Notice that g is not really a polynomial in . As for the other question, a function f is holomorphic at z if it's (complex) differentiable in an open ball around z.
(a) At what points are the functions and and differentiable? At what points are f and g and h holomorphic?
Using Cauchy-Riemann equations (+ showing continuity of partial derivatives) I have found:
f is differentiable only at
For g, solving for x and y in Cauchy-Riemann I end up with and so is in the form . Now, are continuous \{ } and since we must have \{ }. So, g should be differentiable \{ } satisfying .
But I have read that polynomial with coefficients in are differentiable in . Hence, I'm not sure about my answer for g.
h is differentiable only at
As to where the functions are holomorphic, I'm not quite sure I understand the concept very well. What I have found:
f and h are nowhere holomorphic.
if my answer for the first part is right, then g is holomorphic in and . But now that I think of it it doesn't really make much sense.
Can someone please check these answers for me?
I'm not sure I see why g is not a polynomial in .
I know the definition of holomorphic but I'm not sure if I understand how to apply it.As for the other question, a function f is holomorphic at z if it's (complex) differentiable in an open ball around z.
Do my answers to both questions seem right to you? What about the differentiability of g?
Notice that if then is not a lower bound for .
So .
In this way construct a decreasing sequence of distinct terms from converging to .
Consider this set of nonnegative numbers: .
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