反常积分的收敛判别法

l=0+∞∫∞+adxx)(ϕ∫∞+adxxf)([,)a+∞)()(0xKxfϕ≤≤K∫∞+adxx)(ϕ∫∞+adxxf)(∫∞+adxxf)(∫∞+adxx)(ϕCauchy∫∞+adxx)(ϕ0∀εaA≥∃00,AAA≥′∀KdxxAAεϕ∫′)(≤∫′AAdxxf)(εϕ∫′AAdxxK)(∫∞+adxxf)(Cauchy∫∞+adxxf)(00∃εaA≥∀00,AAA≥′∃εKdxxfAA≥∫′)(≥∫′AAdxx)(ϕ0)(1ε≥∫′AAdxxfK∫∞+adxx)(ϕ[,)a+∞0)(,0)(≥≥xxfϕ0)()(lim=+∞→xxfxϕ∫∫∫∞+adxxf)(∞+adxx)(ϕ∫∞+adxxf)(∞+adxx)(ϕ27821)(xxf=)20(1)(=pxxpϕ0)()(lim=+∞→xxfxϕ∫∞+1)(dxxf∫∞+1)(dxxϕ21p10≤p[,)a+∞0)(,0)(≥≥xxfϕ+∞=+∞→)()(limxxfxϕ∫∫∫∞+adxxf)(∞+adxx)(ϕ∫∞+adxxf)(∞+adxx)(ϕxxf1)(=)21(1)(=pxxpϕ+∞=+∞→)()(limxxfxϕ∫∞+1)(dxxf∫∞+1)(dxxϕ121≤p1pCauchy[,)a+∞⊂+∞(,)0fx()≥0KfxKxp()≤p1∫∞+adxxf)(fxKxp()≥p≤1∫∞+adxxf)([,)a+∞⊂+∞(,)0fx()≥0lim()xpxfxl→+∞=0≤+∞lp1∫∞+adxxf)(2790≤+∞lp≤1∫∞+adxxf)()(xϕpx111321xexdxx−++−+∞∫ln;∫∞++131tanarcdxxx;110++∞∫xxdx|sin|;xxdxqp11++∞∫.+∈Rqp,1+∞→x1ln123++−−xexx231x11321xexdxx−++−+∞∫ln2+∞→x31arctanxx+32xπ∫∞++131tanarcdxxx30≥xxxx+≥+11sin11dxx∫∞++011110++∞∫xxdx|sin|4+∞→xpqxx+1qpx−12801−qpxxdxqp11++∞∫xxdxqp11++∞∫fx())cpv(fxdx()−∞+∞∫fxdx()−∞+∞∫fxdx()−∞+∞∫)cpv(fxdx()−∞+∞∫0)(≥xf)cpv(fxdx()−∞+∞∫fxdx()−∞+∞∫)cpv(fxdx()−∞+∞∫+∞→Alim=)(AF+∞→Alim∫−AAdxxf)(Cauchy0∀ε00A∃0,AAA≥′∀ε−)'()(AFAF0,AAA≥′∀0',ABB≥∀≤∫′AAdxxf)(ε−)'()(AFAF≤∫−−BBdxxf')(ε−)'()(BFBF∫∞+0)(dxxf∫∞−0)(dxxffxdx()−∞+∞∫lnlnlnsinxxxdx2+∞∫;sinxxdxp1+∞∫;+∈Rp∫∞+1tanarcsindxxxxp;+∈Rpsin()xdx20+∞∫;∫∞+anmxdxxqxpsin)()(qnpxm()xn()mqx()n),[+∞∈ax1∫=AxdxAF2sin)(xxlnlnln),2[+∞0lnlnlnlim=+∞→xxxDirichletlnlnlnsinxxxdx2+∞∫281≥xxxsinlnlnlnxxx2sinlnlnln)2cos1(lnlnln21xxx−=∫∞+2lnlnlndxxx∫∞+22coslnlnlnxdxxx∫∞+2sinlnlnlndxxxxlnlnlnsinxxxdx2+∞∫21pppxxx1sin≤∫∞+11dxxp1psinxxdxp1+∞∫10≤p∫=AxdxAF1sin)(px1),1[+∞01lim=+∞→pxxDirichletsinxxdxp1+∞∫10≤p∫∞+1|sin|dxxxp10≤psinxxdxp1+∞∫31p≤pxxxarctansinpx2π∫∞+11dxxp1p∫∞+1tanarcsindxxxxp10≤p∫=AxdxAF1sin)(pxxarctan),1[+∞0arctanlim=+∞→pxxxDirichlet∫∞+1arctansindxxxxp10≤p∫∞+1sinarctandxxxxp10≤p∫∞+1arctansindxxxxp42xt==∫∞+02)sin(dxx∫∞+02sindttt∫∞+02sindtttsin()xdx20+∞∫28251+mnxxxqxpnmsin)()(2xK≤1+mn∫∞+anmxdxxqxpsin)()(1+=mn∫=AxdxAF1sin)(x)()(xqxpnm0)()(lim=+∞→xqxpnmxDirichlet∫∞+anmxdxxqxpsin)()(+∞→x)()(xqxpnmxa∫∞+1sin)()(dxxxqxpnm1+=mn∫∞+anmxdxxqxpsin)()(1+mnAxqxpnmx=+∞→)()(limA∞+∞−∫∞+anmxdxxqxpsin)()(fx()[,]abxb='3′Cauchy[,′)abfx()≥0x[b,b−η)b0KfxKbxp()()≤−p1fxdxab()∫fxKbxp()()≥−p≥1fxdxab()∫p1∫−bapdxxb)(1Cauchy2830∀ε0∃δ),0(',δηη∈∀Kdxxbbbpεηη−∫−−')(1≤∫−−')(ηηbbdxxfεηη−∫−−')(bbpdxxbKfxdxab()∫1≥p∫−bapdxxb)(1Cauchy00∃ε0∀δ),0(',δηη∈∃Kdxxbbbp0')(1εηη≥−∫−−≥∫−−')(ηηbbdxxf0')(εηη≥−∫−−bbpdxxbKfxdxab()∫Cauchy[,)abfx()≥0lim()()xbpbxfxl→−−=0≤+∞lp1fxdxab()∫0≤+∞lp≥1fxdxab()∫lim()()xbpbxfxl→−−=+∞≤lp0,10∃δ),(bbxδ−∈∀pxblxf)(1)(−+′lim()()xbpbxfxl→−−=+∞≤≥lp0,10∃δ),(bbxδ−∈∀pxblxf)(2)(−′′fxgxdxab()()∫284Abel[,fxdxab()∫gx())abDirichlet∫−=ηηbadxxfF)()(],0(ab−gx()[,)ab0)(lim=−→xgbx1CauchyGxg≤|)(|fxdxab()∫0∀ε0∃δ),(,bbAAδ−∈′∀GdxxfAA2)(ε∫′∫′AAdxxgxf)()(∫∫′⋅′+⋅≤AAdxxfAgdxxfAgξξ)()()()(∫∫′+≤AAdxxfGdxxfGξξ)()(εεε=+222MF≤|)(|η),[,baAA∈′∀MdxxfAA2)(∫′0)(lim=−→xgbx0∀ε0∃δ),(bbxδ−∈∀Mxg4)(ε∫′AAdxxgxf)()(∫∫′⋅′+⋅≤AAdxxfAgdxxfAgξξ)()()()(|)(|2|)(|2AgMAgM′+≤εεε=+22Cauchy∫∞+adxxgxf)()(112301xxdx()−∫;lnxxdx2011−∫;12202cossinxxdxπ∫;102−∫cosxxdxpπ;|ln|xdxp01∫;xxdpq−−−∫11011()x;∫−−−1011|ln|)1(dxxxxqp.132)1(1xx−321x)0(+→x32)1(1xx−31)1(1x−)1(−→x285112301xxdx()−∫21lnlim21−−→xxx21=10δ01lnlim20=−+→xxxxδ0xδxxx11ln2−lnxxdx2011−∫3xx22sincos121x)0(+→xxx22sincos12)2(1x−π)2(−→πx12202cossinxxdxπ∫4pxxcos1−221−px)0(+→x3p102−∫cosxxdxpπ3≥p102−∫cosxxdxpπ510δp0]|ln|[lim0=+→pxxxδ0xδxxp1lnpxlnpx−−)1(1)1(−→x1−p|ln|xdxp01∫1−≤p|ln|xdxp01∫611)1(−−−qpxxpx−11)0(+→x11)1(−−−qpxxqx−−1)1(1)1(−→x0,0qpxxdpq−−−∫11011()xxxxdpq−−−∫11011()7|ln|)1(11xxxqp−−−qx−−)1(1)1(−→x0|)]ln|)1(([lim11210=−−−−+→xxxxqppx0x21111ln)1(pqpxxxx−−−−1,0−qp∫−−−1011|ln|)1(dxxxxqp∫−−−1011|ln|)1(dxxxxqp286xxxdxpq−−−∫1101ln;+∈Rqp,112230xxxdx()()−−+∞∫;ln()10++∞∫xxdxp;∫∞+0tanarcdxxxp;∫2/0tanπdxxxp;xdpx−−+∞∫10ex;10xxdxpq++∞∫;∫∞+2ln1dxxxqp.1xxxdxpq−−−∫1101ln∫−=2101lndxxxp∫−−2101lndxxxq∫−−−+12111lndxxxxqp0p0q∫−2101lndxxxp∫−2101lndxxxq−→1x=−−−xxxqpln11()[]()[]())1(1ln1)1(11)1(111−+−−+−−−+−−xxxqpqpxxqp−=−−−1)1)((∫−−−12111lndxxxxqpxxxdxpq−−−∫1101ln2=−−∫∞+032)2()1(1dxxxx∫−−1032)2()1(1dxxxx∫−−+2132)2()1(1dxxxx∫∞+−−+232)2()1(1dxxxx32)2()1(1−−xxx313121x⋅−)0(+→x32)2()1(1−−xxx32)1(1−−x)1(−→x∫−−1032)2()1(1dxxxx28732)2()1(1−−xxx32)1(1−−x)1(+→x32)2()1(1−−xxx313)2(121−⋅x)2(−→x∫−−2132)2()1(1dxxxx32)2()1(1−−xxx313)2(121−⋅x)2(+→x32)2()1(1−−xxx341x)(+∞→x∫∞+−−232)2()1(1dxxxx112230xxxdx()()−−+∞∫3=+∫∞+0)1ln(dxxxp++∫10)1ln(dxxxp∫∞++1)1ln(dxxxppxx)1ln(+11−px)0(+→x2p∫+10)1ln(dxxxp2≥p∫+10)1ln(dxxxp1p0)1ln(lim213=⎥⎥⎦⎤⎢⎢⎣⎡+⋅−+∞→ppxxxx0x2131)1ln(−+ppxxx1213−p1p∫∞++1)1ln(dxxxp1≤p∫∞++1)1ln(dxxxp28821p∫∞++0)1ln(dxxxp∫∞++0)1ln(dxxxp4∫∞+0tanarcdxxxp∫=10tanarcdxxxp∫∞++1tanarcdxxxppxxarctan11−px)0(+→x,2p∫10tanarcdxxxppxxarctanpx2π)(+∞→x1p∫∞+1tanarcdxxxp21p∫∞+0tanarcdxxxp∫∞+0tanarcdxxxp5∫2/0tanπdxxxp∫=4/0tanπdxxxp∫+2/4/tanππdxxxppxxtan211−px)0(+→x23p∫4/0tanπdxxxp23≥p∫4/0tanπdxxxppxxtan122()2ppxππ−)2(−→πx∫2/4/tanππdxxxp23p∫2/0tanπdxxxp23≥p∫2/0tanπdxxxp6xdx∫−−=101edxxxppx−−+∞∫10e∫∞+−−+11edxxxp∫∞+−−11edxxxpxpex−−