Suppose that f is holomorphic in a region G(i.e. an open connected set). How can I prove that in any of the following cases
a)R(f) is constant b)I(f) is constant c)|f| is constant d) arg(f) is constant we can conclude that f is constant.
Ps. here R(f) and I(f) are the real and imaginary parts of f.© BrainMass Inc. brainmass.com March 4, 2021, 7:22 pm ad1c9bdddf
We only need the C-R equations.
We know, if f is analytic at a point z=(x,y), f(x,y)=u(x,y)+iv(x,y), then u,v satisfies the C-R equations: du/dx=dv/dy, du/dy=-dv/dx.
Now we come back to this problem. Since f ...
Holomorphisms are investigated. The solution is detailed and well presented.