# conical functions

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##### 1: 14.20 Conical (or Mehler) Functions
###### §14.20(ii) Graphics Figure 14.20.8: Q ^ - 1 2 + i ⁢ τ - 2 ⁡ ( x ) , τ = 0 , 1 2 , 1 , 2 , 4 . Magnify
##### 2: 14.31 Other Applications
###### §14.31(ii) ConicalFunctions
The conical functions $\mathsf{P}^{m}_{-\frac{1}{2}+i\tau}\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). …The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …
##### 3: 14.1 Special Notation
###### §14.1 Special Notation
The main functions treated in this chapter are the Legendre functions $\mathsf{P}_{\nu}\left(x\right)$, $\mathsf{Q}_{\nu}\left(x\right)$, $P_{\nu}\left(z\right)$, $Q_{\nu}\left(z\right)$; Ferrers functions $\mathsf{P}^{\mu}_{\nu}\left(x\right)$, $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P^{\mu}_{\nu}\left(z\right)$, $Q^{\mu}_{\nu}\left(z\right)$, $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$; conical functions $\mathsf{P}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $\mathsf{Q}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $\widehat{\mathsf{Q}}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $P^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $Q^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$ (also known as Mehler functions). …
##### 4: 14.33 Tables
• Žurina and Karmazina (1964, 1965) tabulate the conical functions $\mathsf{P}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ for $\tau=0(.01)50$, $x=-0.9(.1)0.9$, 7S; $P_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ for $\tau=0(.01)50$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7D. Auxiliary tables are included to facilitate computation for larger values of $\tau$ when $-1.

• Žurina and Karmazina (1963) tabulates the conical functions $\mathsf{P}^{1}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ for $\tau=0(.01)25$, $x=-0.9(.1)0.9$, 7S; $P^{1}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)$ for $\tau=0(.01)25$, $x=1.1(.1)2(.2)5(.5)10(10)60$, 7S. Auxiliary tables are included to assist computation for larger values of $\tau$ when $-1.

• ##### 5: 14.32 Methods of Computation
• For the computation of conical functions see Gil et al. (2009, 2012), and Dunster (2014).

• ##### 6: 14.23 Values on the Cut
The conical function defined by (14.20.2) can be represented similarly by
14.23.7 $\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\tfrac{1}{2}e^{% 3\mu\pi i/2}Q^{-\mu}_{-\frac{1}{2}+i\tau}\left(x-i0\right)+\tfrac{1}{2}e^{-3% \mu\pi i/2}Q^{-\mu}_{-\frac{1}{2}-i\tau}\left(x+i0\right).$
##### 7: Bibliography D
• T. M. Dunster (1991) Conical functions with one or both parameters large. Proc. Roy. Soc. Edinburgh Sect. A 119 (3-4), pp. 311–327.
• T. M. Dunster (2013) Conical functions of purely imaginary order and argument. Proc. Roy. Soc. Edinburgh Sect. A 143 (5), pp. 929–955.
• ##### 8: Bibliography G
• A. Gil, J. Segura, and N. M. Temme (2009) Computing the conical function $P^{\mu}_{-1/2+i\tau}(x)$ . SIAM J. Sci. Comput. 31 (3), pp. 1716–1741.
• A. Gil, J. Segura, and N. M. Temme (2012) An improved algorithm and a Fortran 90 module for computing the conical function $P^{m}_{-1/2+i\tau}(x)$ . Comput. Phys. Commun. 183 (3), pp. 794–799.
• ##### 9: 3.8 Nonlinear Equations
For the computation of zeros of Bessel functions, Coulomb functions, and conical functions as eigenvalues of finite parts of infinite tridiagonal matrices, see Grad and Zakrajšek (1973), Ikebe (1975), Ikebe et al. (1991), Ball (2000), and Gil et al. (2007a, pp. 205–213). …
##### 10: Bibliography K
• K. S. Kölbig (1981) A Program for Computing the Conical Functions of the First Kind $P^{m}_{-1/2+i\tau}(x)$ for $m=0$ and $m=1$ . Comput. Phys. Comm. 23 (1), pp. 51–61.