62] 



COMPLEX VARIABLE. 



77 



Points of the plane xy at which the conditions in question break down may 

 be isolated by drawing a small closed curve round each. The rest of the plane 

 is a multiply-connected region, and the value of the integral from ^1 to P 

 becomes a cyclic function of the position of P, as in Art. 50. 



In the hydrodynamical applications, the integral (iii), considered as a 

 function of the upper limit, is taken to be equal to &amp;lt;f)+i\js. If we denote 

 any cyclic constant of this function by K + IH, then K denotes the circulation 

 in the corresponding circuit, and p. the flux across it outwards. 



As a simple example we may take the logarithmic function, considered as 

 denned by the equation 



(v). 



Since z~ l is infinite at the origin, this point must be isolated, e.g. by drawing 

 a small circle about it as centre. If we put 



we have 





so that the value of (v) taken round the circle is 



Hence, in the simply-connected region external to the circle, the function (v) 

 is many-valued, the cyclic constant being - 27rt. 



In the theory referred to, the exponential function is defined as the inverse 

 function of (v), viz. if w = logz, we have e w =z. It follows that e w is periodic, 

 the period being 2^ . The correspondence between the planes of z and w 

 is illustrated by the annexed diagram. The circle of radius unity, described 

 about the origin as centre, in the upper figure, corresponds over and over 



again to lengths 2;r on the imaginary axis of w, whilst the inner and outer 

 portions of the radial line = correspond to a system of lines parallel to 



