_Show that, if f is holomorphic and nonzero in G, then ln|f(x, y)| is harmonic in G._ Assume f = u + iv is holomorphic and nonzero in G. Since f is holomorphic, its real and imaginary parts are harmonic and therefore infinitely differentiable by Proposition 6.4 and Corollary 6.9. The first derivative of ln|f(x, y)| with respect to x is \\ (ln|f(x,y)|)_x &= [ln(+v^{2}})]_x\\ &= {2}[ln(u^{2}+v^{2})]_x\\ &={2(u^{2}+v^{2})}\\ &={u^{2}+v^{2}} And the second derivative with respect to x is (ln|f(x,y)|)_{xx}&= ({u^{2}+v^{2}})_x\\ &= +v_xv_x + vv_{xx})(u^{2}+v^{2})-(uu_x + vv_x)(2uu_x+2vv_x)}{(u^{2}+v^{2})^{2}} Similarly the first and second derivatives of ln|f(x, y)| with respect to y are (ln|f(x,y)|)_y &= [ln(+v^{2}})]_y\\ &= {2}[ln(u^{2}+v^{2})]_y\\ &={2(u^{2}+v^{2})}\\ &={u^{2}+v^{2}} and (ln|f(x,y)|)_{yy}&= ({u^{2}+v^{2}})_y\\ &= +v_yv_y + vv_{yy})(u^{2}+v^{2})-(uu_y + vv_y)(2uu_y+2vv_y)}{(u^{2}+v^{2})^{2}} Since f is nonzero and u and v are harmonic in G, the second partials of ln|f(x, y)| are defined on G. Now consider the Laplace equation, (ln|f(x,y)|)_{xx}+(ln|f(x,y)|)_{yy} &={(u^{2}+v^{2})^{2}}[(uu_x + uu_{xx} +v_xv_x + vv_{xx})(u^{2}+v^{2})-(uu_x + vv_x)(2uu_x+2vv_x)\\ &+ (uu_y + uu_{yy} +v_yv_y + vv_{yy})(u^{2}+v^{2})-(uu_y + vv_y)(2uu_y+2vv_y)] We will show that the numerator expression will be equal to zero and thus the Laplace equation will equal zero. (uu_x + uu_{xx} +v_xv_x + vv_{xx})(u^{2}+v^{2}) &-(uu_x + vv_x)(2uu_x+2vv_x)+(uu_y + uu_{yy} +v_yv_y + vv_{yy})(u^{2}+v^{2})\\ &-(uu_y + vv_y)(2uu_y+2vv_y)\\ &=u^{2}u^{2}_x+u^{3}u_{xx}+u^{2}v^{2}_x+u^{2}vv_{xx}+v^{2}u^{2}_x+v^{2}uu_{xx}+v^{2}v^{2}_x+v^{3}v_{xx}\\ &-2u^{2}u^{2}_x-2uvv_xu_x-2uvu_xv_x-2v^{2}v^{2}_x+u^{2}u^{2}_y+u^{3}u_{yy}+u^{2}v^{2}_y+u^{2}vv_{yy}+\\ &v^{2}u^{2}_y+v^{2}uu_{yy}+v^{2}v^{2}_y+v^{3}v_{yy}-2u^{2}u^{2}_y-2uvv_yu_y-2uvu_yv_y-2v^{2}v^{2}_y\\ &-2u^{2}u^{2}_y-2uvv_yu_y-2uvu_yv_y-2v^{2}v^{2}_y But since u and v are harmonic, uxx + uyy = 0 and vxx + vyy = 0. Using this fact and combining like terms gives us, -u^{2}u^{2}_x+u^{2}v^{2}_x+v^{2}u^{2}_x-v^{2}v^{2}_x-4uvu_xv_x-u^{2}u^{2}_y+u^{2}v^{2}_y+v^{2}u^{2}_y-v^{2}v^{2}_y-4uvu_yv_y And now we can use the Cauchy-Riemann equations to get -u^{2}u^{2}_x+u^{2}v^{2}_x+v^{2}u^{2}_x-v^{2}v^{2}_x-4uvv_yv_x-u^{2}v^{2}_x+u^{2}u^{2}_x+v^{2}v^{2}_x-v^{2}u^{2}_x+4uv_xv_y Which simplifies to 0 and thus (ln|f(x, y)|)xx + (ln|f(x, y)|)yy = 0. Since ln|f(x, y)| has continuous second partials on G and the Laplace equation is satisfied, ln|f(x, y)| is harmonic on G.