​Let a,b be two real numbers such that a < 0 < b . For a positive real number r , define 

If  $$I_{m,n} = \frac{1}{2\pi i} \oint_{C} z^m z^{-n} \, dz \quad \text{where} \quad |z| = 1$$ ,

Which of the following statements is necessarily true ?

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 ?

Let p be a positive integer. Consider theclosed curve $$r(t)=e^{it}, 0\leq t< 2\pi$$​.

Let f be afunction holomorphic in $$\{z:|z|<R\}$$​ whereR >1. If f has a zero only at $$z_0,0<|z_0|<R$$​ ,and it is of multiplicity q , then $$\frac{1}{2\pi i}\int_r \frac{f'(z)}{f(z)}z^pdz$$ equals?​

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

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

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

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Let C be a positively oriented circle in the complex plane of radius 3 centered at the origin.

What is the value of the integral:

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​Let a,b be two real numbers such that a < 0 < b . For a positive real number r , define