
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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:  $xy''+c_1y'+c_2xy=0$ .

Present the answer in LaTex format: \boxed{Your answer}
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