78, 79] COMPLEX VARIABLES. 79 



79. The hydrodynamical interpretation of cases where w is, 

 in the foregoing sense, a many- valued function of z is obvious from 

 the preceding chapter. It is possible however for w to be 

 ambiguous in another way. Let us take for instance the func 

 tion 



+ ism6^ ............... (26). 



.If we start from the point r = 1, = 0, with the value iv=l, 

 the value of the function at any other point is 



w = r* (cos $0 + i sin J 6), 



where 6 must of course be supposed to vary continuously. Hence 

 if the point P (r, 0) describe a closed curve not embracing the 

 origin, w will return to its former value ; but if the path of P 

 encompass the origin this will not be the case; if the motion of P 

 be in the positive direction 6 will have increased by 2?r, and we 

 shall have iu = \. A second circuit round the origin will restore 

 to w its original value. Hence to every point P in the plane of z 

 correspond two values of w, which however pass into one another 

 continuously as P describes a closed curve about the origin, where 

 tlie two values coincide. 



Again, if 



w = J .............................. (27), 



and we start from any point A with the value w ot the value of w 

 at A after one circuit of P round the origin will be OLW O , after a 

 second circuit a 2 w , and after a third a. 3 w , where a is a cube root 

 of unity. Hence to every point P of the plane of z correspond 

 three values of w, forming a cycle which recurs at every third cir 

 cuit of P round the origin. 



A point, such as the origin in the above examples, at which 

 two or more values of a function coincide, is called a branch 

 point . The similarity in their infinitely small parts of the planes 

 of w and z must obviously break down at a branch-point, so that 

 we must have at such points (Art. 76) 



dw A 



-j- = 0, or oc . 

 dz 



