Real analysis: Integrable function

Niles
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Homework Statement


HI all.

In order to perform a Fourier transform on a function f(x), f(x) must be integrable, i.e.

<br /> \int_{-\infty}^{\infty}|f(x)|dx &lt; \infty.<br />

Can you confirm that this also implies that f(x) -> 0 for x -> (+/-) infinity?
 
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Can you find a function that has a finite area beneath it while not being zero in infinity?
 
Cyosis said:
Can you find a function that has a finite area beneath it while not being zero in infinity?

No. But I am wondering if it is a sufficient conditions for the integral to be finite? E.g. f(x) = x-1?
 
To avoid any confusion. Do you mean if f(x)->0 when x->+-infinity then the integral of f(x) over +- infinity converges? This is not true and you already gave a counter example. The function x^-1 goes to zero, but the integral of x^-1 from -infinity to infinity diverges.
 
Great, thanks!
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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