物理数学(一)期中考.pdf

Use Cauchy Riemann equation yxiyxzzzzf Im Re iyyx 2 ivu 1 2 y x u u 1 0 y x v v i analyticnon 1 物理數學 一 期中考 222 22 22222 2 1 2 yx xy yxyx x x u 222 22 2222 2 1 2 yx xy yxyx y y v 222 222 2 2 yx xy x u yx xy y u vyux vxuy analytic 1 22 22 yx y v yx x u ii 22 22 22 2 2 2 2 1 1 yx iyx xyiyx iyxiyx iyx yxyix iyx iyx z zzf Here since is analytic only the last term was checked xyiyx2 22 2 bu au yy xx 2 2 yx vaxu baba 2 22 22xhaxyaxdyv byv vbyu x xy 2 2 cxybaxh yba x xh by x xh ay 2 2 2 2 cbxycxybaaxyv 2 22 caxy cbxy 2 2 2 2222 2 2 iyxa axyiyxaaxyiayaxzf ax x u byaxu 2 22 Ignore the constant we have 3 2 ln 2 1 exp ln 2 1 exp 2 ln 2 exp ln 2 exp 2 ln 2 1 exp ln 2 1 exp 2 ln 2 exp ln 2 exp ln 2 cos ln 2 sin zi zi zi zi zi zii i zi zii i i zi zi zi zi i zi zii i zi zii i zi zii zi zii zi zii z z zi zi zi zi i x x izi zii then x zi zi x x ix x ln 2 arctan 1 1 1 1 11 ln 2 tan ln 2 1 exp set 2 2 2 1 2 1 4 From 3 ln 2 cos ln 2 sin ln 2 tan zi zii zi zii zi zii z ln 2 1 cosh ln 2 1 sinh zi zi i zi zi 1 1 ln 2 1 ln 2 1 arctan ln 2 1 arctan ln 2 1 tanh z z izi izi izih zi zi izh zi zi iz Let izz 5 2 1 0 2 ln 2ln 2ln ln 22 nin a b iArgba iniArgzz inzbia 6 i 2i1 i 2 1 3 2 i 1 22 i 3 1 3 i 1 t dttit t 1 11 t 0 C iz1 t 0 C i1z1 t 0 Im Re 1 0 22 1 0 c 2 1 t i t dtitt t ztitz titz itttzC zzzzf i i i dtttidtttii cc 3 2 3 2 11 22 1 1 1 0 1 0 21 6 ii i i i zi dz iz z dz iz z iz 27 62 9 31 2 3 1 3 9 4 2 3 1 3 2 3 1 3 2 3 3 1 23 3 3 2 2 22 idz z dzzdz z z 20 11 22 iii 6 iv v 31 3 3 2 sin 3 2 cos 3 2 3 2 3 2 3 3 1 3 3 2 3 2 22 i i i i e i e i dz i z e dz iz e i i z zz i i z i dz zz dz zz z 5 2 2 61 1 3 1 2 2 1 2 1 3 1 2 1 372 1 2 1 2 6 vi 2 sin cos2 cossin2 sin 2 22 2 z zzzf zzzf zzf 4 sin 2 2 2 0 2 sin i z cosh 2 2 33 2 3 i z i dz z zz z 7 42 2 22 222 22 22 2 2 2 2 0 2 0 00 2 0 0 2 0 2 1 0 i ei eiee eieei dzezeez dzzeezdzez gdzffgdzfg dzfggdzffgdz fggffg i iii izii i ziziz i zizz z z Here since f z g z is analytic i