In article <firstname.lastname@example.org>, Christopher <email@example.com> wrote: >Hello, >Please tell me how to prove simply that an analytic function is a function of >z only, no z_bar.
First, you have to narrow it down to "continuously differentiable" functions (and centuries ago, people didn't consider other functions anyway). Then you re-write the Cauchy-Riemann equations for w=u+iv (u and v being real functions of real variables x, y): (partial u)/(partial x) = (partial v)/(partial y) (partial v)/(partial x) = - (partial u)/(partial y) substituting formally x+iy = z x-iy = z_bar (inverse substitution: x=(z+z_bar)/2, y=(z-z_bar)/)2*i)
and presto! (partial w)/(partial z_bar) = 0
which, in the minds of the people I mentioned before, indicated that "w is independent of z_bar".
There are gaps in the reasoning; to expose them, I recommend reading "Counterexamples in Analysis" by B. Gelbaum and J. Olmsted, Holden-Day, San Francisco 1964 (newer editions should exist), Chapter 9, Example 11.