdf, png
See also:: Annotations for §14.4(iii), §14.4 and Ch.14

See accompanying text

Figure 14.4.24: $\boldsymbol{Q}^{\mu}_{0}\left(x\right)$ , $\mu=0,2,4,8$ . Magnify

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Symbols:: $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F24
Encodings:: pdf, png
See also:: Annotations for §14.4(iii), §14.4 and Ch.14

See accompanying text

Figure 14.4.25: $P^{-\mu}_{1/2}\left(x\right)$ , $\mu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F25
Encodings:: pdf, png
See also:: Annotations for §14.4(iii), §14.4 and Ch.14

See accompanying text

Figure 14.4.26: $\boldsymbol{Q}^{\mu}_{1/2}\left(x\right)$ , $\mu=0,2,4,8$ . Magnify

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Symbols:: $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F26
Encodings:: pdf, png
See also:: Annotations for §14.4(iii), §14.4 and Ch.14

See accompanying text

Figure 14.4.27: $P^{-\mu}_{1}\left(x\right)$ , $\mu=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F27
Encodings:: pdf, png
See also:: Annotations for §14.4(iii), §14.4 and Ch.14

See accompanying text

Figure 14.4.28: $\boldsymbol{Q}^{\mu}_{1}\left(x\right)$ , $\mu=0,2,4,8$ . Magnify

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Symbols:: $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F28
Encodings:: pdf, png
See also:: Annotations for §14.4(iii), §14.4 and Ch.14

§14.4(iv) Associated Legendre Functions: 3D Surfaces

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Keywords:: associated Legendre functions, graphics
Notes:: These surfaces were produced at NIST.
Permalink:: http://dlmf.nist.gov/14.4.iv
See also:: Annotations for §14.4 and Ch.14

See accompanying text

Figure 14.4.29: $P^{0}_{\nu}\left(x\right)$ , $0\leq\nu\leq 10$ ,

. Magnify 3D Help

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Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F29
Encodings:: Vizualization, pdf, png
See also:: Annotations for §14.4(iv), §14.4 and Ch.14

See accompanying text

Figure 14.4.30: $\boldsymbol{Q}^{0}_{\nu}\left(x\right)$ , $0\leq\nu\leq 10$ ,

. Magnify 3D Help

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Symbols:: $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F30
Encodings:: Vizualization, pdf, png
See also:: Annotations for §14.4(iv), §14.4 and Ch.14

See accompanying text

Figure 14.4.31: $P^{-\mu}_{0}\left(x\right)$ , $0\leq\mu\leq 10$ ,

. Magnify 3D Help

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Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F31
Encodings:: Vizualization, pdf, png
See also:: Annotations for §14.4(iv), §14.4 and Ch.14

See accompanying text

Figure 14.4.32: $\boldsymbol{Q}^{\mu}_{0}\left(x\right)$ , $0\leq\mu\leq 10$ ,

. Magnify 3D Help

ⓘ

Symbols:: $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F32
Encodings:: Vizualization, pdf, png
See also:: Annotations for §14.4(iv), §14.4 and Ch.14

14.3 Definitions and Hypergeometric Representations 14.5 Special Values

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