Your question does not make sense. Do you mean "Given that U(x,y)= x^2- y^2, find a function V(x,y) such that U(x,y)+ iV(x,y) is analytic"? If so, check the "Cauchy-Riemann" equations: https://en.wikipedia.org/wiki/Cauchy–Riemann_equations.
Another possibility is that the question is "Show that U(x, y)= x^2- y^2 is 'real analytic'". If so then this follows immediately from the definition of "real analytic".
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