Evaluate ∫₋∞^∞ sin(x)/x dx using contour integration in the complex plane.
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Step-by-Step Solution
Step 1: Rewrite the integral using Euler's formula
We can express sin(x) using Euler's formula, sin(x)=Im(eix). 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)=zeiz over a contour C. The contour C consists of a large semicircle CR in the upper half-plane with radius R, a small semicircle Cϵ around the origin with radius ϵ, and two segments along the real axis from −R to −ϵ and from ϵ 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→∞, the integral over CR vanishes by Jordan's Lemma. The integral over Cϵ (the small semicircle around the origin) can be evaluated using the formula for a simple pole on the contour: ∫Cϵf(z)dz=−iπRes(f,0). The residue at z=0 is limz→0zzeiz=ei(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 iπ.
Step 6: Extract the imaginary part
Finally, we take the imaginary part of the result to find the value of the original integral.