Integral Representation of the Dottie Number I noticed that a lot of commonly-used mathematical constants that can't be expressed in closed-form can be expressed by integrals, such as
$$\pi=\int_{-\infty}^\infty \frac{dx}{x^2+1}$$
and
$$\frac{1}{1+\Omega}=\int_{-\infty}^\infty \frac{dx}{(e^x-x)^2+\pi^2}$$
I was wondering if anyone knows how to express the Dottie Number $\omega$, or the unique solution to the equation
$$\cos(\omega)=\omega$$
using an integral.
In general, what are some strategies for expressing constants as integrals? I'm also struggling to express the reciprocal fibonacci constant as an integral (but don't tell me how to do that one).

You seem to have provided an answer to your own question. Thank you. I will bookmark this post; it may help me in some of my own work. I'd like to add a little more information; it may suggest other ways to tackle your goal, perhaps direct you to a more concise solution.
The Dottie Number (D) also happens to be the solution of Kepler's Equation of Elliptical Motion (it satisfies the "equal area swept out in equal time" condition) at the quarter period for e = 1.
i.e.,
$$ E - e \sin(E) = M $$
$$ E - \sin(E) = \frac\pi 2 $$
(Aside #1: When e = 1, the conic is a parabola, suggesting D can be expressed in the form
$$ D = a b^2 $$
where a and be are constants yet to be determined.)
Kepler's Equation reduces to
$$ \cos(\sin(E)) = \sin(E) $$
This is the definition of the Dottie Number, where $\sin(E) = D$.
I bring up this connection because Kepler's Equation can be expressed in terms of a Bessel Function of the First Kind ($J_k$):
$$ E = M + 2 \sum_{k=1}^\infty \frac{1}{k} J_k (ke) \sin(kM) $$
It's an infinite series, not an integral, but perhaps it helps you.
(Aside #2: The Dottie Number is just a special case of a more general equation:
$$ \cos(k x) = x $$
where 0 $\le$ k $\le$ 1. The Dottie Number is the solution to the equation for k = 1. A plot of the solutions to this equation for 0 $\le$ k $\le$ 1 is tantalizingly close to a graph of $\text{sech}(0.817326346581 k)$. Since $E$ is periodic and a function of itself, I suppose a solution should include sine and exponential terms, but I haven't found an exact relationship. Perhaps it involves fractional derivatives.)
(Aside #3: There used to be a blog solely about the Dottie Number. Have you come across it? It has  lot of good information about the Dottie Number, for example, Bertrand's Semicircle.)
