含參量積分的分析性質及其應用

.z.- 含參量積分的分析性質及其應用定理1假設二元函數(shù)f(x,y)在矩形區(qū)域R[a,b][c,d]上連續(xù),則函數(shù)c000y0-1y>1y0y1.z.-議)0一1議)00性定理得j1性定理得j122x022議)0003例3研究函數(shù)F(x)=j1yf(x)dx的連續(xù)性,其中f〔*〕在閉區(qū)間[0,1]上是正x2+y2x2+y200000ym0x2+y20x2+y2yy)0+4c(x).z.-a)001+x240tdtjetdt00-y0-y2-y-y0ycdxcc?xxc(x)xfxcxcxc(x)xxF'(y)=djb(y)f(x,y)dx=jb(y)f(x,y)dx+f[b(y),y]b'(y)-f[a(y),y]a'(y).dyaya(y)y0o0a(y)b(y)a(y)0o0i0F'(y)=jb(y0)f(x,y)dx.010a(y)y00由于F(y)=0,所以20-F';(y)=limF(y)-F(y)22o=limF(y)2=limjb(y)f(x,y)dx.20y)yoy-y0y)y0y-y0y)y0b(y0)y-y0by20y)yy-y000F'(y)=b'(y)f[b(y),y].20000yxy解由于〔1〕中被積函數(shù)F(x,t)=(x-t)n-1f(t)及其偏導數(shù)F(x,t)在U上連x于是Q(n)(x)=f(x).a-a4ca這就是說：在f(x,y)連續(xù)性假設下,同時存在求積順序不同的積分：ca.z.-acccaaccca0lnx0lnx0(x)lnx0(x)lnx.z.- (x)lnxalnx (x)lnxalnxx)0+x)1=jbdyj1sin(|ln1)|xydxa0(x)a0=0(x)=j+wf(x,y)dycc0.0-000x)0+0x)0+0x)o+0x)0+0x)0+x)0+0x)o+0x)0+0x)0+00x)0+0aa證明由于j+wf(x,u)dx在[a,b]上一致連續(xù),故對任意c>0,存在A>a,使得a0A30000caa00aa0jA0f(x,u)dx-jA0f(x,u)dx︱+︱j+wf(x,u)dxaa0A00a0-00.解先看函數(shù)()的定義域是什么,即上述積分在什么圍收斂.在*=0附xaxxa1xb1xaxxa1xb1-2<a,arctanx11.而a+3>1,故有比較判別法,積分x(2x3)dx2xa32xa3.z.ncncnnccxabcxC(x)=j+wf(x,y)dy在I上收斂,j+wf(x,y)dy在I上一致收斂,則C(x)在I上可微,ccx且C'(x)=j+wf(x,y)dy.cx0x20x20x0x20y0200因而j+w?(x3e一x2y)dy在[0,10xC(x)=j+wf(x,y)dy在I上收斂,而j+wf(x,y)dy在I上閉一致收斂,則C(x)在I上ccxc可微,且C'(x)=j+wf(x,y)dy..z.-n且I'(x)=xwu'(x)=j+wf(x,y)dy.ncxdxcc?xdxcc?xaa?unana?una?un0x200-02aIaacIbcbIa"(b-a).2.3含參量反常積分的可積性cjbdxj+wf(x,y)dy=j+wdyjbf(x,y)dx.accaaccaacac