Section 12.5 Analytic Functions157The frst equation oF the Cauchy-Riemann equations tells us thatux=vy.Sovy(x, y)=excosy.Integrating with respect toy(treatingxas a constant), we fndv(x, yexsiny+c(x),where the constant oF integrationc(x) is a Function oFx. The second equation oFthe Cauchy-Riemann equations tells us thatuy=-vxvx(x, yexsiny+c±(x);exsiny+c±(xexsiny;c±(x)=0;c(xC.Hence,v(x, yexsiny+C,which matches the previous Formula up to a additive constant.37.(a) The level curves are given byu(x, yyx2+y2=12C,where, For convenience, we have used 1/(2C) instead oF the usualCFor our arbitraryconstant. The equation becomesx2+y2-2Cy=0 orx2+(y-C)2=C2.Thus the level curves are circles centered at (0,C) with radiusC.(b) By Exercise 35, a harmonic conjugate oFu(x, y)isv(x, yxx2+y2.Thus the orthogonal curves to the Family oF curves in (a) are given by the levelcurves oFv,orv(x, yxx2+y2=12Cor (x-C)2+y=C2.Thus the level curves are circles centered at (
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This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.