deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 903c128..e238c0d 100644  --- a/untitled.tex  +++ b/untitled.tex 
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Now consider the Laplace equation,   \begin{align*}  (ln|f(x,y)|)_{xx}+(ln|f(x,y)|)_{yy}  &=\frac{(uu_x &=\frac{1}{{(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) + 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}+v^{2})^{2}} vv_y)(2uu_y+2vv_y)]  \end{align*}  \end{proof}