The discussion revolves around the question of whether there exists any function for which the integral from negative infinity to positive infinity yields an integer value other than 0 or infinity. The scope includes theoretical exploration of integral equations and properties of functions.
Participants do not reach a consensus; multiple competing views and examples are presented regarding the types of functions that can satisfy the integral condition.
Some participants express uncertainty about the definitions and properties of the functions discussed, particularly regarding the impulse function and its classification.
This discussion may be of interest to those exploring integral equations, properties of functions in mathematical analysis, or the conceptual understanding of impulse functions.
Vorde said:Is there any function such that:
[itex]_{-∞}[/itex][itex]\int[/itex][itex]^{∞}[/itex]f(x) dx
Is any integer except 0 and ∞?
Vorde said:I looked it up, apparently is an equation that satisfies the statement that the integral from -∞ to ∞ is 1, but because of the way it's defined it isn't actually a function.
phyzguy said:There are an infinite number of such functions. For example:
[tex]\int^\infty_{-\infty}e^{-x^2}dx = \sqrt{\pi}[/tex]
so if:
[tex]f(x) = \frac{7}{\sqrt\pi}e^{-x^2}[/tex]
then:
[tex]\int^\infty_{-\infty}f(x)dx = 7[/tex]
There are lots of functions like this that one can play this game with.