45] FUNCTIONS OF A COMPLEX VARIABLE. 89 



For example the function w = - is discontinuous at the point 



z = 0. Accordingly the integral I around a closed contour con- 

 taining the origin within' it is not zero, for it may be taken as the 

 difference between the integrals between two points AB on the 

 contour along two paths between which lies the point of dis- 

 continuity of the function w. The integral around any closed 

 contour embracing the origin is however the same as around a 

 circle of radius R with center at the origin, for between the two 

 curves there is no point of discontinuity of the function. Now 

 since z x + iy = re**, if r is constant = R, 



dz = i^ 



and the integral from z = 

 dz 



which taken around the circle is 2-7T*. 



rz fa 



The integral I - is denned as the logarithm of z, and it pos- 



sesses the property that as z describes any closed path enclosing the 

 origin, the function instead of returning to its original value 

 increases by a constant 2?ri. The function is then not uniform, 

 but has at any point an unlimited number of values, depending 

 upon the path by which we arrive at the point. These values all 

 differ by integral multiples of the constant 2?. 



We see that this accords with the ordinary definition of the 

 logarithm, 



log z log (x + iy) = log (re* ( * +anir )) = log r + i</> + 2rwn', 



for if we increase the argument </> of a complex number z by any 

 multiple of 2?r, the number is unchanged. A point such that a 

 function f(z) assumes a new value when the variable traverses a 

 closed circuit about the point is called a critical, or branch point 

 In this case the conformal representation given by the function f(z) 

 is multiple in character, for in the [/"F-plane we are to take a point 

 for each of the values of the function f(z). Each of these repre- 

 sentative points gives a conformal representation of the whole of 

 the JTF-plane on a part of the /"F-plane. 



