treacherous; for example, In [z^/(-s)] ^ In (-2), even In [l^(-l)] ^ In (-1), if the same rule is 

 used for amp (-s) or amp (-1) in both places. The safest procedure here is to calculate In 3 

 from the amplitudes and absolute values of the separate factors, 



22. REGULAR FUNCTIONS OF A COMPLEX VARIABLE 



If, in the complex number, z = x + iy, x and y are allowed to vary, 2 becomes a complex 

 variable. If a value of another complex variable w is associated by means of some rule with 

 each value of 2, then w is a function of 2. It is also possible to regard «; as a complex function 

 of the real variables x and y, of which its real and imaginary parts are likewise functions. Thus 



w ^f{z) = g{x,y) = cli{x,y) + iiij{x,y) [22a] 



where and ih are real functions of x and y. 



Some functions of 2 are single-valued, thac is, there is only one value of the function 

 associated with each value of 2; others are many-valued. A function which, at all points within 

 a certain region on the 2-plane, is both single-valued and differentiable, is said to be regular 

 or analytic or holomorphic within that region.* Such a function is also said to be regular or 

 analytic or holomorphic at any point in the interior of the region. Many functions are regular 

 except at certain points called singular points. 



In dealing with many-valued functions, a particular branch of the function can often be 

 defined so that, taken by itself, it is regular within a certain region. Thus, if 2 = re'^ and 

 6 is kept within the range -77 < < 77, In 2 is an analytic function of 2, except where d = n, since 

 st such points In 2 cannot be differentiated without overstepping the bounds set for d. 



It can be shown that regular frunctions necessarily posses derivatives of all orders. The 

 reason for this special behavior lies in the fact that a point on the s-plane can be ap- 

 proached from many different directions. Thus, in the formula 



df_ lim A/ 



dz A2 -► A2 

 if z is a real variable, A2 can vary only in magnitude, whereas if 2 is a complex variable the 

 increment A2 may vary also in amplitude, or in the direction of the representative vector on the 

 diagram; nevertheless the limit is required to have a fixed value, real or complex. This re- 

 quirement imposes a severe restriction upon the behavior of the function. 



The existence of a derivative with respect to 2 requires, in fact, that c'ertain differential 

 equations in terms of x and y must be satisfied. Consider 



f(z) = f{x + iy) = <p{x,y) + iil,(x,y) 

 where and i// are real. Regarding f(z) on the one hand as a function of 2 and on the other hand 

 as a function of x and y, by the ordinary rule for the differentiation of a function of a function, 



♦Some writers call a function analytic within a region when it has the properties stated except at a finite number 

 of singular points. 



38 



