Legendre Function of the Second Kind

A solution to the Legendre Differential Equation which is singular at the origin. The Legendre functions of the second kind satisfy the same Recurrence Relation as the Legendre Functions of the First Kind. The first few are

$$\begin{eqnarray*} Q_0 &=& {\textstyle{1\over 2}}\ln\left({1+x\over 1-x}\right)\... ...r 4} \ln\left({1+x\over 1-x}\right)- {5x^2\over 2} + {2\over 3}. \end{eqnarray*}$$

The associated Legendre functions of the second kind have Derivative about 0 of

$$\left[{dQ_\nu^\mu(x)\over dx}\right]_{x=0} = {2^\mu\sqrt{\pi... ...over 2}}\nu-{\textstyle{1\over 2}}\mu+{\textstyle{1\over 2}})}$$

(Abramowitz and Stegun 1972, p. 334). The logarithmic derivative is

$$\left[{d \ln Q_\lambda^\mu(z)\over dz}\right]_{z=0} = 2\math... ... 2}}(\lambda+\mu-1)]![{\textstyle{1\over 2}}(\lambda-\mu-1)]!}$$

(Binney and Tremaine 1987, p. 654).

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Legendre Functions.'' Ch. 8 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 331-339, 1972.

Arfken, G. ``Legendre Functions of the Second Kind, $Q_n(x)$.'' Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 701-707, 1985.

Binney, J. and Tremaine, S. ``Associated Legendre Functions.'' Appendix 5 in Galactic Dynamics. Princeton, NJ: Princeton University Press, pp. 654-655, 1987.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 597-600, 1953.

Snow, C. Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. Washington, DC: U. S. Government Printing Office, 1952.

Spanier, J. and Oldham, K. B. ``The Legendre Functions $P_\nu(x)$ and $Q_\nu(x)$.'' Ch. 59 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 581-597, 1987.