We consider the integral of f(z) = (e^(iz))/(z) over a contour C. The contour C consists of a large semicircle C_R in the upper half-plane with radius R, a small semicircle C_ around the origin with radius, and two segments along the real axis from -R to - and from to R. The pole at z=0 lies on the contour, so we must indent around it.

We consider the integral of f(z) = (e^(iz))/(z) over a contour C. The contour C consists of a large semicircle C_R in the upper half-plane with radius R, a small semicircle C_ around the origin with radius, and two segments along the real axis from -R to - and from to R. The pole at z=0 lies on the contour, so we must indent around it.

Evaluate ∫₋∞^∞ sin(x)/x dx using contour integration in the complex plane.

Answer: $\pi$

Step-by-step solution

Step 1: Rewrite the integral using Euler's formula

We can express $\sin(x)$ using Euler's formula, $\sin(x) = \text{Im}(e^{ix})$ . This allows us to rewrite the real integral as the imaginary part of a complex integral, which is often easier to evaluate using contour integration.

Step 2: Define the contour integral

We consider the integral of $f(z) = \frac{e^{iz}}{z}$ over a contour $C$ . The contour $C$ consists of a large semicircle $C_R$ in the upper half-plane with radius $R$ , a small semicircle $C_\epsilon$ around the origin with radius $\epsilon$ , and two segments along the real axis from $-R$ to $-\epsilon$ and from $\epsilon$ to $R$ . The pole at $z=0$ lies on the contour, so we must indent around it.

Step 3: Apply Cauchy's Residue Theorem

Since there are no poles inside the chosen contour $C$ (the pole at $z=0$ is excluded by the indentation), Cauchy's Residue Theorem states that the integral over the closed contour is zero.

Step 4: Evaluate integrals over contour segments

As $R \to \infty$ , the integral over $C_R$ vanishes by Jordan's Lemma. The integral over $C_\epsilon$ (the small semicircle around the origin) can be evaluated using the formula for a simple pole on the contour: $\int_{C_\epsilon} f(z) dz = -i\pi \text{Res}(f, 0)$ . The residue at $z=0$ is $\lim_{z\to 0} z \frac{e^{iz}}{z} = e^{i(0)} = 1$ .

Step 5: Combine results and solve for the principal value

Combining the results, the principal value of the integral along the real axis plus the integral over the small semicircle equals zero. This gives us the principal value of the complex integral as $i\pi$ .

Step 6: Extract the imaginary part

Finally, we take the imaginary part of the result to find the value of the original integral.

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