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F(z) =(| z|/+z) /2is analytic at. |
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Answer» If z=X+iy we have thatf(z)=|z|2=z⋅z¯¯¯=x2+y2 This SHOWS that is a REAL VALUED function and can not be analytic.We can rewrite the above asf(z)=x2+y2+i⋅0 Setu(x,y)=x2+y2 v(x,y)=0 Hencef(x,y)=u(x,y)+i⋅v(x,y) The function f is continuous because u,v are continuous. But Cauchy Riemann holds at the originux=2x,uy=2y ux=vy,uy=−vx x=0,y=0=>z=0 Hence f is differentiable only at the origin, and the derivative is zero. |
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