求泊松积分公式

Question

求泊松积分公式

Answer 1

设 I＝ 泊松积分 ＝ (0,∝ )∫[e^(-x^2)] dx I^2 = {(0,∝ )∫[e^(x^2)] dx }*{(0,∝ )∫[e^(y^2)] dy= (积分区间D )∫∫[e^(-x^2 - y^2 )] dxdy (面积分）=> [ 积分变换 ρ^2 = x^2 + y^2 ,dxdy = ρdρdθ ,D:0 ≤ρ≤ + ∝ ,0 ≤θ≤ π/2 ]= (积分区间D )∫∫[e^(-ρ^2) ] ρdρdθ (面积分）= {(0 ≤θ≤ π/2 )∫dθ}{(0 ≤ρ≤ + ∝ )∫[e^(-ρ^2)ρdρ ] }= (π/2)* (1/2)故 I ＝ 泊松积分 = (√π)/2