UNIVERSITY OF VIRGINIA PUBLICATIONS 



BULLETIN OF THE PHILOSOPHICAL SOCIETY 



SCIENTIFIC SECTION 



Vol. I, No. 15, pp. 335-339 March, 1913 



ON THE ROOT OF A MONOGENIC FUNCTION INSIDE A CLOSED 

 CONTOUR ALONG WHICH THE MODULUS IS CONSTANT.* 



W. H. ECHOLS 



L The basis of Cauchy's theory of functions reposes upon the integral 

 taken around a closed boundary. If the function is regular along and 

 inside this boundary the integral of the function taken around the boundary 

 vanishes. When the closed boundary is a contour or a simple closed curve 

 C along which the modulus ikf of a monogenic function f(z) = u + iv is 

 constant the theory fails owing to the fact that along the contour C 



I / {z) dz= M \ e«V dz, 



in which ,p = e' ''" « is not analytic except when f{z) is a constant. For, 

 in order that the derivative of with respect to z = x + iy shall be inde- 

 pendent of dy/dx it is necessary that 



i^=^, and I- («2 + v^) = i^ (u^ + V'), 

 dx (>y oy ox 



which can only be true when 4) and u^ + v^ are constants. 



When we seek the roots of a function f{z) within the contour C through 

 Cauchy's integral taken around C 



J 



d log fiz) = i \ dip = 2iri, 



* Presented and read before the Scientific Section at its regular meeting in 

 March, 1913. 



335 



