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

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.

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 ?

• 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. – user150203 Nov 28 '18 at 23:25
• No log is taken DavidG. Just a name. Thanks – mick Nov 29 '18 at 14:49
• Thanks for the clarification. Wanted to be sure before I proceeded. – user150203 Nov 29 '18 at 23:12