# Functions of Complex Variables

by alchemistoff
Tags: complex, functions, variables
 P: 7 1. The problem statement, all variables and given/known data {Q 6.2.2 from Arfken "Mathematical Methods for Physicists"} Having shown that the real part $$u(x,y)$$ and imaginary part $$v(x,y)$$ of an analytic function $$w(z)$$ each satisfy Laplace's equation, show that $$u(x,y)$$ and $$v(x,y)$$ cannot have either a maximum or a minimum in the interior of any region in which $$w(z)$$ is analytic. (They can have saddle points) 2. Relevant equations Cauchy-Riemann (CR) relations for analyticity of the function $$u_x=v_y$$ and $$u_y=-v_x$$ where subscript stands for partial differentiation with respect to that variable. $$\nabla^2u=0$$ and $$\nabla^2v=0$$ (it follows from CR relations and proves that analytic function satisfies Laplace's equation) 3. The attempt at a solution The local minimum/maximum points are to satisfy $$u_x=0$$ and $$u_y=0$$ and $$M=u_{xx}u_{yy}-(u_{xy})^2>0$$ $$\nabla^2u=u_{xx}+u_{yy}=0\therefore u_{xx}=-u_{yy}$$ $$M=-u_{yy}^2-u_{xy}^2\leq0$$ ...and it looks like totally wrong direction...
 P: 48 i think you proved it, for a function of two variables to have a min or max $$-u_{yy}^2-u_{xy}^2 > 0$$ but for a saddle point $$-u_{yy}^2-u_{xy}^2 < 0$$ since the square of two real numbers is always positive, the condition will give only a number <= 0. if it is = 0 you cannot conclude anything if it is <0 it is a saddle point so u(x,y) and v(x,y) cannot have either a maximum or a minimum in the interior of any region in which w(z) is analytic.
 P: 7 Functions of Complex Variables Precisely, the fact that $$M$$ can be zero doesn't allow proof to be completed. The only way to complete it is to show that for points where $$u_x=0$$ and $$u_y=0$$ $$M$$ cannot be zero. or there is probably different approach which I cannot see...
 P: 7 From Gaussian theorem we have $$\int _V\nabla^2 u \, dV=\int_S \nabla u \cdot n\, dS$$ $$0=\int_S \nabla u \cdot n\, dS$$ I intuitively see that zero flux of $$\nabla u$$ implies that $$u$$ cannot have minimum or maximum but do not fully grasp it. If it is to be true, then in the enclosed region where $$u$$ is minimum/maximum $$\nabla u$$ must be all negative/positive but cannot see why this is true.