captain black can you or someone else give me a hand to solve this problem. spent hours trying to do it but finding it very very difficult so if someone could help me would be really grateful and would really help my understanding
thanx in advance
(a) Verify that Im(z) and z do not satisfy the Cauchy-Riemann equations
at any point (so neither function is differentiable anywhere).
(b) Consider the function f(x+iy) = p|xy|, where x, y ∈ R. Show that
f satisfies the Cauchy-Riemann equations at the origin, yet f is not
holomorphic at 0.
Edgar has emailed me with the question he had meant to post when he
posted the above by mistake:
Suppose that f is holomorphic in an open set D. Use the Cauchy-Riemann
equations to prove that in any of the following cases:
(a) Re(f) is constant;
(b) Im(f) is constant;
(c) |f| is constant;
one can conclude that f is constant.
Hint for part (c): calculate the partial derivatives of |f|2 = u2+v2 = const.
RonL
(a) Let ,
then the Cauchy-Riemann equations tell us that:
,
and:
But we have is a constant, so:
,
and
So along any curve such that ; is a constant.
But a holomorphic function is determined every where by its values on a
curve, but is holomorphic and equal to on ,
so every where in .
Part (b) is similar.
RonL
Now this last one is fiddly, and so prone to error, so you will need to
check this carefully.
constant implies is a constant, so:
and:
Hence we have:
and .
But and also satisfy the Cauchy-Riemann equations:
and .
So with a bit of jiggery pokery we have:
and .
Now multiply the first of these by and the second by to get:
and ,
so:
,
Now chose a vertical line segment on which has no zeros (such a segment exists because the zeros of a non-zero holomorphic function are isolated), then on this segment we have (as if this were non zero at some point we would have which would mean that at that point - a contradiction), so is a constant on this segment.
Now I don't have the patience to do this myself, but I would expect a similar argument to show that on this segment , which implies that is a constant on the segment, which with our previous result shows that is constant on this segment, but a holomorphic function is determined everywhere by its values on a curve (of any kind) and so determined by its values on the segment. But on this segment is equal to a constant function which is also holomorphic so is constant on .
RonL