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Conical Function

Functions which can be expressed in terms of Legendre Functions of the First and Second Kinds. See Abramowitz and Stegun (1972, p. 337).

$\displaystyle P^\mu_{-1/2+ip}(\cos\theta)$ $\textstyle =$ $\displaystyle 1+{4p^2+1^2\over 2^2} \sin^2({\textstyle{1\over 2}}\theta)$  
  $\textstyle \phantom{=}$ $\displaystyle + {(4p^2+1^2)(4p^2+3^2)\over 2^24^2}\sin^4({\textstyle{1\over 2}}\theta)+\ldots$  
  $\textstyle =$ $\displaystyle {2\over\pi} \int_0^\theta {\cosh(pt)\,dt\over \sqrt{2(\cos t-\cos\theta)}}$  
$\displaystyle Q^\mu_{-1/2\mp ip}(\cos\theta)$ $\textstyle =$ $\displaystyle \pm i\sinh(p\pi)\int_0^\infty {\cos(pt)\,dt\over\sqrt{2(\cosh t+\cos \theta)}}$  
  $\textstyle \phantom{=}$ $\displaystyle + \int_0^\infty {\cosh(pt)\,dt\over \sqrt{2(\cos t-\cos\theta)}}.$  

See also Toroidal Function


Abramowitz, M. and Stegun, C. A. (Eds.). ``Conical Functions.'' §8.12 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 337, 1972.

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1464, 1980.

© 1996-9 Eric W. Weisstein