How to specify integration borders for Integro-Differential equation?

[lhapp27]

Hello, first-time user here.
I plan to solve the integro-differential equation:

$$\frac{d^2}{d\alpha^2} \phi(\alpha,\rho) - [\beta+f(\alpha,\rho)]\phi(\alpha,\rho) - f(\alpha,\rho) \int_{|\pi/3 - \alpha|}^{\pi/2 - |\pi/6-\alpha|} d\alpha' \phi(\alpha',\rho) = 0$$
My goal is to find the eigenvalue $\beta = \beta(\rho)$, and the eigenfunction $\phi(\alpha,\rho)$, where $\rho$ is an external parameter, and $\alpha \in [0,\pi/2]$
For now, let's forget about the function $f$, i.e. $f = 1$.

In the documentation (https://juliaapproximation.github.io/ApproxFun.jl/latest/usage/operators/#Algebraic-manipulation-of-operators), there is some information on how to deal with integrals, however the borders seem to be fixed at -1 and 1. Is there a way (and how) to provide the integration borders as in my case?

Any help is appreciated.

[jishnub]

You need to set the domain to one that you want to integrate over. By default, the documentation mentions -1 to 1 because this is the domain for the canonical Chebyshev polynomials.

[lhapp27]

Thank you for the quick reply. I understand now partially, however I think the documentation doesn't mention it explicitly. Also one needs to pay attention that the integrand is defined in a space at least as large as the integration borders. Maybe that is obvious, but newcomers might struggle. Here an example:

x = Fun(0..pi/2)
u = 1/(1+x^2)
integral1 = DefiniteIntegral(Chebyshev(0..pi/2))
i1 = integral1 * u;
println("i1 = ",i1(0)) # i1 = 1.0038848218538874 (correct)

x = Fun(0..pi/2)
u = 1/(1+x^2)
integral2 = DefiniteIntegral(Chebyshev(0..pi))
i2 = integral2 * u;
println("i2 = ",i2(0)) # i2 = 1.0038848218539356 (incorrect), there is a warning though about "Maximum number of coefficients reached"

x = Fun(0..pi)
u = 1/(1+x^2)
integral3 = DefiniteIntegral(Chebyshev(0..pi))
i3 = integral3 * u;
println("i3 = ",i3(0)) # i3 = 1.2626272556789127 (correct)

So this works for constant integration borders, but any help on how to define the borders such that they depend on alpha, which is the dynamical variable in my equation?