B1.19

Methods of Mathematical Physics
Part II, 2002

State the Riemann-Lebesgue lemma as applied to the integral

abg(u)eixudu\int_{a}^{b} g(u) e^{i x u} d u

where g(u)g^{\prime}(u) is continuous and a,bRa, b \in \mathbb{R}.

Use this lemma to show that, as x+x \rightarrow+\infty,

ab(ua)λ1f(u)eixuduf(a)eixaeiπλ/2Γ(λ)xλ\int_{a}^{b}(u-a)^{\lambda-1} f(u) e^{i x u} d u \sim f(a) e^{i x a} e^{i \pi \lambda / 2} \Gamma(\lambda) x^{-\lambda}

where f(u)f(u) is holomorphic, f(a)0f(a) \neq 0 and 0<λ<10<\lambda<1. You should explain each step of your argument, but detailed analysis is not required.

Hence find the leading order asymptotic behaviour as x+x \rightarrow+\infty of

01eixt2(1t2)12dt\int_{0}^{1} \frac{e^{i x t^{2}}}{\left(1-t^{2}\right)^{\frac{1}{2}}} d t