1/8 平成 23 年3 月24 日午後 6 時11 分 複素微分： …

1 1/8 23 3 24 6 11 z ()fz z 2 ,xy zxiy=+ ()fz 2 ()(),, ,uxyxyv () ( )(),,fzuxyixy=+v z z 2 ,xy zixy=+ z z ()fz 2 ,xy 2 ()(),, ,uxyxyv ()()(),,fzuxyxiy=+v ()fz ()(),, ,uxyxyv I. z ()fz () (),,iuxyxy+v ()fz ()(),, ,uxyxyv z ()fz ()fz ()()()0limzfzzfzfzz + = ( ) ()()(),,zxiyzxiyfzuxy xy =+ = + =+ v ( ) ()()()()()0,,,,limzu x xy y i x xy yu xy i xyfzxiy + + + + + + = + vv ( )

2 2/8 23 3 24 6 11 0z z 0z 0y = 1a ()()()()()0,,,lim,xux xiixyuxyyzxxxyf ++ + + = vv ()()()()()()00limli,,m,,,,xxxiixxyuxyxyu x xy uxyuxyxxx = = ++ vv ()()(),,uxyxixzyfx =+ v ( ) z 0z 0x = 1b ()()()()()0,,,lim,yxyuxy yuyxiiyfzixyy + + = ++ vv ()()()()()()()()1100,,, ,limlim,,,,iiiiyyiiyuxy y uxyuxyuxyyyixxyuxyxyxyyiiiyiy = = + == + == vvv ()()(),,uxyyyfz ixy = + v ( ) ( ) ( ) ()fz ()()()()(),, ,,u xyxyu xyxyfziixxyy =+ = + vv ( ) ()()()(),,,,xyuxyxuxyxyxyyii+= + vv ( ) xiy+ 00xy = 1b 1a xiy+ 00xy = 3/8 23 3 24 6 11 ()()()(),,,,xyu xuxyyyyyxxx ==vv ( )

3 Cauchy-Riemann ()()(),,uxyxyfzixx =+ v ( ) ()fz z ()(),, ,uxuxyv 1 z ()(),, ,uxuxyv z () (),,uxy i xy+v ()zxiy=+ ()(),,uxy i xy+v x,y z,z* z ()()()()()PP()212,22222,2 22,,22izzuxy x yiixiy yix xiyyixxyx yxuxxyxiyxiiiyziyxyyz = =+ + =+ =+++=+ + =+ =+ + =+++ vv ( ) () (),,wuxy i xy=+v ( ) w ()(),,uxyxydwidzxx =+ v ( ) w () (),,wuxy i xy=+v ( )

4 W wz 0limzdwwdzz = ( ) w 4/8 23 3 24 6 11 ()(),,wwuxxyyixxyy+ = + + + + + v ( ) ()()()(),,,,ww uxxyyixwwuxyixyxyy+ + + + + + = = + vv ( ) w wz w w Real part Imarginary part ()()()()()()()(),,,, ,,,,w u x xy y i x xy yu xy i xyu x xy y u xyxxyxyiy = + + + + + + =+ + + + + vvvv ( ) ()()()()()()ReIm,,,,xxy yxyux xy y uxyww = + + =+ + vv ( ) ()Rew ()(),,0uxyyuxyy + ++ = ()()()()()()()0,Re,,,=,,uxwuxxyyuxyuxxyy uyyuxyyxy= + + = + + + ++ ( )()()()

5 =,,,,u x xy y u xy y u xy y u xy+ + + ++ ( ) 1c A x xx+ () 01xtxt+ < < 01ttxxtx x x== + + ()fxtx + 1d ()()( )fxxfxfxtxx + =+ + ()()()() 01fx x fxttfx xx+ + = << x y 2 ()fx xxx+ () 01xtxt+ x ()fx()fx()fxtxx + ()fxx+ xxx+ () 01xtxt+ 0t=1t=()fxtx= + x ()fxxxxt =+ A 1c 1d5/8 23 3 24 6 11 ()()()(),,,,yxuyuyuxuyyyxxxx = = ( ) ()()()()()()()(),,, 01,,, 01xxyyxyuyyuyyyyyt yyyuyytuxuxuxxxxt xxxt+ + + =< <+ + =< + <+ ( ) ( ) ()()()()()()()()Re 0,1,,,,,,xyxyxuxy y uxyuxyty yux xy y uxy yuxtwttxy y x =+=+ + +<+ + + ++ < ( ) () ()()()Re,, 0,1xxyxyw uxtxy yxuxytyytt = + + + + < < ( ) ( ) ()Imw () ()()()Im,, 0,1xxyyxywxsxyyxxysyyss = + + + + < <vv ( ) ()()ReImwwiw = + ()()()()()()

6 Im ,,Re,,xxyxxyywuxtxy y xuxytwxsxy y xxyyywsiyiy + + ++ + + ++=+ =+ vv ( ) ,0, 0xy xdxydywdw ( ) ,0,0lim limxyxxxydwwux tx = + =,yy+ (),yux tyxy ++ ()xxix xys+ + v,yy+ (),yyxy s yx ++ v()() ()() () ,,,,xyx ydxydyduxyuxyidxyxyxy =++ + vv ( ) ()(),, ,uxuxyv 6/8 23 3 24 6 11 ()()()()() ()() ()()()()(),,,,,,, ,,,,,xyxxyyyxuxyxyuxyxyuxyxyxyxuxuxyxyyu xyxyyyx = == = = = vvvvvv ( ) ,yyuv ,xxuv ()()()()()()()()() () () ()() ()P() () ()()() ()1,, ,, ,,,, ,,,, ,,,,, , ,xxxxxx x xxxyxxxxxxxyxdw u x y dxdy ix y dxdyu x y dxdy ix y dxdyu xydxxydy i xydx iu xydyuxydx ii xydyi xydxiuxyduxyxyxyuxyixyuxyiuydxdyxxdiyi =++ + =+ + + = + += +++ +=+vvvvvvvvvv()()()()()() () () () ,,,,,,,dzdxzxx xxxdydx idydx idyu xydz i xydz u xy i xy dxxzyuyixy=++ +=+ =+ vvv ( ) () (),,xxdwu x yix y dz=+ v ( )

7 Dz () (),,xxdwuxy i xydz=+v ( ) ()(),, ,xxuxyxyv ( ) ()(),,uxyxydwidzxx =+ v ( ) ( ) ()(),,wuxy i xy=+v z () (),,uxy i xy+v z ()()(),,fzuxyixy=+v z ()fz ()()()(),,, ,, uxyxyxyuxyxyx y == vv z ()()()(),,, ,, uxyxyxyuxyxyx y == vv 7/8 23 3 24 6 11 () (),,uxy i xy+v ()zxiy=+ II.

8 2 ()2fzz= ()2fzzz z == z ()2fzz= z ()2fzzz z == ()2fzz= 2 ()( ),, ,uxyxyv ()( )( )()()2222222,,2fz zx iy xxiy ixyyiixxuxyyy==+ =+ + =+= + v ( ) ()()222,,xuxyyyxyx = = v ()()()()()()()()()()()()()2222222,2,,,2, 2,,,22222uxyxuxyxyxxxxyxyxyyyxyxyyxxyxyx xxxyu xyxyuxyyyyxyyyyxyxy === = === === = ==== vvvv ( ) ()2fzz= z ()()2fzzz z == 8/8 23 3 24 6 11 2 ()(),, ,uxyxyv ()( )()()()()()222,,fzz xiyxx iy iyxyuxyixzxiyiyxiyy ==+= ++=+= + v ( ) ()()22,,0uxy x yxy =+ = v ()()()()()()()()()()()()222222,2,,,00,00 ,,,2xyuxyxxuxyxyxxxxyxyyyxyxxxyu xyxyxyuxyyyyyy + === = == == = + === vvvv ( ) ()()2fzzz z == z