ON THE ROOT OF A MONOGENIC FUNCTION 337 



and therefore f{z) must be zero at x, y. The discruninant 



bx" dy- \ dxciy j 



at .r, )/ has the value 



which is positive when/(^) = and /'(a) =(= 0, and since then 



^ bx'- \bxl \by) 



is positive there is a true minimum at x, y. If f{z) is different from zero 

 and/'(s) = the discriminant is negative and there is neither maximum nor 

 minimum at x, y unless also/" (2) = 0, m which case the discriminant van- 

 ishes and it is necessary to proceed generally and investigate the condition 

 at X, y when/''(2) = for r = 1, 2, . . ., n — 1. 

 4. Consider the general ?!th derivative of /(s) as to z. 



(d^^ + dy^Yu + i(dx^ + dy^ 

 , . _ d"{u+iv) _\ bx by / V ^^ ^2/ 



{dx+i dy) " (dx + idy)" 



Let dx = Idr = dr cos d, dy =mdr = dr sin d. Since f^{z) is independent 

 of dy/dx or 8 there results for all values of 9 



„ , • • M /d"M , . b''v 



(co? nd + I sm nd) — — + i ^ — 



\ bx" bx" 



Hence 



, d , b\" b"u . b"v . 



I — 4- 7)1 —- ] u — - — cos nd — - — sm nd, 

 bx by/ bx" bx" 



, d , b\" b"u . . , b"v 



I —- + m z— ] V = -r — sm 7id + TT — cos nO. 

 bx by/ bx" bx" 



Now at the point x, y all the derivative of /(s) below the ?ith vanish, there- 

 fore for p = 1, . . .,71 — 1, 



