dx dz dx dz dy dz dy dz 



Hence 



dz dx dy 



or 



^=^+t#=-z^ + #. [22b] 



dz dx dx dy dy 



and, equating real and imaginary parts separately in this last equation, 



^ = #, ^=_M [22c,d] 



dx dy dy dx 



These equations are known as the Cauchy-Riemann relations. They hold necessarily 

 wherever f{z) is differentiable; and it can be shown that they guarantee the existence of a der- 

 ivative with respect to z wherever the derivatives of <^ and ;// are continuous functions of x and y. 



From [22b] and [22c, d] it follows that 



where | | denotes as usual the absolute value. 



If to = f{z), then 2 = F {w) where F denotes another function known as the inverse of 

 the function /. If f{z) is a regular function at any point 3, so is F {w) at the corresponding 

 value of w. As in real variables, 



dz JdwY"^ 

 dw \dz ) 



23. CONFORMAL REPRESENTATION OR MAPPING 



Assume that 3 = x + iy 

 and 



w = f(z) = (^ + i i// [23a] 



where f{z) denotes a regular function of z and <p and i/r are real functions <f>{x,y), tp(x,y). 

 Suppose that the values of w are plotted on the same plane with those of s, with a common real 

 axis. Then the transformation from 3 to to displaces each point on the plane, representing a 

 value of 2, into another position where it represents a value of w. Curves are displaced and, 

 in general, changed in shape. 



Often, however, it is more convenient to plot w on a separate plane called the it-plane. 

 Then, to each point or curve on one plane there corresponds a point or curve on the other. The 

 configuration on the 2-plane is said to be transformed into that on the tc-plane, or to be repre- 

 sented by it, by means of the transformation w = f{z). A diagram on the w-plane can be re- 

 garded as a kind of map of the corresponding 2 diagram. The comparison is facilitated if the 

 two planes are thought of as parallel, and with parallel axes for real and imaginary numbers. 



Corresponding curves on the two planes will usually differ both in linear scale and in 

 direction. Let 2 undergo a small increment 8z along a curve, as from Pj to Pj Ji Figure 22. 



39 



