82 FUNCTIONS OF A COMPLEX VARIABLE. [INT. IV. 



is in general dependent on - , that is on the direction in which 



we leave the point z. The value of the derivative will not then be 

 determined for the point z irrespective of the direction of leaving 

 it unless the numerator is a multiple of the denominator and the 



expression containing -^ divides out. 



In order that this may be true we must have 



fdu .dv\ /du .dv\ . 



- -M 3- 1 : 1" = ( 57 -|- i - 1 : tj 



\dx dxJ \dy dyj 



. . . /du .dv\ du .dv 



that is Mo~ + *o~ == 3~~ + i o~' 



\dx dxJ dy dy 



Putting real and imaginary parts on both sides equal, 



du _ dv dv _ du 

 dx dy' dx~ dy' 



IS t/ 



dw du .dv dv . du 



and -j- = r- + * 5- = 5 l 6~ > 



dz ox ox oy oy 



i dw 2 /du\ 2 /du\ 2 fdv\* fdv \ 2 



In this case the function w has a definite derivative, and it is 

 only when the functions u and v satisfy these conditions that u+iv 

 is said to be an analytic function of z. This is Riemann's definition 

 of a function of a complex variable*. (Cauchy says monogenic 

 instead of analytic.) The real functions u and v are said to be 

 conjugate functions of the real variables x, y. 



It is obvious that if w is given as an analytic expression 

 involving z, w f(z\ then w always satisfies this condition. For 



dw_ df(z)dz_j,,^ dw_dj\2)dz_ 

 d^~~d7~dx~^ (Z) ' d~ dz d-V 



* Biemann, Mathematische Werke, p. 5. 



