40 42] FUNCTIONS OF A COMPLEX VARIABLE. 81 



representing w f(z) describes another curve, if f(z) is continuous, 

 otherwise the point u, v may jump from one point to another. 

 The definition of continuity is that two points on the function 

 curve may be made to approach each other as nearly as we please 

 by taking the corresponding points on the curve of the variable 

 sufficiently near. Or, a function is continuous in a region of the 

 ,3-plane continuing Z Q if to every real positive quantity e as small 

 as we please, we can find a corresponding quantity 8 such that 



I/W-/WI <* if | *-*!< 



In considering the representation by means of curves, it is of 

 importance to inquire whether, if the curve of z starting from an 

 arbitrary point z , returns to it after describing a closed curve, the 

 curve representing w =f(z) also returns to its point of departure. 

 If this is the case, the function f(z) within the region in which 

 this property holds, is said to be uniform, or single-valued, for to 

 every value of z corresponds one value of w. 



42. Derivative. Analytic Function. Let us examine 

 the relation between an infinitesimal change in z and the corre- 

 sponding change in f(z). The change dz = dx -t- idy has the 



modulus | dz \ = */dx* -f dy 2 , and the argument w = tan" 1 . 

 The change dw = du + idv has the modulus | dw \ = Vdu 2 -f dv 2 

 and the argument 6 = tan" 1 -j- . 



Also 



, du j du j 

 du = dx dx + ^j dy ' 



-, dv 1 dv 7 

 dv = 5- dx + dy, 



dx dy ' 



7 7 . 7 du -, da 7 . (dv 7 dv 

 dw = du + idv = -dx + ^- dy + i-<~-dx + ^-c 

 ox dy ' (ox oy 



The ratio 



du du j . (dv j dv 

 7 7 . 7 TT~ dx -f -^r- dy -f- 1 < ^- dx 4- dy 

 , . dw _ du + idv _ dx dy * (dx dy ' 



I dz i =0 dz dx + idy ~ dx -f idy 





du .dv_ du 

 ~ + l 



*Tx 



W. E. 



