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Comment by andrewla | Hacker News Reader

The trouble is that complex conjugation is not holomorphic (analytic) over the complex numbers.

It's incredibly useful as an operator, and it appears all over the place in Hilbert spaces and other complex-probability situations, but fundamentally it defies attempts to use it for analytic purposes (like differential equations or contour integration).

Useful when you need to treat the complex plane as a vector space or are interested in the topology of a complex function, but a pain to deal with in almost any other context.

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For those of you wondering why on Earth such a simple function wouldn't be well-behaved, it can help to think about what you can do with the conjugate.

In particular, since sums of holomorphic functions are holomorphic, the sum f(z) = z + z* of the identity function g(z) = z and conjugate function h(z) = z* would need to be holomorphic. But z + z* cancels out the imaginary part of z and therefore maps the complex plane to the real line - and it should be obvious that a map from C to R cannot possibly preserve the properties of C in any intuitive way.

Roughly speaking, the conjugate fails to be holomorphic because it maps C to its mirror image, rather than C itself, in much the same way that reflecting a circle is different from rotating it.

Constant functions are holomorphic without preserving the properties of C

The issue is that the Cauchy Riemann equations don't hold. (You would need 1==-1)

Picard's theorem tells us constant functions are the only example of that, though (more or less because 0 == -0 [1] and that's only true for 0).

[1] I sincerely apologize to anyone who works with floats for this statement.

No need to apologize, 0 == -0 even if we are talking about floats.

Is there an analytic function that approximates the conjugate, and can this approximation be made better and better by making the function more complicated?

Edit: no, it can't be done because the Cauchy-Riemann equations make no sense here.

For any smooth function (like the conjugate) it makes sense to ask whether there exist holomorphic functions that approximate it arbitrarily well.

In fact, if a sequence of holomorphic functions converges uniformely on compact sets, the limit is itself holomorphic because of Cauchy's theorem.

> The trouble is that complex conjugation is not holomorphic (analytic) over the complex numbers.

Even more generally, a real-differentiable function is complex differentiable iff its (Wirtinger) derivative with respect to z* is 0.

Does the problem boil down to selecting the real part of a complex number? Correct me if I'm wrong, but the conjugate of z is merely 2*Re(z) - z. For example, the conjugate of a + bi is 2*a - (a + bi) = a - bi.

There is no problem. This is just a non holomorphic function of C to C. Differentiable functions are the exception not the norm (no pun intended).

Maybe worth noting that it is real-analytic.

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