ON HARMONIC FUNCTIONS IN TWO DIMENSIONS. 

 By Maxime BdCHER. 



Presented January 10, 1906. ReceiTed January 9, 1906. 



A function u (x, y) is said to be harmonic throughout a certain region 

 T of the a^-plane if 1) it is analytic there, and 2) it satisfies Laplace's 

 equation. It is well known that the first of these requirements may be 

 replaced by the apparently much less restrictive one that the function u 

 be continuous and have continuous first and second partial derivatives. 

 In fact, it is sufficient to demand even less than the continuity of the 

 second partial derivatives ; but some demand beyond the mere existence 

 of these derivatives must be made in order that it should be possible to 

 apply Green's theorem. 



It is my object in the present note to show how the theory of harmonic 

 functions in two dimensions may he established without demanding even 

 the existence of the second partial derivatives. This makes it necessary 

 for us to take as our point of departure not Laplace's equation, since this 

 involves the second partial derivatives whose existence we do not wish 

 to demand, but some other characteristic property of harmonic functions. 

 We select for this purpose the fundamental property that the integral of 

 the normal derivative of a harmonic function, extended around a closed 

 curve, is zero, and we thus have as our starting point not a differential 

 equation of the second order, but a differentio-integral equation which 

 involves only first partial derivatives.! It is needless to insist on the fact 

 that it is this differentio-integral equation and not Laplace's equation 

 which forms the starting point in most physical applications of harmonic 

 functions. 



We will lay down the following 



Definition : u (x, y) is said to be harmonic throughout a region T of 

 the x, y-plane if it is single valued and continuous there, has continuous 

 first partial derivatives, and, for any circle which lies wholly within T, 2 

 satisfies the equation : 



1 We note in passing that a pure integral equation for harmonic functions is 

 given by Poisson's Integral. 



2 That i9, all points within the circle or on its circumference are to lie within the 

 region T'and not on its boundary. 



vol xli. — 37 



