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Every now and then, I encounter an equation (in analysis, usually) that needs to be interpreted as, "if one side exists, then the other side exists, and then the two sides are equal."

This prompted me to ask, are there examples of equations where one side might exist when the other side does not, but if both sides exist, then they are equal? I can't recall ever encountering such a thing in practice.

Timothy Chow
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  • https://en.wikipedia.org/wiki/Natural_density – Will Jagy Nov 22 '19 at 03:29
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    If a function on $[a,b]$ is Riemann integrable, then it is Lebesgue integrable, and their integrals agree. But the Lebesgue integral might exist while the other does not. Is that what you mean? – pancini Nov 22 '19 at 03:35
  • @ElliotG : Yes, that's a pretty good example, but I should have clarified that I'd like an example where either side could exist but the other does not. – Timothy Chow Nov 22 '19 at 03:37
  • @WillJagy : I don't quite understand your suggestion. If the upper and lower densities both exist, they are not necessarily equal. But maybe that's not what you're saying? – Timothy Chow Nov 22 '19 at 03:38
  • My memory is that the natural density, if it exists, is the same as the Dirichlet density. – Will Jagy Nov 22 '19 at 03:40
  • Refinement: natural density of a set of primes: https://mathoverflow.net/questions/230424/if-the-natural-density-relative-to-the-primes-exists-then-the-dirichlet-densi – Will Jagy Nov 22 '19 at 03:47
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    Analytic continuation affords a number of such examples (the Riemann zeta function comes to mind). – hardmath Nov 22 '19 at 03:48
  • Once again, not an example where two things can exist independently but only equal when they both exist – pancini Nov 22 '19 at 03:58
  • What does it mean here for something to exist? – Allawonder Nov 22 '19 at 04:54
  • @Allawonder : I have in mind expressions that are defined in terms of some kind of limit. A limit fails to exist if, for example, we get oscillatory behavior. Sometimes one also says that a limit fails to exist if the limit is infinite. – Timothy Chow Nov 22 '19 at 16:02
  • @hardmath: Ah, that's a good thought! Maybe make this an answer, with an explicit example of two nontrivial expressions with different but overlapping domains of convergence? – Timothy Chow Nov 22 '19 at 16:06

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These might be too trivial, but how about

$$\left(\sqrt{x}\right)^2 = \sqrt{x^2}$$

or

$$2 \log x = \log x^2$$

or

$$\frac{x^2-1}{x+1}=x-1$$

over $\,\Bbb R?$

If you need either side to exist when the other does not, perhaps

$$\frac{x^2-x}{x-1} = \frac{x^2+x}{x+1}$$

For something slightly less trivial, for the $\,f(x)\,$ shown below

f(x)

the equation

$$\sqrt{f(x)\hphantom{,}} = \sqrt{f(-x)\hphantom{,}}$$

would be an example. I feel there should be a relatively simple way to write f(x) without resorting to piecewise, but it's very late and my brain's not cooperating.

A.J.
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