State the Riemann-Lebesgue lemma as applied to the integral
∫abg(u)eixudu
where g′(u) is continuous and a,b∈R.
Use this lemma to show that, as x→+∞,
∫ab(u−a)λ−1f(u)eixudu∼f(a)eixaeiπλ/2Γ(λ)x−λ
where f(u) is holomorphic, f(a)=0 and 0<λ<1. You should explain each step of your argument, but detailed analysis is not required.
Hence find the leading order asymptotic behaviour as x→+∞ of
∫01(1−t2)21eixt2dt