76, 77.] COMPLEX VARIABLES. 75 



The meaning of the integral 



wdz (15), 



taken along any assigned path from Z Q to z, is defined as follows. 

 Supposing the path divided into infinitesimal portions, we form 

 the product wdz for each of them, and add the results. It is easily 

 shewn that the value of the integral is to a certain extent inde 

 pendent of the nature of the path joining Z Q and z. The theorem 

 of Art. 40, when applied to a plane space (xy) becomes 



dy 



The double integral extends over any portion of the plane xy 

 throughout which u and v are finite and continuous, and their first 

 derivatives finite ; whilst the single integral extends in the positive 

 direction (see Art. 39) round the boundary of that portion. Now 

 if we write u = w, v = iw, it follows by (14) that f(wdx + iwdy), or 

 Jwdz, is zero when taken round the boundary of any portion of the 

 plane xy throughout which w is finite and continuous, and its first 

 derivative finite. 



We infer, as in Art. 41, that the integral (15) is the same for 

 any two paths joining z, Z Q) so long as these paths do not include 

 between them any points at which w is infinite or discontinuous, 



dw . 



or -=-^ infinite. 

 dz 



Any points at which these conditions are violated may be 

 isolated by drawing an infinitely small closed curve around each. 

 The rest of the plane xy then forms a multiply-connected region. 

 The value of the integral fwdz taken round any evanescible circuit 

 drawn in this region is zero; and the integral (15) is the same for 

 any two reconcileable paths. The values of (15) corresponding 

 to two irreconcileable paths differ by a quantity of the form 



Pi K i+Pi**+ where p v p 2 , are integers, and /e lt * 2 , 



denote the values of fwdz taken round the several circuits sur 

 rounding the above-mentioned points*. It is unnecessary to 



* In the analytical theory ^ , K 2 , . . . axe called the moduli of periodicity of the 

 integral (15). 



