197] COMPLEX VARIABLE. 523 



In this case the function w is said to have a definite derivative defined by 



j, f , N j. dw 



dy = 



and it is only when the functions u and v satisfy these conditions 118) 

 that u + iv is said to be an analytic function of g. This is Riemann's 

 definition of a function of a complex variable. 1 ) 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 0, w = f(&), then w always satisfies this condition. For 



dw df(z] dz f'C\ dw df(z] dz . ~, / x 



dx dz dx I \ )i fly d% fly I \ )' 



Accordingly 



.dw . (du . dv\ dw cu . dv 



cu _ dv dv _ du 



dx dy' ex dy 



If we differentiate the equations 118), the first by x, and the second 

 by y and add, since 



d*v d z v 



dxdy ~ dydx' 

 we obtain 



120^ 4- 



dx*^ dy*-"' 



Differentiating the second by x and the first by y and subtracting, 

 we find that v satisfies the same equation 



Thus every function of a complex variable gives a pair of solutions 

 of Laplace's equation, either one of which may be taken for the 

 velocity potential, representing two different states of flow. 



It is to be noticed that the question here dealt with is simply 

 one of kinematics, since Laplace's equation is simply the equation of 

 continuity and there is no reference to the dynamical equations. 



The question arises whether any two solutions of Laplace's 

 equation will conversely give us the function of a complex variable. 

 It obviously will not answer to take any two harmonic functions, 

 for they must be related so as to satisfy the equations 118) or be 

 mutually conjugate. In order to avoid confusion with the velocity 



1) Riemann, Mathematische Werke, p. 5. 



