Lub classical interior uniqueness theorem rau holomorphic (uas yog, ib leeg-valued analytic) ua haujlwm ntawm D hais tias yog ob txoj haujlwm holomorphic f(z) thiab g(z) hauv D coincide ntawm qee qhov teeb tsa E⊂D uas muaj tsawg kawg yog ib qho kev txwv hauv D, ces f(z)≡g(z) txhua qhov chaw hauv D.
Puas yog holomorphic ua haujlwm tag nrho?
A holomorphic muaj nuj nqi uas nws lub npe yog tag nrho cov dav hlau complex hu ua tag nrho cov haujlwm Cov kab lus "holomorphic ntawm ib qho z0" txhais tau tias tsis yog txawv ntawm z0, tab sis txawv txhua qhov chaw hauv qee thaj tsam ntawm z0 hauv dav dav.
Puas yog txhua qhov kev ntsuas kev ua haujlwm sib txawv?
Txhua yam kev tshuaj ntsuam xyuas yog du, uas is, infinitely differentiable. Kev sib tham tsis muaj tseeb rau kev ua haujlwm tiag tiag; qhov tseeb, nyob rau hauv ib qho kev txiav txim siab, qhov kev tshuaj ntsuam tiag tiag muaj qhov sib txawv piv rau txhua qhov kev ua haujlwm tiag tiag infinitely txawv.
Dab tsi yog qhov txawv ntawm kev ua haujlwm holomorphic thiab analytic functions?
A function f:C →C tau hais tias yog holomorphic hauv qhov qhibteeb A⊂C yog tias nws txawv ntawm txhua qhov chaw ntawm qhov teeb tsa A. Cov haujlwm f: C → C tau hais tias yuav tshuaj xyuas yog tias nws muaj lub zog sawv cev.
Vim li cas holomorphic functions yog infinitely txawv?
The existence of ib tug complex derivative txhais tau hais tias hauv zos muaj nuj nqi tsuas yog tig thiab nthuav. Ntawd yog, nyob rau hauv qhov txwv, disks yog mapped rau disks. Qhov rigidity no yog dab tsi ua rau ib tug complex sib txawv function infinitely txawv, thiab txawm ntau, analytic.
