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How can i say that analytic function must be free from $bar{z}?$ I verified it for many examples like $bar{z}, z^2+bar{z} $ etc and found that the results seems to be true according to me . I am thinking like as if any complex function contains $bar{z}$ term in any form(obviously in nontrivial form unlike $bar{z}-bar{z}$) then that terms is never analytic and hence whole function is also never analytic. Please suggest . Thanks .

This is an interesting question, which is usually glossed over when doing complex calculus in $z$ and $bar z$ terms. It is also not so clear what the actual claim is. E.g., the function $q(z):={rm conjugate}(bar z)$ is analytic $ldots$

I'm going to argue in the following setup: Let
$$Phi:quadOmegato{mathbb C},qquad(z,zeta)toPhi(z,zeta)$$
be defined on the domain $Omegasubset{mathbb C}^2$. Assume that $Phi$ is (i) a $C^1$ function of its four real variables $(x, y, xi,eta)$, and is (ii) separately analytic in each of the two complex variables $z$ and $zeta$, meaning that the partial functions
$$zmapstoPhi(z,zeta_0),qquad {rm resp.,}qquad zetamapstoPhi(z_0,zeta)$$
are analytic functions of $z$, resp., of $zeta$, as long as $(z,zeta_0)inOmega$, resp., $(z_0,zeta)inOmega$. Such a $Phi$ could be given as a series, or as an "analytic expression" in terms of the variables $z$ and $zeta$.

Given such a $Phi$ we consider the function
$$f(z):=Phi(z,bar z)qquad(zinOmega') ,tag{1}$$
whose domain is the open set $Omega':=bigl{zin{mathbb C}bigm| (z,bar z)inOmegabigr}subset{mathbb C}$. We now pose the following question:

What are necessary and sufficient conditions on $Phi$ making $f$ an analytic function on $Omega'$?

Claim. The function $f$ is analytic on $Omega'$ iff $$Phi_{.2}(z,bar z)=0qquadforall>zinOmega' .$$
Here $Phi_{.2}$ denotes the partial derivative of $Phi$ with respect to its second complex variable.

Proof. Fix a $zinOmega'$, and let $h$ be a complex increment variable. Then
$$eqalign{f(z+h)-f(z)&=Phi(z+h,bar z+bar h)-Phi(z,bar z)cr &=Phi_{.1}(z,bar z)>h+Phi_{.2}(z,bar z)>bar h +obigl(|h|bigr)qquad(hto0in{mathbb C}) .cr}tag{2}$$
Here we just have used the assumptions on $Phi$ and the definition of derivative. Now the main part of the right hand side of $(2)$ is a complex linear function of $h$, i.e., of the form $hmapsto lambda h$ for some $lambdain{mathbb C}$, iff $Phi_{.2}(z,bar z)=0$.$quadsquare$

When we are applying this principle there will be given a function $f$ defined in the form $(1)$. By abuse of notation one then does not explicitly introduce an auxiliary variable $zeta$. Instead one just considers the $bar z$ appearing in $(1)$ as an "independent complex variable" and formally differentiates the expression $Phi(z,bar z)$ with respect to this variable. In this way one arrives at the handy condition
$${partialPhi(z,bar z)overpartialbar z}=0qquadforall zinOmega' .$$