1、一个公式:(b>a>0) A=∫(0,∞)(e-ax-e-bx)(1/x)dx=lnb-lna. 2、发散变换:(不同解) A=∫(0,∞)e-ax(1/x)dx-∫(0,∞)e-bx(1/x)dx =∫(0,∞)e-y(1/y)dy-∫(0,∞)e-y(1/y)dy =0(出现错误,因为两个积分是发散积分) 3、收敛变换:(同解) (1)当k>0时, A=(k→0+)∫(0,∞)xk-1(e-ax-e-bx)dx =(k→0+)[∫(0,∞)xk-1e-axdx-∫(0,∞)xk-1e-bxdx] =(k→0+)[a-k∫(0,∞)yk-1e-ydy-b-k∫(0,∞)yk-1e-ydy] =(k→0+)(a-k-b-k)Γ(k)=(k→0+)(1/k)(a-k-b-k)Γ(k+1) (2)当(-1<k<0)时, A=(k→0-)∫(0,∞)xk-1(e-ax-e-bx)dx =(k→0-)[∫(0,∞)xk-1(e-ax-1)dx-∫(0,∞)xk-1(e-bx-1)dx] =(k→0-)[a-k∫(0,∞)yk-1(e-y-1)dy-b-k∫(0,∞)yk-1(e-y-1)dy] =(k→0-)(a-k-b-k)Γ(k)=(k→0-)(1/k)(a-k-b-k)Γ(k+1) 4、推广: (1)当(k>-1)时,A(k)=∫(0,∞)xk-1(e-ax-e-bx)dx=(a-k-b-k)Γ(k). (2)A(-1)不存在。由此得出,当(k≤-1)时,(1)中的A(k)积分式是发散的。 (3)当(-2<k<-1)时,A(k)=∫(0,∞)xk-1(e-ax+ax-e-bx-bx)dx=(a-k-b-k)Γ(k). (4)以此类推,由Γ(k)的定义式,可得出A(k)的定义式。A(k)=(a-k-b-k)Γ(k). |
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来自: toujingshuxue > 《数学》