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 A 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 0+i^>. If we denote 

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

 in the corresponding circuit, and /z the flux across it outwards. 



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

 defined 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 



2= r (cos + i sin 0), 



we have = h idd, 



z r 



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?r 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 Zn on the imaginary axis of w, whilst the inner and outer 

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



