I am currently reading Hamming's Numerical Methods for Scientists and Engineers. On pg. 79 he discusses the topic of finding the zeros of a complex analytic function.
He then proceeds to discuss different types of complex conjugation for a function $w(z)$. Here are examples of the three types for $w(z)=\sin z = \frac{e^{iz} - e^{-iz}}{2i}$
- $\overline{w}(z)$ : replace $i$ with $-i$ in $w$, e.g. $\overline{w}(z)= \frac{e^{-iz} - e^{iz}}{-2i} = \sin z$.
- $w(\overline{z})$ : conjugate the argument, $w(\overline{z}) = \frac{e^{i\overline{z}} - e^{-i\overline{z}}}{2i}$
- $\overline{w(z)}$ : Hamming describes this as conjugating the values which I take to mean the conjugate of the image of $w(z)$ in the codomain. Although, I am not clear on this.
These three definitions have left me a bit confused. 2 and 3 seem relatively straight forward. But 1 leaves me a bit baffled in that it doesn't appear to be an actual complex conjugate of anything.
It appears that the first version can only be applied to functions of $z$, because if I rewrite
$w(z)=\sin z$ as $w(x + iy) = \sin(x+iy) = \sin x \cosh y + i \cos x \sinh y$
then
$\overline{w}(x + iy) = \sin x \cosh y - i \cos x \sinh y \neq w(x+iy) = \sin(x + iy)$
gives what I would expect to be the value of case 3 and a different answer then in the $z$-form where $w(z)=\overline{w}(z) = \sin z$.
Also, he later appears to state that $\overline{w(z)} = \overline{w}(\overline{z})$ although it is not clear whether that is true for all analytic functions or just those that have the property of $w(z) = \overline{w}(z)$ like $\sin z$.
My question is several-fold. Is the definition of conjugation given in 1 standard? What is its meaning? Also, is my interpretation of 3 correct and is $\overline{w(z)} = \overline{w}(\overline{z})$ the proper definition of 3.
Addendum:
The motivation behind understanding definition 1 is that Hamming uses it in the proof that analytic complex functions have zeros that are conjugate pairs if the function is real over the real domain.
He states without proof that if $w(z)$ is real for real $z$ then $w(z)=\overline{w}(z)$. He then provides the following proof for the above.
$w(a+bi)=0=\overline{w(a+bi)}=\overline{w}(a-bi)=w(a-bi)$