第6章 6.21

$u = yf_x - xf_y$

$\Delta f = 0$ から

$\frac{d^2f}{dx^2} + \frac{d^2f}{dy^2} = f_{xx} + f_{yy} = 0$

$\frac{d^2u}{dx^2} = \frac{d}{dx}(yf_{xx} - f_y - xf_{xy}) = \frac{d}{dx}(-yf_{yy} - f_y - xf_{xy}) = -yf_{xyy} - f_{xy} - f_{xy} - xf_{xxy}$

$= -yf_{xyy} - 2f_{xy} - xf_{xxy}$

$\frac{d^2u}{dy^2} = \frac{d}{dy}(yf_{xy} - f_x - xf_{yy}) = \frac{d}{dy}(yf_{xy} + f_x + xf_{xx}) = yf_{xyy} + f_{xy} + f_{xy} + xf_{xxy}$

$= yf_{xyy} + 2f_{xy} + xf_{xxy}$

これから $\Delta f = 0$ となり調和関数である