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- Thread starter Mappe
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fzero

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fzero

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If a derivative depends on the path, that derivative does not exist, and the function is not differentiable. If a function is differentiable, all derivatives exist.

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*Since the derivatives exist, they must necessarily be the same regardless of which direction you approach the point from.

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fzero

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*Since the derivatives exist, they must necessarily be the same regardless of which direction you approach the point from.

If f is analytic, u and v are continuous and differentiable. The use of "where" is a bit misleading, since it follows from analyticity and is not an additional assumption.

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The situation is:

1. Necessary condition: f(z) is analytic => C-R equations satisfied (True)

2. Sufficient condition: C-R equations statisfied => f(z) is analytic (False)

The first one is the necessary condition and it must hold if f(z) is analytic.

The second one is false in general. However if we also assume that all four partials derivatives of u(x,y) and v(x,y) exist and are themselves continuous, then the second condition is true. The proof of this is a good bit more tricky than the necessary proof that you have referred to. It involves the mean value theorem, and this applies to continuous, differentiable functions.

Hence, if these extra conditions are valid we have a two-way implication (aka a necessary and sufficient condition):

f(z) is analytic <=> C-R equations satisfied

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the converse of this is less trivial. i.e. if we assume instead the cauchy riemann equations hold on an open set, and also that the partials are continuous, then it follows that the function is holomorphic. i.e. in the presence of continuity of the partials, the equality of the limits in two directions implies the equality in all directions. this takes a little more work to prove.

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