14 Legendre and Related Functions Real Arguments 14.19 Toroidal (or Ring) Functions 14.21 Definitions and Basic Properties

§14.20 Conical (or Mehler) Functions

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Referenced by:: §14.2(ii), §14.20(i)
Permalink:: http://dlmf.nist.gov/14.20
See also:: Annotations for Ch.14

§14.20(i) Definitions and Wronskians
§14.20(ii) Graphics
§14.20(iii) Behavior as $x\to 1$
§14.20(iv) Integral Representation
§14.20(v) Trigonometric Expansion
§14.20(vi) Generalized Mehler–Fock Transformation
§14.20(vii) Asymptotic Approximations: Large $\tau$ , Fixed $\mu$
§14.20(viii) Asymptotic Approximations: Large $\tau$ , $0\leq\mu\leq A\tau$
§14.20(ix) Asymptotic Approximations: Large $\mu$ , $0\leq\tau\leq A\mu$
§14.20(x) Zeros and Integrals

§14.20(i) Definitions and Wronskians

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Keywords:: Wronskians, conical functions, connection formulas, definitions, differential equation, notation
Notes:: (14.20.3) follows from (14.9.10) and (14.20.2). (14.20.4) and (14.20.5) follow from (14.2.3) and (14.20.3). (14.20.6) follows from (14.3.10) and (14.9.12).
Referenced by:: §14.32
Permalink:: http://dlmf.nist.gov/14.20.i
See also:: Annotations for §14.20 and Ch.14

Throughout §14.20 we assume that $\nu=-\frac{1}{2}+i\tau$ , with $\mu\geq 0$ and $\tau\geq 0$ . (14.2.2) takes the form

14.20.1 $\left(1-x^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}-2x\frac{% \mathrm{d}w}{\mathrm{d}x}-\left(\tau^{2}+\frac{1}{4}+\frac{\mu^{2}}{1-x^{2}}% \right)w=0.$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : real variable, $\tau$ : real variable and $\mu$ : general order
Referenced by:: §14.20(i), §14.20(i), 2nd item
Permalink:: http://dlmf.nist.gov/14.20.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(i), §14.20 and Ch.14

Solutions are known as conical or Mehler functions. For and $\tau>0$ , a numerically satisfactory pair of real conical functions is $\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$ and $\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(-x\right)$ .

Another real-valued solution $\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ of (14.20.1) was introduced in Dunster (1991). This is defined by

14.20.2 $\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\Re\left(e^{\mu% \pi i}\mathsf{Q}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)\right)-\tfrac{1}{2}% \pi\sin\left(\mu\pi\right)\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right).$

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Defines:: $\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)$ : conical function
Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\Re$ : real part, $\sin\NVar{z}$ : sine function, : real variable, $\tau$ : real variable and $\mu$ : general order
Referenced by:: §14.20(i), §14.23
Permalink:: http://dlmf.nist.gov/14.20.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(i), §14.20 and Ch.14

Equivalently,

14.20.3 $\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)=\frac{% \pi e^{-\tau\pi}\sin\left(\mu\pi\right)\sinh\left(\tau\pi\right)}{2({\cosh}^{2% }\left(\tau\pi\right)-{\sin}^{2}\left(\mu\pi\right))}\mathsf{P}^{-\mu}_{-\frac% {1}{2}+\mathrm{i}\tau}\left(x\right)+\frac{\pi(e^{-\tau\pi}{\cos}^{2}\left(\mu% \pi\right)+\sinh\left(\tau\pi\right))}{2({\cosh}^{2}\left(\tau\pi\right)-{\sin% }^{2}\left(\mu\pi\right))}\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left% (-x\right).$

$\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ exists except when $\mu=\frac{1}{2},\frac{3}{2},\dots$ and $\tau=0$ ; compare §14.3(i). It is an important companion solution to $\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ when $\tau$ is large; compare §§14.20(vii), 14.20(viii), and 10.25(iii).

14.20.4 $\mathscr{W}\left\{\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right% ),\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(-x\right)\right\}=\frac% {2}{|\Gamma\left(\mu+\frac{1}{2}+\mathrm{i}\tau\right)|^{2}(1-x^{2})}.$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathscr{W}$ : Wronskian, $\mathrm{i}$ : imaginary unit, : real variable, $\tau$ : real variable and $\mu$ : general order
Referenced by:: §14.20(i)
Permalink:: http://dlmf.nist.gov/14.20.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(i), §14.20 and Ch.14

14.20.5 $\mathscr{W}\left\{\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right% ),\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)% \right\}=\frac{\pi(e^{-\tau\pi}{\cos}^{2}\left(\mu\pi\right)+\sinh\left(\tau% \pi\right))}{|\Gamma\left(\mu+\frac{1}{2}+\mathrm{i}\tau\right)|^{2}({\cosh}^{% 2}\left(\tau\pi\right)-{\sin}^{2}\left(\mu\pi\right))(1-x^{2})},$

provided that $\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ exists.

Lastly, for the range $1<x<\infty$ , $P^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$ is a real-valued solution of (14.20.1); in terms of $Q^{\mu}_{-\frac{1}{2}\pm\mathrm{i}\tau}\left(x\right)$ (which are complex-valued in general):

14.20.6 $P^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{ie^{-\mu\pi i}}{\sinh\left(% \tau\pi\right)\left|\Gamma\left(\mu+\frac{1}{2}+i\tau\right)\right|^{2}}\*% \left(Q^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)-Q^{\mu}_{-\frac{1}{2}-i\tau}% \left(x\right)\right),$ $\tau\neq 0$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, : real variable, $\tau$ : real variable and $\mu$ : general order
Referenced by:: §14.20(i)
Permalink:: http://dlmf.nist.gov/14.20.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(i), §14.20 and Ch.14

§14.20(ii) Graphics

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Keywords:: conical functions, graphics
Notes:: These graphs were produced at NIST.
Permalink:: http://dlmf.nist.gov/14.20.ii
See also:: Annotations for §14.20 and Ch.14

Figure 14.20.1: $\mathsf{P}^{0}_{-\frac{1}{2}+i\tau}\left(x\right)$ , $\tau=0,1,2,4,8$ . Magnify

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathrm{i}$ : imaginary unit, : real variable and $\tau$ : real variable
Permalink:: http://dlmf.nist.gov/14.20.F1
Encodings:: pdf, png
See also:: Annotations for §14.20(ii), §14.20 and Ch.14

Figure 14.20.2: $\widehat{\mathsf{Q}}^{0}_{-\frac{1}{2}+i\tau}\left(x\right)$ , $\tau=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)$ : conical function, $\mathrm{i}$ : imaginary unit, : real variable and $\tau$ : real variable
Permalink:: http://dlmf.nist.gov/14.20.F2
Encodings:: pdf, png
See also:: Annotations for §14.20(ii), §14.20 and Ch.14

Figure 14.20.3: $\mathsf{P}^{-1/2}_{-\frac{1}{2}+i\tau}\left(x\right)$ , $\tau=0,1,2,4,8$ . Magnify

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathrm{i}$ : imaginary unit, : real variable and $\tau$ : real variable
Permalink:: http://dlmf.nist.gov/14.20.F3
Encodings:: pdf, png
See also:: Annotations for §14.20(ii), §14.20 and Ch.14

Figure 14.20.4: $\widehat{\mathsf{Q}}^{-1/2}_{-\frac{1}{2}+i\tau}\left(x\right)$ , $\tau=\tfrac{1}{2},1,2,4$ . (This function does not exist when $\tau=0$ .) Magnify

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Symbols:: $\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)$ : conical function, $\mathrm{i}$ : imaginary unit, : real variable and $\tau$ : real variable
Permalink:: http://dlmf.nist.gov/14.20.F4
Encodings:: pdf, png
See also:: Annotations for §14.20(ii), §14.20 and Ch.14

Figure 14.20.5: $\mathsf{P}^{-1}_{-\frac{1}{2}+i\tau}\left(x\right)$ , $\tau=0,1,2,4,8$ . Magnify

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathrm{i}$ : imaginary unit, : real variable and $\tau$ : real variable
Permalink:: http://dlmf.nist.gov/14.20.F5
Encodings:: pdf, png
See also:: Annotations for §14.20(ii), §14.20 and Ch.14

Figure 14.20.6: $\widehat{\mathsf{Q}}^{-1}_{-\frac{1}{2}+i\tau}\left(x\right)$ , $\tau=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)$ : conical function, $\mathrm{i}$ : imaginary unit, : real variable and $\tau$ : real variable
Permalink:: http://dlmf.nist.gov/14.20.F6
Encodings:: pdf, png
See also:: Annotations for §14.20(ii), §14.20 and Ch.14

Figure 14.20.7: $\mathsf{P}^{-2}_{-\frac{1}{2}+i\tau}\left(x\right),\tau=0,1,2,4,8$ . Magnify

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathrm{i}$ : imaginary unit, : real variable and $\tau$ : real variable
Permalink:: http://dlmf.nist.gov/14.20.F7
Encodings:: pdf, png
See also:: Annotations for §14.20(ii), §14.20 and Ch.14

Figure 14.20.8: $\widehat{\mathsf{Q}}^{-2}_{-\frac{1}{2}+i\tau}\left(x\right)$ , $\tau=0,\tfrac{1}{2},1,2,4$ . Magnify

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Symbols:: $\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)$ : conical function, $\mathrm{i}$ : imaginary unit, : real variable and $\tau$ : real variable
Permalink:: http://dlmf.nist.gov/14.20.F8
Encodings:: pdf, png
See also:: Annotations for §14.20(ii), §14.20 and Ch.14

§14.20(iii) Behavior as $x\to 1$

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Keywords:: behavior at singularities, conical functions
Notes:: (14.20.7) and (14.20.8) may be derived from §14.8(i) and (14.20.3).
Permalink:: http://dlmf.nist.gov/14.20.iii
See also:: Annotations for §14.20 and Ch.14

The behavior of $\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(\pm x\right)$ as $x\to 1-$ is given in §14.8(i). For $\mu>0$ and $x\to 1-$ ,

14.20.7 $\widehat{\mathsf{Q}}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)\sim\tfrac{1}{2}% \Gamma\left(\mu\right)\left(\frac{2}{1-x}\right)^{\mu/2},$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)$ : conical function, $\sim$ : asymptotic equality, $\mathrm{i}$ : imaginary unit, : real variable, $\tau$ : real variable and $\mu$ : general order
Referenced by:: §14.20(iii)
Permalink:: http://dlmf.nist.gov/14.20.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(iii), §14.20 and Ch.14

14.20.8 $\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)\sim\frac{\pi% \Gamma\left(\mu\right)(e^{-\tau\pi}{\cos}^{2}\left(\mu\pi\right)+\sinh\left(% \tau\pi\right))}{2({\cosh}^{2}\left(\tau\pi\right)-{\sin}^{2}\left(\mu\pi% \right)){\left|\Gamma\left(\mu+\frac{1}{2}+\mathrm{i}\tau\right)\right|}^{2}}% \*\left(\frac{2}{1-x}\right)^{\mu/2}.$

§14.20(iv) Integral Representation

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Keywords:: conical functions, integral representation
Notes:: (14.20.9) may be derived from (14.12.1).
Referenced by:: 3rd item
Permalink:: http://dlmf.nist.gov/14.20.iv
See also:: Annotations for §14.20 and Ch.14

When $0<\theta<\pi$ ,

14.20.9 $\mathsf{P}_{-\frac{1}{2}+i\tau}\left(\cos\theta\right)=\frac{2}{\pi}\int_{0}^{% \theta}\frac{\cosh\left(\tau\phi\right)}{\sqrt{2(\cos\phi-\cos\theta)}}\,% \mathrm{d}\phi.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\tau$ : real variable and $0<\theta<\pi$ : variable
A&S Ref:: 8.12.3
Referenced by:: §14.20(iv), §14.20(v)
Permalink:: http://dlmf.nist.gov/14.20.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(iv), §14.20 and Ch.14

§14.20(v) Trigonometric Expansion

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Keywords:: conical functions, trigonometric expansion
Notes:: See Erdélyi et al. (1953a, p. 174).
Permalink:: http://dlmf.nist.gov/14.20.v
See also:: Annotations for §14.20 and Ch.14

14.20.10 $\mathsf{P}_{-\frac{1}{2}+i\tau}\left(\cos\theta\right)=1+\frac{4\tau^{2}+1^{2}% }{2^{2}}{\sin}^{2}\left(\tfrac{1}{2}\theta\right)+\frac{\left(4\tau^{2}+1^{2}% \right)\left(4\tau^{2}+3^{2}\right)}{2^{2}\cdot 4^{2}}{\sin}^{4}\left(\tfrac{1% }{2}\theta\right)+\cdots,$ $0\leq\theta\leq\pi$ .

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{i}$ : imaginary unit, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\sin\NVar{z}$ : sine function and $\tau$ : real variable
A&S Ref:: 8.12.1
Referenced by:: §14.20(v)
Permalink:: http://dlmf.nist.gov/14.20.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(v), §14.20 and Ch.14

From (14.20.9) or (14.20.10) it is evident that $\mathsf{P}_{-\frac{1}{2}+i\tau}\left(\cos\theta\right)$ is positive for real $\theta$ .

§14.20(vi) Generalized Mehler–Fock Transformation

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Keywords:: Mehler–Fock transformation, conical functions, generalized, generalized Mehler–Fock transformation
Notes:: See Braaksma and Meulenbeld (1967).
Referenced by:: §14.31(ii)
Permalink:: http://dlmf.nist.gov/14.20.vi
See also:: Annotations for §14.20 and Ch.14

14.20.11 $f(\tau)=\frac{\tau}{\pi}\sinh\left(\tau\pi\right)\Gamma\left(\tfrac{1}{2}-\mu+% i\tau\right)\*\Gamma\left(\tfrac{1}{2}-\mu-i\tau\right)\int_{1}^{\infty}P^{\mu% }_{-\frac{1}{2}+i\tau}\left(x\right)g(x)\,\mathrm{d}x,$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, : real variable, $\tau$ : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.20.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(vi), §14.20 and Ch.14

where

14.20.12 $g(x)=\int_{0}^{\infty}P^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)f(\tau)\,% \mathrm{d}\tau.$

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Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{i}$ : imaginary unit, $\int$ : integral, : real variable, $\tau$ : real variable and $\mu$ : general order
Permalink:: http://dlmf.nist.gov/14.20.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(vi), §14.20 and Ch.14

Special cases:

14.20.13 $P_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{\cosh\left(\tau\pi\right)}{\pi}\int% _{1}^{\infty}\frac{P_{-\frac{1}{2}+i\tau}\left(t\right)}{x+t}\,\mathrm{d}t,$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : Legendre function of the first kind, : real variable and $\tau$ : real variable
Permalink:: http://dlmf.nist.gov/14.20.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(vi), §14.20 and Ch.14

14.20.14 $\pi\int_{0}^{\infty}\frac{\tau\tanh\left(\tau\pi\right)}{\cosh\left(\tau\pi% \right)}P_{-\frac{1}{2}+i\tau}\left(x\right)P_{-\frac{1}{2}+i\tau}\left(y% \right)\,\mathrm{d}\tau=\frac{1}{y+x}.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\cosh\NVar{z}$ : hyperbolic cosine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : Legendre function of the first kind, : real variable and $\tau$ : real variable
Permalink:: http://dlmf.nist.gov/14.20.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(vi), §14.20 and Ch.14

§14.20(vii) Asymptotic Approximations: Large $\tau$ , Fixed $\mu$

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Keywords:: asymptotic approximations, conical functions, large degree
Notes:: For (14.20.15) see Olver (1997b, p. 473). (14.20.16) may be derived from (14.20.18).
Referenced by:: §14.20(i), 1st item, §2.8(iv)
Permalink:: http://dlmf.nist.gov/14.20.vii
See also:: Annotations for §14.20 and Ch.14

For $\tau\to\infty$ and fixed $\mu$ ,

14.20.15 $\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(\cos\theta\right)=\frac{1}{\tau^{% \mu}}\left(\frac{\theta}{\sin\theta}\right)^{1/2}I_{\mu}\left(\tau\theta\right% )\*\left(1+O\left(\ifrac{1}{\tau}\right)\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\cos\NVar{z}$ : cosine function, $\mathrm{i}$ : imaginary unit, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\sin\NVar{z}$ : sine function, $\tau$ : real variable, $\mu$ : general order and $0<\theta<\pi$ : variable
Referenced by:: §14.20(vii)
Permalink:: http://dlmf.nist.gov/14.20.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(vii), §14.20 and Ch.14

14.20.16 $\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}\left(\cos\theta\right)=\frac{% 1}{\tau^{\mu}}\left(\frac{\theta}{\sin\theta}\right)^{1/2}K_{\mu}\left(\tau% \theta\right)\*\left(1+O\left(\ifrac{1}{\tau}\right)\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)$ : conical function, $\cos\NVar{z}$ : cosine function, $\mathrm{i}$ : imaginary unit, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\sin\NVar{z}$ : sine function, $\tau$ : real variable, $\mu$ : general order and $0<\theta<\pi$ : variable
Referenced by:: §14.20(vii)
Permalink:: http://dlmf.nist.gov/14.20.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(vii), §14.20 and Ch.14

uniformly for $\theta\in(0,\pi-\delta]$ , where and are the modified Bessel functions (§10.25(ii)) and $\delta$ is an arbitrary constant such that $0<\delta<\pi$ . For asymptotic expansions and explicit error bounds, see Olver (1997b, pp. 473–474). See also Žurina and Karmazina (1966).

§14.20(viii) Asymptotic Approximations: Large $\tau$ , $0\leq\mu\leq A\tau$

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Keywords:: asymptotic approximations, conical functions, large degree
Notes:: See Dunster (1991).
Referenced by:: §14.20(i), §2.8(vi)
Permalink:: http://dlmf.nist.gov/14.20.viii
See also:: Annotations for §14.20 and Ch.14

In this subsection and §14.20(ix), and $\delta$ denote arbitrary constants such that and $0<\delta<2$ .

As $\tau\to\infty$ ,

14.20.17 $\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\sigma(\mu,\tau)\left(% \frac{\alpha^{2}+\eta}{1+\alpha^{2}-x^{2}}\right)^{1/4}I_{\mu}\left(\tau\eta^{% 1/2}\right)\*\left(1+O\left(\ifrac{1}{\tau}\right)\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathrm{i}$ : imaginary unit, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, : real variable, $\tau$ : real variable, $\mu$ : general order, $\alpha$ , $\sigma(\mu,\tau)$ and $\eta$
Permalink:: http://dlmf.nist.gov/14.20.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(viii), §14.20 and Ch.14

14.20.18 $\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\sigma(\mu,\tau% )\left(\frac{\alpha^{2}+\eta}{1+\alpha^{2}-x^{2}}\right)^{1/4}K_{\mu}\left(% \tau\eta^{1/2}\right)\*\left(1+O\left(\ifrac{1}{\tau}\right)\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)$ : conical function, $\mathrm{i}$ : imaginary unit, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, : real variable, $\tau$ : real variable, $\mu$ : general order, $\alpha$ , $\sigma(\mu,\tau)$ and $\eta$
Referenced by:: §14.20(vii)
Permalink:: http://dlmf.nist.gov/14.20.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(viii), §14.20 and Ch.14

uniformly for $x\in[-1+\delta,1)$ and $\mu\in[0,A\tau]$ . Here

14.20.19 $\alpha=\mu/\tau,$

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Symbols:: $\tau$ : real variable, $\mu$ : general order and $\alpha$
Permalink:: http://dlmf.nist.gov/14.20.E19
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(viii), §14.20 and Ch.14

14.20.20 $\sigma(\mu,\tau)=\frac{\exp\left(\mu-\tau\operatorname{arctan}\alpha\right)}{% \left(\mu^{2}+\tau^{2}\right)^{\mu/2}}.$

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Symbols:: $\exp\NVar{z}$ : exponential function, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\tau$ : real variable, $\mu$ : general order, $\alpha$ and $\sigma(\mu,\tau)$
Permalink:: http://dlmf.nist.gov/14.20.E20
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(viii), §14.20 and Ch.14

The variable $\eta$ is defined implicitly by

14.20.21 ${\left(\alpha^{2}+\eta\right)^{1/2}+\tfrac{1}{2}\alpha\ln\eta-\alpha\ln\left(% \left(\alpha^{2}+\eta\right)^{1/2}+\alpha\right)}={\operatorname{arccos}\left(% \frac{x}{\left(1+\alpha^{2}\right)^{1/2}}\right)+\frac{\alpha}{2}\ln\left(% \frac{1+\alpha^{2}+\left(\alpha^{2}-1\right)x^{2}-2\alpha x\left(1+\alpha^{2}-% x^{2}\right)^{1/2}}{\left(1+\alpha^{2}\right)\left(1-x^{2}\right)}\right)},$

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Symbols:: $\operatorname{arccos}\NVar{z}$ : arccosine function, $\ln\NVar{z}$ : principal branch of logarithm function, : real variable, $\alpha$ and $\eta$
Permalink:: http://dlmf.nist.gov/14.20.E21
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(viii), §14.20 and Ch.14

where the inverse trigonometric functions take their principal values. The interval is mapped one-to-one to the interval $0<\eta<\infty$ , with the points and corresponding to $\eta=\infty$ and $\eta=0$ , respectively.

§14.20(ix) Asymptotic Approximations: Large $\mu$ , $0\leq\tau\leq A\mu$

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Keywords:: asymptotic approximations, conical functions, large order
Notes:: See Dunster (1991, Eqs. (5.11) and (5.14) have been corrected).
Referenced by:: §14.20(viii), 1st item, Erratum (V1.0.10) for References
Permalink:: http://dlmf.nist.gov/14.20.ix
Addition (effective with 1.0.10):: A sentence and reference to Dunster (2013) were added at the end of this section.
See also:: Annotations for §14.20 and Ch.14

As $\mu\to\infty$ ,

14.20.22 $\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{\exp\left(\mu\beta% \operatorname{arctan}\beta\right)}{\Gamma\left(\mu+1\right)\left(1+\beta^{2}% \right)^{\mu/2}}\frac{e^{-\mu\rho}}{\left(1+\beta^{2}-x^{2}\beta^{2}\right)^{1% /4}}\left(1+O\left(\frac{1}{\mu}\right)\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\operatorname{arctan}\NVar{z}$ : arctangent function, : real variable, $\tau$ : real variable, $\mu$ : general order, $\beta$ and $\rho$
Referenced by:: §14.20(ix), Erratum (V1.2.1) for Equation (14.20.22)
Permalink:: http://dlmf.nist.gov/14.20.E22
Encodings:: TeX, pMML, png
Correction (effective with 1.2.1):: On the right-hand side, we have removed an incorrect multiplicative factor of $\beta$ .
Suggested 2024-03-18 by Tianye Liu
See also:: Annotations for §14.20(ix), §14.20 and Ch.14

uniformly for $x\in(-1,1)$ and $\tau\in[0,A\mu]$ . Here

14.20.23 $\beta=\tau/\mu,$

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Symbols:: $\tau$ : real variable, $\mu$ : general order and $\beta$
Permalink:: http://dlmf.nist.gov/14.20.E23
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(ix), §14.20 and Ch.14

and the variable $\rho$ is defined by

14.20.24 $\rho=\frac{1}{2}\ln\left(\frac{\left(1-\beta^{2}\right)x^{2}+1+\beta^{2}+2x% \left(1+\beta^{2}-\beta^{2}x^{2}\right)^{1/2}}{1-x^{2}}\right)+\beta% \operatorname{arctan}\left(\frac{\beta x}{\sqrt{1+\beta^{2}-\beta^{2}x^{2}}}% \right)-\frac{1}{2}\ln\left(1+\beta^{2}\right),$

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Symbols:: $\operatorname{arctan}\NVar{z}$ : arctangent function, $\ln\NVar{z}$ : principal branch of logarithm function, : real variable, $\beta$ and $\rho$
Permalink:: http://dlmf.nist.gov/14.20.E24
Encodings:: TeX, pMML, png
See also:: Annotations for §14.20(ix), §14.20 and Ch.14

with the inverse tangent taking its principal value. The interval is mapped one-to-one to the interval $-\infty<\rho<\infty$ , with the points , , and corresponding to $\rho=-\infty$ , $\rho=0$ , and $\rho=\infty$ , respectively.

With the same conditions, the corresponding approximation for $\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(-x\right)$ is obtainable by replacing $e^{-\mu\rho}$ by $e^{\mu\rho}$ on the right-hand side of (14.20.22). Approximations for $\mathsf{P}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$ and $\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$ can then be achieved via (14.9.7) and (14.20.3).

For extensions to complex arguments (including the range $1<x<\infty$ ), asymptotic expansions, and explicit error bounds, see Dunster (1991). For the case of purely imaginary order and argument see Dunster (2013).

§14.20(x) Zeros and Integrals

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Keywords:: conical functions, integrals with respect to degree, zeros
Permalink:: http://dlmf.nist.gov/14.20.x
See also:: Annotations for §14.20 and Ch.14

For zeros of $\mathsf{P}_{-\frac{1}{2}+i\tau}\left(x\right)$ see Hobson (1931, §237).

For integrals with respect to $\tau$ involving $\mathsf{P}_{-\frac{1}{2}+i\tau}\left(x\right)$ , see Prudnikov et al. (1990, pp. 218–228).

14.19 Toroidal (or Ring) Functions 14.21 Definitions and Basic Properties

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§14.20 Conical (or Mehler) Functions

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