Can any on help mt to solve problem in harnonic function

Question

Let U be a harmonic function on a region B and let F:A→B be analytic ,then prove that U o F is harmonic?

Answer 1

Assume $g(u,v)$ is harmonic, $f(x,y)= u + iv$ analytic and $h=g\circ f$. Then $$h_x=g_u u_x + g_v v_x $$ $$h_{xx} = g_{uu} u_x^2 + g_{uv} v_x u_x + g_u u_{xx} + g_{uv}v_x u_x+ g_{vv}v_x^2 + g_v v_{xx}$$ Similarly $$h_{yy} = g_{uu} u_y^2 + g_{uv} v_y u_y + g_u u_{yy} + g_{uv}v_y u_y+ g_{vv}v_y^2 + g_v v_{yy} $$ Now add. Observe that $g_{uu} + g_{vv}= 0$ is the factor of $u_x^2+u_y^2+g_u+g_v$ and vanishes. Then use Cauchy-Riemann for $u_x, v_y$ etc to kill the four remaining terms.

Answer 2

If $U$ is harmonic, then there is a holomorphic function $G$ whose real part is $U$ and whose imaginary part is unique up to an additive constant.

Let $H = G \circ F$. Now we note that that the real part of $H$ is $U \circ F$, and $H$ is the composition of two holomorphic functions, so $H$ is holomorphic. Hence, $U \circ F$ is the real part of a holomorphic function, so it is harmonic.