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We know for $ u > 1 $ $$ \int_{-\pi/2}^{+\pi/2} \ln(\sin(x) + u) dx = \pi \left(\ln\left(u + \sqrt{u^2 -1}\right) - \ln(2)\right) $$

Usually this is shown by using differentiation under the Integral sign or contour integration.

This made me wonder :

Consider the log-transform

For a given real continuous function $F(u)$ find a real continuous function $g(x) $ that only depends on $x$ ( not on $u$ ) such that

$$ F(u) = \int_{-\pi/2}^{+\pi/2} \ln(g(x) + u) dx $$

So

$$ \mathcal{L}(F(u)) = g(u) $$

Where $\mathcal{L}$ stands for “ log-transform “.

For instance $$ \mathcal{L}\left[ \pi \left(\ln\left(u + \sqrt{u^2 -1}\right) - \ln(2)\right) \right] = \sin(u). $$

Notice that in this case $\sin(2u),\sin(3u),\sin(4u),... $ and $\sin(-u),\sin(-2u),\sin(-3u),\sin(-4u),... $ are also solutions ! Perhaps uniqueness comes from functions that are strictly increasing ? ( $\mathcal{L} ... = exp(u) $? )

What is known about these (inverse) integral transforms ?

Is there an Integral representation for them ?

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  • $\begingroup$ Pardon my ignorance, when you say 'log-transform' do you mean to take the log of the function? (Just want to be 100% clear before I give this a go). Btw - Great question. $\endgroup$ – user150203 Nov 28 '18 at 23:25
  • $\begingroup$ No log is taken DavidG. Just a name. Thanks $\endgroup$ – mick Nov 29 '18 at 14:49
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    $\begingroup$ Thanks for the clarification. Wanted to be sure before I proceeded. $\endgroup$ – user150203 Nov 29 '18 at 23:12

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