(adsbygoogle = window.adsbygoogle || []).push({}); 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 [tex]u(x,y)[/tex] and imaginary part [tex]v(x,y)[/tex] of an analytic function [tex]w(z)[/tex] each satisfy Laplace's equation, show that [tex]u(x,y)[/tex] and [tex]v(x,y)[/tex]cannot have either a maximum or a minimumin the interior of any region in which [tex]w(z)[/tex] is analytic. (They can have saddle points)

2. Relevant equations

Cauchy-Riemann (CR) relations for analyticity of the function [tex]u_x=v_y[/tex] and [tex]u_y=-v_x[/tex] where subscript stands for partial differentiation with respect to that variable.

[tex]\nabla^2u=0[/tex] and [tex]\nabla^2v=0[/tex] (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 [tex]u_x=0[/tex] and [tex]u_y=0[/tex]

and

[tex]M=u_{xx}u_{yy}-(u_{xy})^2>0[/tex]

[tex]\nabla^2u=u_{xx}+u_{yy}=0\therefore u_{xx}=-u_{yy}[/tex]

[tex]M=-u_{yy}^2-u_{xy}^2\leq0[/tex]

...and it looks like totally wrong direction...

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# Functions of Complex Variables

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