I'm just starting to learn complex analysis and currently reading the book "Complex analysis" by Stein and Shakarchi.
Since the fact of the function being holomorphic appears to be pretty important, I'm starting with it.
Here's the definition:
The function $f$ is holomorphic at the point $z_0 \in \Omega$ if the quotient $$\frac{f(z_0+h)-f(z_0)}{h}$$ converges to a limit when
$h \to 0$.
Then we are given two examples. First of the function $f(z)=z$ being holomorphic on any open set in $\Bbb C$, and $f'(z) = 1$. And second of the function $f(z)=\bar z$ being not holomorphic.
While I think I understand first example, I cannot see how the second will be different, how the fact of the complex conjugate changes the set up? Basically, why $f(z)=\bar z$ is not holomorphic by definition?
I am aware of the Cauchy-Riemann equations, but I want to see the proof based on the definition (especially that the fact is claimed at this point in the book).
I tried to search for an answer but didn't find anything particularly related. I also think I might miss some very basic concept. With all that said I will really appreciate your input.
Best Answer
A good rule of thumb for complex analysis is that, if something has gone wrong, $0$ is almost certainly to blame.
Here is the difference quotient for $\overline{z}$ at $0$ taken along two different paths in the complex plane:
Tending to $0$ along the positive imaginary axis produces:
$\lim_{h\rightarrow0}\frac{\overline{\left(0+ih\right)}-\overline{0}}{ih}=\lim_{h\rightarrow0}\frac{-ih}{ih}=-1$
On the other hand, tending to $0$ along the positive real axis produces:
$\lim_{h\rightarrow0}\frac{\overline{\left(0+h\right)}-\overline{0}}{h}=\lim_{h\rightarrow0}\frac{h}{h}=1$
which is completely different. Since the limits do not agree, the limit of this difference quotient as $h$ tends to $0$ in $\mathbb{C}$ does not exist. Thus, the conjugate function is not complex differentiable at $0$.
Generally, when working "from first principles" as it were, one has to rely on the two-dimensionality of $\mathbb{C}$ (along with its algebraic/"arithmetic" properties) to show that things are different than the case of single real-variable analysis. Indeed, it is exactly because the condition of holomorphy requires these two-dimensional limits to exist that holomorphic functions are so marvelously well-behaved (in stark contrast to functions of multiple real variables).