Mathematics, 24.04.2020 18:10, musicaljay8320

Exercise II: Gaussian integral For K > 0, define RK = [−K, K] × [−K, K], DK = {(x, y) ∈ R 2 , x2 + y 2 ≤ K2 }. 1. Compute Z Z DK e −x 2−y 2 dA. 2. Argue that Z Z DK e −x 2−y 2 dA ≤ Z Z RK e −x 2−y 2 dA ≤ Z Z D√ 2K e −x 2−y 2 dA. 3. Deduce the value of Z +[infinity] −[infinity] e −x 2 dx. Recall that, by definition, Z +[infinity] −[infinity] e −x 2 dx = lim K→+[infinity] Z K −K e −x 2 dx. Be clear and precise and use a famous result to compute limits!

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Exercise II: Gaussian integral For K > 0, define RK = [−K, K] × [−K, K], DK = {(x, y) ∈ R 2 , x2...