Problem 1377 (difficulty: 10/10)

For \(\displaystyle x\in\R\) let \(\displaystyle \displaystyle I(x) = \int_{-\infty}^\infty \frac{e^{-t^2/2}}{\sqrt{2\pi}}\cos(xt)\dt\).

(a) Prove that \(\displaystyle I(x) \cdot I(y) = I\big(\sqrt{x^2+y^2}\big)\).

(b) Describe the behavior of \(\displaystyle I\) near \(\displaystyle 0\).

(c) \(\displaystyle I(x)=?\)

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