6.6

6.6
Show that, if \(f\) is holomorphic and nonzero in \(G\), then \(ln\mid f(x,y) \mid\) is harmonic in \(G\).
————————————————————
Hoping to use Definition 6.1, ie., if \(f\) has second partial derivatives that satisfy \(f_{xx}+f_{yy}=0\), then \(f\) is harmonic. \[\begin{aligned} f_{xx}+f_{yy}&=0\\ f&=ln\mid f(x,y) \mid\\ f_x&=\frac{1}{f(x,y)}*f_x(x,y)\\ f_{xx}&=\frac{-1}{f(x,y)^2}f_x(x,y)+\frac{1}{f(x,y})f_{xx}(x,y)^2\\ f_y&=\frac{1}{f(x,y)}f_y(x,y)\\ f_{yy}&=\frac{1}{f(x,y)f_{yy}}+\frac{-1}{f(x,y)^2}f_y(x,y)f_y(x,y)\\ &=\frac{1}{f(x,y)}f_{yy}(x,y)+\frac{-1}{f(x,y)^2}f_y(x,y)^2\\ f_{xx}+f_{yy}&=\frac{-1}{f(x,y)^2}f_x(x,y)+\frac{1}{f(x,y})f_{xx}(x,y)^2+\frac{1}{f(x,y)}f_{yy}(x,y)+\frac{-1}{f(x,y)^2}f_y(x,y)^2 \overset{?}{=}0\end{aligned}\] Dosen’t look like these are equal, most likely an error somewhere.