Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 30D (Asymptotic Methods)

Write the contour near the simple saddle aswith its given orientation. Since ${\varphi }^{\prime }(z_0)=0$,Along a steepest-descent tangent the quadratic coefficient is real and negative, soMoreover $dz=e^{i\alpha }ds$. Only a neighbourhood of width $O(x^{-1/2})$ contributes at leading order, and there $f(z)=f(z_0)+o(1)$. The local integral is consequentlyEvaluating the Gaussian proves the simple-saddle contribution in steepest descent:

Factor the integrand in the Schläfli contour integral for Legendre polynomials asThus one may takewith a branch of the logarithm chosen locally along the deformed contour. Differentiation givesFor $t=\cos \theta$, its saddle points solveand hence

At a saddle the numerator of ${\varphi }^{\prime }$ vanishes, so differentiating the quotient there givesSincewe obtain

The tangent angle $\alpha$ must satisfyAn anticlockwise deformation of the original contour through both saddles can be oriented withAngles differing by $\pi$ describe the same unoriented tangent; the displayed choices give the orientation needed to pass through the lower saddle and then the upper saddle while still enclosing $t$.

At the two saddles,andApplying the method of steepest descent with the oriented angles from part (ii), and including the prefactor $1/(2^{n+1}\pi i)$, gives two conjugate contributions. Iftheir sum isTherefore the Debye asymptotic for Legendre polynomials isThus