Question
Complex anaylsis, cite all theorems used. Y are one Consider the real valued function ulx,y) with...
Complex anaylsis, cite all theorems used.
Y are one Consider the real valued function ulx,y) with x and real variables. For each definition of ucxy) below, find whether there cette exists real-valued function v(x,y) such that f(2)= u(x, y) tivcx,y) is a function analytic in some DEC. If such such v(x,y) and determine V(X,Y) the domain of analyticity D for fcz). It such a not exist, prove that it does not exist. (i) u (x,y)= xy2-x²y (ii) ucx, y) = y3 - 3x²4 Gm ulx,y)= Arg (xtiys exists, find ucxys does
Answers
it il Given analyhe then both u (ny)= ngx If f(&l=utir be u&.v must be Batisfies u Harmonic.ire, both u av enna Laplacian equ.. My = any-n? 22 Now y² Myy un = zny --22 unn 20-2 -270 Uant Myy So N (n will becom u does not satisfies Laplaciau eat. venry) does not exist for which fautir analytie u my-g²-30% 34 ) us 372 -bny 64 = Un nyt uan of a vam Unntuy Some Lalinfies the Laplacian eanation. Therefore we can find for which fauti will becomes analghe in o's Now, fo analyſe too by Cauchy Riemann Ecuation hane & nyany 7 unary vg v (p) -bay Inlegmaling ung 7 x = 34 34 sagt de vinsia x² - 3 ng T is Apicol 3 Thenbere, ser x-snytk v no Constant Scanned with CamScanner (Ame Here the analytic on &
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