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I don't quite understand one example in my notes it says.

Exercise with answer showing that conjugation is not complex differentiable

My query is this: I don't understand what the significance of $\theta$ is. Why does it matter that $\theta \in (-\pi,\pi]$? I see the argument as this, clearly anyway in which we approach $0$ must each give the same limit otherwise the limit doesn't exist since the limit is independent of $w$ the limit only depends on $\theta$ hence for any two distinct values of $\theta$ of which there are infinitely many the limit is different $\implies$ the limit does not exist.

Is this correct?

Also what is $\theta$ changes as the complex number $w$ approaches zero what is to say we need to approach along a straight line with constant $\theta$?

Could anyone clear up my misconceptions?

Thanks.

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The significance of $\theta$ comes from the exponential/polar representation of a complex number. Here we have $$ w \in \mathbb{C} \iff \exists r \in \mathbb{R}^+ : \exists \theta \in \mathbb{R} : w = re^{i \theta}$$ However the author of the answer has limited our investigation to a subset of the possible values of $\theta$ such that $\theta \in (-\pi , \pi]$. I believe this is a roundabout way of saying that there exist values of $\theta$ for which the limit is different and by inspection of this subset of values we may find them. The argument does not change if we let $ \theta \in \mathbb{R}$. Perhaps the author was trying to get your attention to some specific values in the interval? For instance $$ \theta = 0 \implies w = r, \ \overline{w} = r \implies \frac{\overline{w}}{w} = 1$$ $$ \theta = \frac{\pi}{2} \implies w = ir, \ \overline{w} = -ir \implies \frac{\overline{w}}{w} = -1 $$ Both these values of $\theta$ are in the interval described, and as can be seen now the limit is not unique and hence does not exist.

As to the second part. You are stumbling upon a very nice property of complex analysis. The fact that "direction" is intrinsically built in! What I mean by this is that as opposed to the standard formulation of differentiation on $\mathbb{R}^2$ in which we deal with linear transformations and sorts, we instead have a formulation of the derivative which is not entirely dependent on this "distance" from the point we are investigating. This can be seen by considering the mapping $$ (x,y) \to (x, -y)$$ This corresponds to a reflection along the x-axis, and furthermore in complex analysis this is the act of conjugation! However one will find out that although reflection in $\mathbb{R}^2$ is differentiable, in complex analysis it is not(holomorphic that is)!

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It is not necessary for $w$ to approach $0$ along straight lines. But if the limit were to exist, it wouldn't matter what path one takes to $0$. So finding a contradiction among the straight line paths to $0$ is enough to conclude the limit doesn't exist.

This is just like ordinary limits. Or perhaps this feels like ordinary limits in multivariable calculus/analysis.

I do not understand what you say after My query is this:

The crux of the problem is that we can approach $0$ along different paths at an angle $\theta$. The resulting limit depends on $\theta$, which means the limit doesn't exist. This is akin to the function $$ f(x) = \begin{cases} -1 & x < 0 \\ 1 & x > 0\end{cases}$$ and asking for the limit as $x \to 0$. If we approach from the right, we get $1$. If we approach from the left, we get $-1$. But for the limit to exist, it must not matter from which direction we approach.

So the limit does not exist.

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  • $\begingroup$ I understand that now and the examples given above really helped concrete it for me thank you. $\endgroup$ – David P Jul 2 '15 at 11:39

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