Integrals with Trig Factors

Complex Analysis · Axiom Academy

LESSON Integrals with Trig Factors A real integral carrying a or is hard head-on — but routed through the complex exponential e^ iax , it becomes one residue. 1. Package Both Waves Into One Exponential Euler's identity says . So a single complex integrand secretly carries both trig integrals at once — the cosine version is its real part , the sine version its imaginary part : Sine integral = imaginary part 2. The Exponential Dies Upward Promote x to a complex z=x+iy . With a>0 , the modulus of e^ iaz depends only on the height y — and it collapses toward zero as we climb into the upper half-plane: That decay kills the big closing arc, so we may close the contour upward when a>0 . The exponential's decay does extra work, so we only need for f=P/Q — milder than the +2 a bare rational function needs. 3. Close the Contour, Collect the Residue Now run the full method on . Replace with e^ ix and close with an upper-half-plane arc of radius R . The arc dies (Step 2), the bottom edge becomes the real integral, and only the pole z=i sits inside: Residue at the enclosed pole z=i Residue Theorem, then take the real part Reading off the answer: is already real, so its real part is itself — the cosine integral equals . Its imaginary part is 0 , which says — exactly as the odd integrand demands. Real, nonzero — compute it from the residues. Odd integrand ⇒ the integral is 0 by symmetry. For , use e^ 2iz with the pole z=2i : , so — and the real part gives .

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