# Are there any known approaches of generalizing functions that do not have a limit at infinity to values at infinity?

Let's consider the affinely extended real line. The functions that have a limit on positive or negative infinity $\lim_{x\to+\infty} f(x)$ or $\lim_{x\to-\infty} f(x)$ can be generalized to the values at infinity $f(+\infty)$, $f(-\infty)$ based on these limits.

But what about functions that do not have such limits?

For instance, if we take these integrals following Borel (or Abel) generalization, we get:

$\int_0^\infty \cos x=0$

and

$\int_0^\infty \sin x=1$

Since integral of $\cos x$ is $\sin x$ and integral of $\sin x$ is $-\cos x$, we can probably generalize these functions to infinity (assuming the fundamental theorem of calculus should still hold): $\sin (\infty)=\cos(\infty)=0$ (notice, this would apparently break the main trigonometric identity $\sin^2 x+\cos^2 x =1$ at this point but one of the books linked in the comments resolves the paradox by claiming that $(\sin \infty)^2\ne \sin^2 \infty$ and the same for cosine).

Still, I wonder whether there were other attempts at such generalization? Possibly, something along Cesaro lines (limit of mean value of the function) or sometyhing else?

• See the notion of "Banach limit" en.wikipedia.org/wiki/Banach_limit – Gerald Edgar Jul 5 '17 at 12:57
• Of possible historical interest are the 1800s items I cite in this 14 January 2009 sci.math post. – Dave L Renfro Jul 5 '17 at 14:36
• @Dave L Renfro the links from your post do not work :( – Anixx Jul 5 '17 at 16:04
• For the JFM reviews (whose links in my post don't work), here's the JFM look-up site itself. Enter author last name, part of title, etc. as needed. Alternatively, you can also obtain JFM reviews at the Zbl reviews look-up site (select "Structured Search"). – Dave L Renfro Jul 5 '17 at 16:35
• @Dave L Renfro interestingly, they claim $\sin \infty=\cos \infty=0$ but $\sin^2\infty=\cos^2\infty=1/2$ And, indeed the value at infinity is the mean value (so the Cesaro method applies). – Anixx Jul 5 '17 at 17:34

In physics we use the Sokhotski–Plemelj theorem to evaluate integrals from zero to infinity of cosine and sine in the form:

$$F(k)=\int_0^\infty (\cos kx +i\sin kx)\,dx=\pi\delta(k)+i\,{\cal P}\frac{1}{k}$$

where $\delta(k)$ is the Dirac delta function and ${\cal P}$ is a reminder that when the right-hand-side is integrated over $k$ one should take the Cauchy principal value of the integral: $$\int_{-\infty}^\infty F(k)g(k)\,dk=\pi g(0)+i\,{\cal P}\,\int_{-\infty}^\infty \frac{g(k)}{k}\,dk.$$

Incidentally, from $F(1)=i$ I would indeed associate $\int_0^\infty \cos x\,dx$ and $\int_0^\infty \sin x\,dx$ with $0$ and $1$, respectively, as in the OP.

• Still not clear, principal value of what integral?... – Anixx Jul 5 '17 at 12:03
• If you get -1, you definitely made some mistake, I think. It is still unclear though how the first and second formulas are related (what is g(x) for instance). – Anixx Jul 5 '17 at 13:11
• indeed, my mistake, corrected, thanks; $g(x)$ is an arbitrary function. – Carlo Beenakker Jul 5 '17 at 13:25
• Purely logically the integral should be positive because sine function is initially positive. It is analogious to the series 1-1+1-1+... which sums up to 1/2, while if the first term is -1, the sum would be -1/2 – Anixx Jul 5 '17 at 15:36
• the Wikipedia article has several applications of the Sokhotski–Plemelj integral relation, and a Google search gives many more pointers; a Fourier-transform formulation is here (appendix). – Carlo Beenakker Jul 5 '17 at 16:04

Many people developed a theory of limits of distributions at finite or infinite values in the 50's and 60's and for these all of your formulae hold. A particularly elementary version was developed by J. Sebastião e Silva and a complete description can be found online by googling has name and "limits of distributions at infinity". There is also a text book version by J. Campos Ferreira et al.---"Introduction to the theory of distributions".

• I could not find anything using your search suggestion. – Anixx Jul 5 '17 at 17:19
• The article I am referring to appeared in "Theory of Distributions", the proceedings of a summer institute published by the Gulbenkian Foundation, Lisbon, 1964. There is a site in Portugal which contains the complete works of Silva online. I will try to dig out the details for you. – cart Jul 5 '17 at 17:28
• Unfortunately, I cannot read in Partugal... – Anixx Jul 5 '17 at 17:33
• the article is not in portuguese. The site is "sebastiãoesilva100anos.org. – cart Jul 5 '17 at 17:34
• then click on "publicações". There is a french and an english version, as I recall. Anyway, the text of Campos Ferreira is in english. – cart Jul 5 '17 at 17:36