14 Z05anb 枝点z=0和z=∞,割线如图红线 上岸θ≡ag=丌,下岸θ=-r 积分回路如图,回路内有三个单极点 z=±ai=ae2和z=1 Res glai= lim (=ai=aetna) Res g(-ai)= limI/In a、4r =2+cy 2(aeiar) ai(2 Ina-im RsgD=lm2+).-1,:=1是hnE的阶0点,是的单极点 xd3,|g(=)d: (x2+a2)(nx+in) Jo(x2+a2)(nx+ir) x2+a2)(nx-i) (r2+a)(nx-ir) g()d=+ g()d==-2xil 由大圆弧、小圆弧引理:f(=)d==0,|f()d== 12丌i 综上:= ai 41n2a+72 a2+1 番 Mathematica(至少102版)只能做数值积分,不妨试试更高版本,iany INtegrate /.a→a0,x,0,∞}, NorkingPrecision→20 (a2+x2)(Log【x]2+m2) /.a→a0 a立4og[a]2+n2a2+1 0.26572188849733879316 5m2+4Log[2] 0.26572188849733879318枝点 z = 0 和 z = ∞，割线如图红线 ， 上岸 θ ≡ arg z = π，下岸 θ = -π 积分回路如图 ，回路内有三个单极点 z = ±a  = a  ± π 2 和 z = 1 L1 +Cr +L2 +CR = 2 π  k Res g(zk) Res g(a ) = lim za  1/ln z z2 + a2 ′ z = a  = a  π/2 = 1 ln a +  π 2 2 a  π/2 = 1 a  (2 ln a +  π) x y CR -R L1 R L2 Cr a  -a  1 Res g(-a ) = lim z-a  1/ln z z2 + a2 ′ = 1 ln a -  π 2 2 a - π/2 = - 1 a  (2 ln a -  π) Res g(1) = lim z1 1 z2 + a2 (ln z)′ = 1 a2 + 1 ，z = 1 是 ln z 的一阶 0 点，是 g(z) 的单极点 L1：z = x  π, L1 g(z) z = ∞ 0  π x x2 + a2 (ln x +  π) = 0 ∞ x x2 + a2 (ln x +  π) L2：z = x - π, L2 g(z) z = 0 ∞ - π x x2 + a2 (ln x -  π) = -0 ∞ x x2 + a2 (ln x -  π) L1 g(z) z + L1 g(z) z = -2 π  I 由 大圆弧、小圆弧引理 ：CR f (z) z = 0， Cr f (z) z = 0 综上： I = 1 a  2 π  4 ln2 a + π2 - 1 a2 + 1  Mathematica 至少 10.2 版 只能做数值积分 ，不妨试试更高版本 ，if any a0 = 1 / 2; NIntegrate 1 (a2 + x2) Log[x]2 + π2 /. a  a0, {x, 0, ∞}, WorkingPrecision  20 1 a  2 π  4 Log[a]2 + π2 - 1 a2 + 1 /. a  a0 N[%, 20] 0.26572188849733879316 - 4 5 + 4 π π2 + 4 Log[2]2 0.26572188849733879318 14 z05a.nb