Is the function f(z) = Re(z2) analytic? Explain. Solution A necessary condition

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A necessary condition that w =u +i v be analytic in a region D is that in D, u and v satisfy Cauchy-Riemann equations if u and v have continuous first partial derivatives that satisfy everywhere in D : ux = vy and uy = -vx ux = ?u/?x , uy = ?u/?y , vx = ?v/?x and vy = ?v/?y for instance, consider z= x+yi then x is called the real part of z,Re{z} = x and y is called the coefficient of the imaginary part of z,Im{z}=y: f(z) = z² => f(z) = x ² - y² +2xyi is analytic u = x ² - y² v=2xy ux =2x , vy=2x uy =-2y , vx =2y so CRE are satisfied hence f is analytic ( actually ,Cauchy-Riemann conditions are necessary and sufficient for analyticity ) Now, in my understanding,for f(z) = Im ( z² ) f = 2xy is a real valued function of complex variable that is f= u+0i , u=2xy and v=0 => => ux=2y ,uy=2x ,vy=0 and vx=0 The partial derivatives are continuous ,however, CRE are not satisfied and f is not analytic.

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