It is quite time-consuming to check a complicated looking function's analyticity. Is there any trick that we can use? or is there an online tool that can check analyticity?
If two analytic function multiplied together, do we still get an analytic function?
If $f(z)$ and $g(z)$ are both analytic, then is $f(g(z))$ also analytic?
Are there any rules like these? I can not find any material.
From Wikipedia's holomorphic function page:
Because complex differentiation is linear and obeys the product, quotient, and chain rules; the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero.
... which addresses your particular examples. In fact, the material in the rest of that section is much of the first material in a Complex Analysis course. With the immediately following examples, you get a starting point of several holomorphic functions.
Following zhw's comment -- try Saff's & Snider's, $Fundamentals~of~Complex~Analysis$ with Applications to Engineering, Science, and Mathematics, 3rd Ed.
I'll post some theorems that may help you. The analyticity property of a function $z=x+iy\longmapsto^{^{\!\!\!\!\!f}} u(x,y)+iv(x,y)$ exhibits a connection between the real and imaginary parts of said function.
First, we recall the Cauchy-Riemann Equations. To keep it simple (I'm going to use capital letters here to distinguish from the notation in the theorems below), we consider the (complex-valued) function $F(z)=U(x,y)+iV(x,y)$, where $z\in\text{Dom}(f)\subseteq\mathbb{C}$, and we assume that $F$ is differentiable at some $z_{0}\in\text{Dom}(F)$. If you would like the derivation, I'm happy to provide it; however, we can derive:
$\dfrac{\partial U}{\partial x}=\dfrac{\partial V}{\partial y}~~$ as well as $~~\dfrac{\partial U}{\partial y}=-\dfrac{\partial V}{\partial x}$ which are called the Cauchy-Riemann equations.
Now, here are some theorems regarding analyticity as a sufficient and necessary condition, respectively, in connection with these equations. The first of two theorems is as followed.
Theorem 1 A necessary condition for a function $f(z)=u(x,y)+iv(x,y)$ to be differentiable at a point $z_{0}$ is that the Cauchy-Riemann equations must hold at $z_{0}$. Consequently, if $f$ is analytic in an open set $G$, then the Cauchy-Riemann equations must hold at all points of $G$.
The second theorem is as followed.
Theorem 2 Let $f(z)=u(x,y)+iv(x,y)$ be defined in some open set $G$ containing a point $z_{0}$. If the first partial derivatives of $u,v$ exist in $G$, are continuous at $z_{0}$, an satisfy the Cauchy-Riemann equations at $z_{0}$, then $f$ is differentiable at $z_{0}$. Consequently, if the first partial derivatives are continuous and satisfy the Cauchy-Riemann equations at all points of $G$, then $f$ is analytic in $G$.
If you would like proofs of these theorems, I'm also happy to provide them (I omitted the proofs and the derivation of the Cauchy-Riemann equations in the interest of saving space -- I found a free, downloadable *.pdf of the book I referenced above here...see page 73, 74, etc., which is where this material, and their proofs, can be found). To be mathematically precise, the concept with the second theorem is that the Cauchy-Riemann equations alone are not sufficient to ensure differentiability -- we need continuity of the first partials of $u=\text{Re}(f)$ and $v=\text{Im}(f)$ in addition to the Cauchy-Riemann equations holding to gives us analyticity in our open set $G$. On the other hand, the Cauchy-Riemann equations give a necessary condition for differentiability (namely analyticity).
Lastly and for example regarding the contrapositive of the first theorem, if you can find at least one point contained in some fixed, open set such that the Cauchy-Riemann equations do not hold, then the function can not be analytic there (to which this point may be a singularity and the given function is not analytic in this open set).
