(pp. 109-134)

A function of*x*and*y,*$f(x,y)$can be written as a function of*z*,$\overline z $through the identity

(11.1)$F(z,\overline z ) = f\left( {\frac{{z + \overline z }} {2},\frac{{z - \overline z }} {{2i}}} \right)$

Since

$\frac{{\partial F}} {{\partial z}} = \frac{{\partial F}} {{\partial x}}\frac{{\partial x}} {{\partial z}} + \frac{{\partial F}} {{\partial y}}\frac{{\partial y}} {{\partial z}} = \frac{{\partial F}} {{\partial x}}\left( {\frac{1} {2}} \right) + \frac{{\partial F}} {{\partial y}}\left( {\frac{1} {{2i}}} \right)$,

we have

(11.2)$\frac{{\partial F}} {{\partial z}} = \frac{1} {2}\left( {\frac{{\partial F}} {{\partial x}} - i\frac{{\partial F}} {{\partial y}}} \right)$

Similarly,

(11.2’)$\frac{{\partial F}} {{\partial \overline z }} = \frac{1} {2}\left( {\frac{{\partial F}} {{\partial x}} + i\frac{{\partial F}} {{\partial y}}} \right)$

This suggests that it will be convenient to introduce the operators

(11.3)$\frac{\partial } {{\partial z}} = \frac{1} {2}\left( {\frac{\partial } {{\partial x}} - i\frac{\partial }{{\partial y}}} \right)$,$\frac{\partial } {{\partial \overline z }} = \frac{1} {2}\left( {\frac{\partial } {{\partial x}} + i\frac{\partial } {{\partial y}}} \right)$.

Inversely,

(11.4)$\frac{\partial } {{\partial x}} = \frac{\partial } {{\partial z}} + \frac{\partial } {{\partial \overline z }}$,$\frac{\partial } {{\partial y}} = i\left( {\frac{\partial } {{\partial z}} - \frac{\partial } {{\partial \overline z }}} \right)$.

The directional derivative of$f(x,y)$in the direction a is defined by

${D_\alpha }f(x,y) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h\cos \alpha ,y + h\sin \alpha )}} {h}$

$ = \frac{{\partial f}} {{\partial x}}\cos \alpha + \frac{{\partial f}} {{\partial y}}\sin \alpha $

Using (11.4), one obtains

(11.5)${D_\alpha }F(z,\overline z ) = {F_z}{e^{i\alpha }} + {F_{\overline z }}{e^{ - i\alpha }}$

To obtain directional derivatives along and normal to an analytic arc*C*with Schwarz Function$S(z),$we proceed as follows. Let${z_0} \in C$and let$\psi $be the angle the tangent to*C*...