Let f be an entire function . Which of the following statements is FALSE ?

Let $Y=\{1,2,3,\dots ,100\}$ and let $h:Y\to Y$ be a strictly increasing function. The total number of functions $g:Y\to Y$ satisfying $g(h(j))=h(g(j))$ for all $j\in Y\; are\; ?$​

Let  f  be a holomorphic function on the disc $$\{z \in \mathbb{C}: |z| < 2\}$$​ 

Assume that the only zero of  f  in the closed unit disc  $$\{z \in \mathbb{C}: |z| \leq 1\}$$​​

is a simple zero at the origin. Let  $$\gamma$$​ be the positively oriented circle  $$\{z \in \mathbb{C}: |z| = 1\}.$$​​

The integral  $$\int_\gamma \frac{dz}{f(z)}$$​  equals ?

Sum of the infinite series 

​$$S = \frac{1}{2} - \frac{1}{3 \times 1!} + \frac{1}{4 \times 2!} - \frac{1}{5 \times 3!} + \cdots$$

is equal to ?

A function f: ℂ → ℂ is said to be analytic at ∞, if the function g defined by

g(w) = f ($$\frac{1}{w}$$​) is analytic at 0 with an appropriate value given for g(0).

Which of the following statements is true?

If | $$e^{e^z}$$ | = 1 for a complex number z = x + iy, x, y ∈ R,then.

which of thefollowing is true ?

Let f be a non-constant entire function such that |f(z)| = 1 for |z| = 1.

Let U denote the open unit disk around 0.

Which of the following is FALSE ?

​$$\text{Define } f : \mathbb{C} \to \mathbb{C} \text{ by } f(z) = ||z|^2 - 1|^2. \\\text{ Which of the following statements is true?}$$​​

​$$\text{Let } \gamma \text{ be the positively oriented circle in the complex plane given by:} \\\{z \in \mathbb{C} : |z - 1| = \frac{1}{2}\}. \\\text{The line integral:} \int_{\gamma} \frac{z e^{1/z}}{z^2 - 1} \, dz \ is\ ?$$​​

​$$\text{For a positive integer } p, \text{ consider the holomorphic function:} \\f(z) = \frac{\sin z}{z^p}, \quad \text{for } z \in \mathbb{C} \setminus \{0\}.\text{For which values of } p \text{ does there exist}\\\text{ a holomorphic function } g: \mathbb{C} \setminus \{0\} \to \mathbb{C} \\\text{such that } f(z) = g'(z), \text{ for } z \in \mathbb{C} \setminus \{0\}?$$​​

​$$\text{Let } \gamma \text{ be the positively oriented circle in the complex plane given by:} \\\{z \in \mathbb{C} : |z - 1| = 1\}. \\\text{Then:} \\\frac{1}{2\pi i} \int_{\gamma} \frac{dz}{z^3 - 1} \quad \text{equals.}$$​​