Let f(2) = u(x, y) + i v(x, y) = U (r, 0) + iV (r, 0). By exploiting the relations x = r cos0 , y =r sin 0, we proved that if the Cauchy-Riemann equations hold true, that is, = Vy Uy Vx (1) then, r Up...

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Extracted text: Let f(2) = u(x, y) + i v(x, y) = U (r, 0) + iV (r, 0). By exploiting the relations x = r cos0 , y =r sin 0, we proved that if the Cauchy-Riemann equations hold true, that is, = Vy Uy Vx (1) then, r Up = Ve r Vr = -U . (2) One can actually show that the viceversa is also true, that is, if f satisfies (2) then f satisfies also (1). The equations in (2) are the polar form of the Cauchy-Riemann equations. Recalling that f' (2) : = Ux +i vr, check that f'(2) = e-i0 (U, + i V,).