I'm considering here the fact that
limR→∞∫ΓReizz2+1dz=0
, where \Gamma is a contour defined as a semicircle centred about the origin, of radius R>1, in the upper half-plane of \mathbb{C}, positively oriented.
Now, I think, one could express \int\limits_{-\infty}^{\infty} \frac {sin(t)}{t^2+1} dt as \lim\limits_{R\to\infty} \int_{\Gamma_R} \frac {sin(t)}{t^2+1} dt But here's where I don't know what to do next. Would appreciate a hint.
On a side note,
\int\limits_{-\infty}^\infty \frac {e^{iz}}{z^2+1} dz\ne 0
, so I'm not sure what I'm not seeing here because \lim\limits_{R\to\infty} \int_{\Gamma_R} \frac {e^{iz}}{z^2+1} dz=0, so where's the analogy?
Answer
The integrand is and odd function of t and the integral is absolutely convergent, so
\int_{-\infty}^{\infty}\frac{\sin t}{t^2+1}dt=0
EDIT: Writing out the theorem longhand,
\begin{align}\int_{-\infty}^{\infty}\frac{\sin t}{t^2+1}dt&=\lim_{M\rightarrow-\infty}\int_M^0\frac{\sin t}{t^2+1}dt+\lim_{N\rightarrow\infty}\int_0^N\frac{\sin t}{t^2+1}dt\\ &=\lim_{M\rightarrow-\infty}\int_{-M}^0\frac{\sin(-u)}{u^2+1}(-du)+\lim_{N\rightarrow\infty}\int_0^N\frac{\sin u}{u^2+1}du\\ &=\lim_{M\rightarrow-\infty}-\int_0^{-M}\frac{\sin u}{u^2+1}du+\lim_{N\rightarrow\infty}\int_0^N\frac{\sin u}{u^2+1}du\\ &=-\int_0^{\infty}\frac{\sin u}{u^2+1}du+\int_0^{\infty}\frac{\sin u}{u^2+1}du=0\end{align}
Where we have used the substitution u=-t in the left integral and u=t in the right integral. Notice that all that was required for this proof to work was that the integrand was an odd function, the interval of integration was (-\infty,\infty), although an interval of (-a,a) would have worked and that the function was Riemann integrable over the interval. This theorem can reduce some perfectly horrible-looking integrals to zero with relatively little effort.
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