In my signal and systems course, my professor claims that the energy of a signal $x(t)$ over an infinite and continuous time interval is: $$ E_{\infty}=\lim _{T \rightarrow \infty} \int_{-T}^{T}|x(t)|^{2} d t=\int_{-\infty}^{+\infty}|x(t)|^{2} d t $$ My concern is about the substitution of $\infty$ in place of $T$ before evaluating the integral. Is this always correct or is it mathematically more correct to evaluate the integral then compute the limit?
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3$\begingroup$ Look back at the definition of $\int_{-\infty}^{+\infty}$ - there's less here than meets the eye! $\endgroup$– Noah SchweberFeb 2, 2021 at 4:54
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$\begingroup$ Check out en.wikipedia.org/wiki/Integral#Improper_integrals $\endgroup$– Amaan MFeb 2, 2021 at 4:55
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1$\begingroup$ Thank you so much I think my question was embarrasing and hence I will delete it oh but before I do, I noticed in the wikipedia page that they said this is an abuse of notation to replace the upper bound with a $\infty$ $\endgroup$– Maria KuznetsovFeb 2, 2021 at 4:57
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Many people write $\int_{-\infty}^{\infty} |x(t)|^2dt$ as $\lim \limits_{T \to \infty} \int_{-T}^{T} |x(t)|^2dt$ because they believe that using infinity as a number is not proper, so they write it as a limit. However, since $$\int_{-\infty}^{\infty} |x(t)|^2dt = \lim \limits_{T \to \infty} \int_{-T}^{T} |x(t)|^2dt$$
is true, that must mean that the converse, $$\lim \limits_{T \to \infty} \int_{-T}^{T} |x(t)|^2dt =\int_{-\infty}^{\infty} |x(t)|^2dt $$
is true as well.
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