Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 336 / 1

Write the Schläfli contour integral for Legendre polynomials asThe saddle points satisfyFor $\mu =\cos \theta$, they are $z_{\pm }=e^{\pm i\theta }$. Deforming the contour through both conjugate saddles and using the local Gaussian contributions from the method of steepest descent gives conjugate exponentials. Their sum is the Debye asymptotic for Legendre polynomialsBoth saddles are required for the real oscillatory answer.

Put $s=|\xi |$. Since $\cosh \xi =\cosh s$, the saddle points are $e^{\pm s}$. The contour deformation relevant to the prescribed contour passes through the dominant saddle $z=e^s$. There,The quadratic saddle factor and the remaining amplitude give the Hyperbolic Debye asymptotic for Legendre polynomialsThe second saddle gives an exponentially subdominant contribution on this contour. At $\xi =0$ the saddles coalesce and this formula is nonuniform; the exact endpoint value is $P_n(1)=1$.