第二十八讲 Green函数( 第5页 当积分体积V内包含原点r=0时,由于函数1/r在r=0点不可导,上面的结果不成立这 时不妨将V就取为整个(三维)空间.容易得到 / r2 sin eded 令r=atanθ,即可证明上面的积分与a无关,且 20 cos 0de 2 12m·sin3Wu Chong-shi ❩❬❭❪❫ Green èä (❴) é 5 ê ➪Ú➩➇ Ú V ➡ ➈➉➊ò r = 0 ï✚❅➹➾➚ 1/r ↕ r = 0 òÜ★ ◆✚➞⑦ ➦➌➍Ü❍➎✛ ✲ ï Ü➏➐ V à❂ ➲➑➳ (⑨⑩) ❶ ➠ ✛➒➓ðñ ZZ Z ∇2 1 r dxdydz = lim a→0 Z ZZ ∇2 1 √ r 2 + a 2 dxdydz = − lim a→0 ZZ Z 3a 2 (r 2 + a 2) 5/2 r 2dr sin θdθdφ = − 12π lima→0 Z ∞ 0 a 2 (r 2 + a 2) 5/2 r 2dr, ➔ r = a tan θ ✚✽★ ❻ ❼➞⑦ ➦Ú➩✖ a ➙✛ ✚õ ZZ Z ∇ 2 1 r dxdydz = −12π Z π/2 0 tan2 θ ￾ 1 + tan2 θ
3/2 dθ = −12π Z π/2 0 sin2 θ cos θdθ = −12π · 1 3 sin3 θ     π/2 0 = −4π.