Calculation of γ is related to survival probability of a random walk,
originating at the tip, and in the presence of absorbing boundaries.
"A guide to first-passage processes," Sidney Redner, (CUP, NY, 1983)
"Kinetics of first-passage in a cone," E. Ben-Naim, P.L. Krapivsky, J. Phys. A 43, 495007 (2010)
The exponent η is the solution to the eigenvalue equation
with ψ(θ)=0 on the absorbing surfaces. The solution, related to zeros of associated Legendre functions, yields