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AGMC 2021 Prelim Q5

Source:

January 10, 2023

calculuscomplex analysisOdedifferential equation

Problem Statement

For the complex-valued function

f(x)

which is continuous and absolutely integrable on

\mathbb{R}

, define the function

(Sf)(x)

on

\mathbb{R}

:

(Sf)(x)=\int_{-\infty}^{+\infty}e^{2\pi iux}f(u)du

. (a) Find the expression for

S(\frac{1}{1+x^2})

and

S(\frac{1}{(1+x^2)^2})

. (b) For any integer

k

, let

f_k(x)=(1+x^2)^{-1-k}

. Assume

k\geq 1

, find constant

c_1

,

c_2

such that the function

y=(Sf_k)(x)

satisfies the ODE with second order:

xy''+c_1y'+c_2xy=0

.

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