Figure 28 - The transformation ic = z^^^. 



onto half of the w-plane; and w maps it again onto the other half. So long as the motion of 



2 is restricted in this manner, w^ and w^ behave as distinct single-valued and regular functions 



of 3. 



Yet Wy and w^ cannot be regarded as completely separate functions. For, if $ is 

 allowed to vary without limit, and if 2 goes completely around the origin and returns to its 

 starting point, as along curve efg (Figure 28), w^ and w^ will have just changed places; and, 

 if 2 then explores the plane as before, w and w^ interchange roles. It is thus clear that, as 

 2 moves freely, both w and w^ move continuoiely and freely on the whole M)-plane. Further- 

 more, the location of the half plane on which the entire 2-plane is mapped by either value of 

 2^/-^ can be varied at will by changing the position of the line, or curve, along which the 

 2-plane is cut. 



At 2 = 0, w. = w = 0, so that the two branches come together. For this reason the 

 point 2 = is called a branch point for the function 2^^^. If 2 actually passes through the 

 branch point along a continuous curve, the function 2^/^, approaching along a given branch, 

 may be assumed to emerge without discontinuity along either branch. 



The point 2 = is also a singular point of a certain kind, and at this point angles are 

 not preserved in the transformation from 2 to w. 



The inverse transformation 2 = w^ is single-valued. But each value of 2 except 

 2=0 and 2 = 00 occurs twice among the possible values of 3 = 1^^, 



The function In 2 is discussed in Section 40, and 2" in Section 39. 



25. RELATION OF REGULAR FUNCTIONS TO TWO-DIMENSIONAL 

 IRROTATIONAL FLOW 



Consider the regular transformation [22a] 



W = f(z) = + t (/f 



46 



