It's curious that you can have different constants of integration on intervals. E.g. if $$f(x) = \left\{ \begin{array}{llr} \frac{-1}{x}+a, & x>0\\ \frac{-1}{x}+b, & x<0\\ \end{array} \right. $$ then $$ \frac{\mathrm{d}}{\mathrm{d}x}f(x) = \frac{1}{x^2}.$$ Are there situations/applications where different constants of integration on different intervals are needed/required?
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2$\begingroup$ You are computing piecewise derivatives. Your domain is not connected, it is separated by the point 0. So there is no coupling condition between the left and the right part. $\endgroup$– Bertoldo BaccalàCommented Oct 7 at 12:06
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2$\begingroup$ Indeed! A point often missed, or quietly passed over in textbooks. My question asks whether this is ever needed or used? $\endgroup$– Chris SangwinCommented Oct 7 at 12:30
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1$\begingroup$ Not a research-level question. More appropriate for MSE $\endgroup$– Robert IsraelCommented Oct 7 at 14:06
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2$\begingroup$ Thanks @RobertIsrael for suggesting MSE. I'm undertaking an educational research project and whether this technical detail from analysis "matters". It would be helpful to ask in a research forum (i.e. here) whether this opportunity with constants on different connected domains is needed/requires in particular situations/applications. I hope I'm not asking an elementary question in the wrong forum. $\endgroup$– Chris SangwinCommented Oct 7 at 14:22
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2$\begingroup$ Thanks @MichaelHardy, yes there are a couple of tutorial examples in textbooks which cover this. E.g. taking $ \tan^{-1}\left(\frac{ax+1}{a-x}\right)$ to be an antiderivative of $ \frac{1}{1+x^2}$ is interesting, and related. All the examples/discussion I've found so far are from a pure mathematics perspective. I'm interested in applications (including pure mathematics techniques) where different constants are needed. $\endgroup$– Chris SangwinCommented Oct 8 at 12:57
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