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 ?

Correct option is B

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

Option 1.

Let m $$\geq$$​ n  take m = n = 2 Then,

​$$I_{m,n} = \frac{1}{2\pi i} \oint_{C} z^2 z^{-2} \, dz=\frac{1}{2\pi i} \oint_{c}1dz =0\neq1$$

$$\implies \textbf{Option 1 is incorrect.}$$

Option 2.

Let m+1=n , Then

​$$I_{m,n} = \frac{1}{2\pi i} \oint_{C} z^{n-1} z^{-n} \, dz=\frac{1}{2\pi i} \oint_{c}\frac{1}{z}dz =\frac{1}{2\pi i} 2\pi i=1 \text{ ( Using cauchy integral formula}$$)

$$\implies \textbf{Option 2 is correct.}$$​​

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