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double integral with exp is polynomial

Source: Putnam 1982 B2

September 27, 2021
calculusintegrationalgebrapolynomial

Problem Statement

Let A(x,y)A(x,y) be the number of points (m,n)(m,n) in the plane with integer coordinates mm and nn satisfying m2+n2≤x2+y2m^2+n^2\le x^2+y^2. Let g=∑k=1∞e−k2g=\sum_{k=1}^\infty e^{-k^2}. Express ∫−∞∞∫−∞∞A(x,y)e−x2−y2dxdy\int^\infty_{-\infty}\int^\infty_{-\infty}A(x,y)e^{-x^2-y^2}dxdy as a polynomial in gg.
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