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

§14.4 Graphics

Permalink:: http://dlmf.nist.gov/14.4

§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

Notes:: These graphs were produced at NIST.
Keywords:: Ferrers functions
Permalink:: http://dlmf.nist.gov/14.4.i

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

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F1
Encodings:: pdf, png

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

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F2
Encodings:: pdf, png

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

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F3
Encodings:: pdf, png

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

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F4
Encodings:: pdf, png

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

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F5
Encodings:: pdf, png

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

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F6
Encodings:: pdf, png

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

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F7
Encodings:: pdf, png

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

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F8
Encodings:: pdf, png

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

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F9
Encodings:: pdf, png

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

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F10
Encodings:: pdf, png

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

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F11
Encodings:: pdf, png

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

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F12
Encodings:: pdf, png

§14.4(ii) Ferrers Functions: 3D Surfaces

Notes:: These surfaces were produced at NIST.
Keywords:: Ferrers functions
Permalink:: http://dlmf.nist.gov/14.4.ii

Visualization Help

Figure 14.4.13: $\mathop{\mathsf{P}^{{0}}_{{\nu}}\/}\nolimits\!\left(x\right),0\leq\nu\leq 10,-% 1<x<1$ .

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F13
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 14.4.14: $\mathop{\mathsf{Q}^{{0}}_{{\nu}}\/}\nolimits\!\left(x\right),0\leq\nu\leq 10,-% 1<x<1$ .

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F14
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 14.4.15: $\mathop{\mathsf{P}^{{-\mu}}_{{0}}\/}\nolimits\!\left(x\right),0\leq\mu\leq 10,% -1<x<1$ .

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F15
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 14.4.16: $\mathop{\mathsf{Q}^{{\mu}}_{{0}}\/}\nolimits\!\left(x\right),0\leq\mu\leq 6.2,% -1<x<1$ .

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, : real variable and $\mu$ : general order
Keywords:: Ferrers functions
Permalink:: http://dlmf.nist.gov/14.4.F16
Encodings:: VRML, X3D, pdf, png

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

Notes:: These graphs were produced at NIST.
Keywords:: associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.4.iii

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

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F17
Encodings:: pdf, png

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

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F18
Encodings:: pdf, png

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

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F19
Encodings:: pdf, png

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

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F20
Encodings:: pdf, png

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

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F21
Encodings:: pdf, png

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

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F22
Encodings:: pdf, png

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

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F23
Encodings:: pdf, png

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

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F24
Encodings:: pdf, png

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

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F25
Encodings:: pdf, png

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

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F26
Encodings:: pdf, png

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

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F27
Encodings:: pdf, png

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

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F28
Encodings:: pdf, png

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

Notes:: These surfaces were produced at NIST.
Keywords:: associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.4.iv

Visualization Help

Figure 14.4.29: $\mathop{P^{{0}}_{{\nu}}\/}\nolimits\!\left(x\right),0\leq\nu\leq 10,1<x<10$ .

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F29
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 14.4.30: $\mathop{\boldsymbol{Q}^{{0}}_{{\nu}}\/}\nolimits\!\left(x\right),0\leq\nu\leq 1% 0,1<x<10$ .

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : real variable and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.4.F30
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 14.4.31: $\mathop{P^{{-\mu}}_{{0}}\/}\nolimits\!\left(x\right),0\leq\mu\leq 10,1<x<10$ .

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F31
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 14.4.32: $\mathop{\boldsymbol{Q}^{{\mu}}_{{0}}\/}\nolimits\!\left(x\right),0\leq\mu\leq 1% 0,1<x<10$ .

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.4.F32
Encodings:: VRML, X3D, pdf, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 14.3 Definitions and Hypergeometric Representations 14.5 Special Values

§14.4 Graphics

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