Questions without answers in Complex variables

For the complex function$$f(z)=f(x+i y)=x^{3}-4 x y^{2}+i\left(4 x^{2} y-2 y^{3}\right)$$consider the following two statements$P: f$ is not analytic at $(0,0)$.$Q: f$ doe...

Let $C$ be the circle $(x-1)^{2}+y^{2}=25$ oriented counterclockwise. Then, the value of the line integral$$\oint_{C}\left[\left(x^{5}-3 y\right) d x+\left(-2 x+e^{y^{2}}...

​​​​​Let $f(z)$ be an analytic function such that$$ {Re}\left(f^{\prime}(z)\right)=3 x^{2}-4 y-3 y^{2}, f(i)=0, f^{\prime}(0)=0, $$where $i=\sqrt{-1}$. Then the value of ...

Consider $f(z)=e^{z}$, where $z=x+i y$ and $i=\sqrt{-1}$. Which of the following statements is correct?$f$ is periodic$f$ is not periodic$|f|=1$$\arg (f)=y \pm n \pi$ for...

Which one of the following functions is differentiable at $z=0$ but NOT differentiable at any other point in the complex plane $\mathbb{C}$?$f(z)=z|z|, \quad z \in \mathb...

Let $\mathbb{C}=\{z=x+i y: x$ and $y$ are real numbers, $i=\sqrt{-1}\}$ be the set of complex numbers. Let the function $f(z)=u(x, y)+i v(x, y)$ for $z=x+i y \in \mathbb{...

Let $\Gamma$ be the positively oriented circle $x^2+y^2=9$ in the $x y$-plane. If$$\oint_{\Gamma}\left(3 y+e^{x \sin x}\right) d x+\left(7 x+\sqrt{e^y+2}\right) d y=\alph...