- #1

QIsReluctant

- 37

- 3

## Homework Statement

... if f

_{j}are holomorphic on an open set U and f

_{j}[itex]\stackrel{uniformly}{\rightarrow}[/itex] f on compact subsets of U then δ/δz(f

_{j}) [itex]\stackrel{uniformly}{\rightarrow}[/itex] δ/δz(f) on compact subsets of U.

__Give an example to show that if the word "holomorphic" is replaced by "infinitely differentiable" then the result is false.__

## Homework Equations

The above.

## The Attempt at a Solution

I've used the disk D(0, 1) and all of the obvious choices: |z|, [itex]\overline{z}[/itex], etc. None of them work. It seems that their derivatives exhibit similar behaviour to the functions themselves, i.e., converge uniformly on that disk. I've considered some wackier functions like ln(z) but finding the real and imaginary parts of the function and doing partial derivatives is, shall we put it mildly, a chore. Can anyone hrlp me plz?