Laplace-Mehler Integral

$$P_n(\cos\theta) = {1\over\pi}\int_0^{2\pi} (\cos\theta+i\sin\theta\cos\phi)^n\,d\phi$$

$$= {\sqrt{2}\over\pi}\int_0^\theta {\cos[(n+{\textstyle{1\over 2}})\phi]\over\sqrt{\cos\phi-\cos\theta}}\,d\phi$$

$$= {\sqrt{2}\over\pi}\int_\theta^\pi {\sin[(n+{\textstyle{1\over 2}})\phi]\over\sqrt{\cos\theta-\cos\phi}}\,d\phi.$$

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1463, 1980.