ANALYTIC FUNCTIONS and complex Analysis

Suppose f is an entire function of the form f (x, y) = u(x)+iv(y). Show that f is a linear polynomial.
thanks for your help

Use the Cauchy-Riemann Equations.

Since u is only a function in x, and v is only a function in y, we obtain
u_x = v_y ==> u'(x) = v'(y)
u_y = -v_x ==> 0 = 0.

From u'(x) = v'(y), we conclude that u'(x) = v'(y) = C for some constant C, because that is the only way in which a function of x can equal a function in y.

So, u(x) = Cx + A and v(y) = Cy + B for some constants A and B.
==> f(z) = u(x) + i v(y) = C(x + iy) + (A + Bi) = Cz + (A + Bi).
That is, f is linear.

I hope this helps!