To find the value of the given contour integral, we can use the Cauchy's Integral Formula.

Step 1: Determine the function f(z) that we need to integrate. In this case, f(z) = ez / (z^2 + 9).

Step 2: Check if the function f(z) is analytic within the region enclosed by the contour C. In this case, f(z) is analytic everywhere except at z = ±3i, which are outside the contour C. Therefore, f(z) is analytic within the region enclosed by C.

Step 3: Apply Cauchy's Integral Formula, which states that for a function f(z) that is analytic within a region enclosed by a simple closed contour C, the value of the contour integral ∮ f(z) dz over C is equal to 2πi times the sum of the residues of f(z) at its isolated singularities within C.

Step 4: Find the residues of f(z) at its isolated singularities within C. In this case, the singularities of f(z) are at z = ±3i. To find the residues, we can use the formula Res(f, z0) = lim(z→z0) [(z - z0) * f(z)].

For z = 3i, the residue is Res(f, 3i) = lim(z→3i) [(z - 3i) * (ez / (z^2 + 9))] = (3i * ez / (3i)^2) = (3i * ez / -6) = -i/2 * ez.

For z = -3i, the residue is Res(f, -3i) = lim(z→-3i) [(z + 3i) * (ez / (z^2 + 9))] = (-3i * ez / (-3i)^2) = (-3i * ez / -6) = i/2 * ez.

Step 5: Calculate the sum of the residues. The sum of the residues is Res(f, 3i) + Res(f, -3i) = (-i/2 * ez) + (i/2 * ez) = 0.

Step 6: Multiply the sum of the residues by 2πi. 2πi * 0 = 0.

Therefore, the value of the contour integral ∮ f(z) dz over C is 0.

Hence, the correct answer is (a) 0.

Question

To find the value of the given contour integral, we can use the Cauchy's Integral Formula.

Step 1: Determine the function f(z) that we need to integrate. In this case, f(z) = ez / (z^2 + 9).

Step 2: Check if the function f(z) is analytic within the region enclosed by the contour C. In this case, f(z) is analytic everywhere except at z = ±3i, which are outside the contour C. Therefore, f(z) is analytic within the region enclosed by C.

Step 3: Apply Cauchy's Integral Formula, which states that for a function f(z) that is analytic within a region enclosed by a simple closed contour C, the value of the contour integral ∮ f(z) dz over C is equal to 2πi times the sum of the residues of f(z) at its isolated singularities within C.

Step 4: Find the residues of f(z) at its isolated singularities within C. In this case, the singularities of f(z) are at z = ±3i. To find the residues, we can use the formula Res(f, z0) = lim(z→z0) [(z - z0) * f(z)].

For z = 3i, the residue is Res(f, 3i) = lim(z→3i) [(z - 3i) * (ez / (z^2 + 9))] = (3i * ez / (3i)^2) = (3i * ez / -6) = -i/2 * ez.

For z = -3i, the residue is Res(f, -3i) = lim(z→-3i) [(z + 3i) * (ez / (z^2 + 9))] = (-3i * ez / (-3i)^2) = (-3i * ez / -6) = i/2 * ez.

Step 5: Calculate the sum of the residues. The sum of the residues is Res(f, 3i) + Res(f, -3i) = (-i/2 * ez) + (i/2 * ez) = 0.

Step 6: Multiply the sum of the residues by 2πi. 2πi * 0 = 0.

Therefore, the value of the contour integral ∮ f(z) dz over C is 0.

Hence, the correct answer is (a) 0.

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