Final answer:
∫γ₂ eᶦᶻ / z dz | → 0 as R → ∞.The behavior of the exponential function and the geometric growth of the contour's length elucidate why the absolute value of the integral tends to zero as the contour size expands infinitely in the complex plane.
Step-by-step explanation:
The given contour integral represents the integral of the complex function eᶦᶻ / z along the contour γ₂ defined as z = Reᶦᵗ where 0 ≤ t ≤ π, as R approaches infinity. To understand why the absolute value of this integral tends to zero, we can use the Estimation Lemma in complex analysis.
When considering the contour γ₂, as R becomes very large, the function eᶦᶻ / z tends to zero faster than the increase in the arc length of the contour. This is due to the exponential function's behavior in the complex plane; it grows at a slower rate compared to the polynomial increase in the length of the contour as R increases.
Considering the contour's circular shape in the complex plane, the magnitude of eᶦᶻ / z tends to zero uniformly as R → ∞. This behavior causes the integral's absolute value over the contour γ₂ to approach zero, satisfying the condition stated in the question.
In essence, the integral of the given function over the contour γ₂ decreases faster than the contour's length grows as R approaches infinity, ultimately resulting in the limit of the absolute value of the integral approaching zero.
The behavior of the exponential function and the geometric growth of the contour's length elucidate why the absolute value of the integral tends to zero as the contour size expands infinitely in the complex plane.