Let a, b be two real numbers such that a < 0 < b. For a positive real number r, define γ_r(t) = re^it (where t ∈ |0, 2π|) and I_r =  Which of the following statements is necessarily true? 

Concept:

Residue Theorem:
 

The integral of a function around a closed contour in the complex plane is  times the sum of the residues

of the function inside the contour. Hence,  will depend on whether  and /or  lie inside the contour defined by  .

Explanation: The contour  is a circle of radius  centered at the origin, and the integrand has singularities

(poles) at  and  .

Poles of the integrand:

The function  has two poles

at  and at 

, where  for  and  are real numbers such that  . 

The poles of the function  within the contour , which is a circle of radius   centered at the origin.

The poles of the function are at and . Since  and  , the two poles are located on opposite sides of the origin.

Depending on the radius r , the contour  may or may not enclose one or both of the poles.

Conditions for :

If the radius r is smaller than |a| (the absolute value of a ), then the contour does not enclose

any poles, so by the Cauchy integral theorem,   = 0 .

If the radius r is greater than  , the contour encloses both poles, and by the residue theorem,  will be non-zero.

If r is between |a| and b , the contour may enclose exactly one pole, which will also make  non-zero.

Option 3: Ir = 0 if r > max {|a|, b} and |a| = b

This condition is true because if r > , the contour encloses both poles, leading to a situation

where the integral sums the residues at both poles, potentially canceling each other out.

Additionally, if , both poles are symmetrically placed, reinforcing the cancellation.

Thus, Option 3) is the correct answer.