Let f(z) = (x 2 + y 2) + i2xy and g(z) = 2xy + i(y 2 – x 2) for z

Let f(z) = (x² + y²) + i2xy and g(z) = 2xy + i(y² – x²) for z = x + iy ϵ C. Then, in the complex plane C.

A. f is analytic and g is NOT analytic

B. f is NOT analytic and g is analytic

C. Neither f nor g is analytic

D. both f and g are analytic

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Explanation:

Given:

f(z) = (x² + y²) + i 2xy

g(z) = 2xy + i (y² – x²)

To check analyticity of a function, we need to check CR equations.

u_x = v_y, u_y = - v_x

f(z) = (x² + y²) + i 2xy

u = x² + y², v = 2xy

u_x= 2_x

u_y = 2_y

v_x = 2_y

v_y = 2_x

u_x = v_y but u_y ≠ -v_x

Hence, f(z) is not analytic

g(z) = 2xy + i (y² – x²)

u = 2xy, v = y²– x²

u_x = 2y

u_y = 2x

v_x = -2x

v_y = 2y

u_x = v_y and u_y = -v_x

Hence g(z) is analytic.