Exams › GATE › Engineering Mathematics

The values of the integral (1)/(2π j)∮_c (e^z)/(z-2) dz along a closed contour c in anti-clockwise direction for (i) the point z0 = 2 inside the contour c, and (ii) the point z0 = 2 outside the contour c, respectively, are

1. (A) (i) 2.72, (ii) 0
2. (B) (i) 7.39, (ii) 0
3. (C) (i) 0, (ii) 2.72
4. (D) (i) 0, (ii) 7.39

Correct answer: (B) (i) 7.39, (ii) 0

Solution

By the Cauchy integral formula, (1/2 pi j)*contour-integral of e^z/(z-2) with z0=2 inside equals e^2 = 7.39. With z0=2 outside the contour the integrand is analytic inside, so the integral is 0. Hence (i) 7.39, (ii) 0.

Related GATE Engineering Mathematics questions

• z = (2 - 3i)/(-5 + i) can be expressed as
• If the semi-circular contour D of radius 2 is as shown in the figure, then the value of the integral ∫_D 1/(s² - 1) ds is
• The equation sin(z) = 10 has
• The residue of the function f(z) = 1 / ((z + 2)² (z - 2)²) at z = 2 is
• If f(z)=c₀+c₁ z⁻¹, then ∮ (1+f(z))/z dz is given by unit circle
• An AC source of RMS voltage 20 V with internal impedance Zs = (1 + 2j) Ω feeds a load of impedance ZL = (7 + 4j) Ω in the figure below. The reactive power consumed by the load is

⚔️ Practice GATE Engineering Mathematics free + battle 1v1 →