Cone function

The cone functions are special spherical functions that were introduced by Gustav Ferdinand Mehler in 1868 as a solution to the problem of determining the potential of electrical charges distributed on a conical surface .

definition

The functions originally introduced by Mehler can be represented by spherical functions, namely assigned Legendre polynomials of the first and second kind with a special complex index: ${\displaystyle K^{\mu }(x)}$

${\displaystyle P_{-(1/2)+i\lambda }^{\mu }(x)}$and . ${\displaystyle Q_{-(1/2)+i\lambda }^{\mu }(x)}$

It is real and arbitrary. Accordingly, these two special Legendre functions are called cone functions today. The following applies in particular with and : ${\displaystyle \lambda }$ ${\displaystyle \mu }$ ${\displaystyle x=\cos(\theta )}$ ${\displaystyle \mu =0}$

${\displaystyle P_{-(1/2)+i\lambda }(\cos \theta )={\frac {2}{\pi }}\int _{0}^{\theta }{\frac {\cosh(\lambda t)}{\sqrt {2(\cos t-\cos \theta )}}}dt}$

${\displaystyle Q_{-(1/2)\mp i\lambda }(\cos \theta )=\pm i\sinh \lambda \pi \int _{0}^{\infty }{\frac {\cos(\lambda t)}{\sqrt {2(\cosh t+\cos \theta )}}}dt+\int _{0}^{\infty }{\frac {\cosh(\lambda t)}{\sqrt {2(\cosh t-\cos \theta )}}}dt}$

literature

• Abramowitz, Stegun: Handbook of Mathematical Functions . Dover 1972, p. 337 (Section 8.12)
• GF Mehler: About the distribution of static electricity in a body delimited by two spherical caps . In: Journal for pure and applied mathematics , Volume 68, 1868, p. 134, online
• GF Mehler: About a function related to the ball and cylinder functions and its application in the theory of electricity distribution . In: Mathematische Annalen , Volume 18, 1881, p. 161, online
• Carl Gottfried Neumann : About Mehler's cone functions and their application to electrostatic problems . In: Mathematische Annalen , Volume 18, 1881, p. 195, online
• G. Leonhardt: Integral properties of the adjoint cone functions . In: Mathematische Annalen , Volume 19, 1882, p. 578, online

Web links

• Eric W. Weisstein : Conical Function . In: MathWorld (English).