双重广义积分

Overview of "two-step" improper integrals

We now understand "one-step" improper integrals: those with a single infinite endpoint (but no infinite singularity), or a single infinite singularity at a finite endpoint (but no infinite endpoint). We now discuss "two-step" improper integrals, which can be broken into two simpler, one-step improper integrals. For instance, if the integrand $f(x)$ has a single infinite singularity at an interior point $c\in (a,b)$ (where interval $[a,b]$ is assumed bounded), then the improper integral $\int_a^b f(x)\,dx$ is defined as the sum of the one-step improper integrals $\int_a^c f(x)\,dx$ and $\int_{c}^b f(x)\,dx$. Similarly, if each endpoint $a,b$ is "bad" (either infinite, or a finite point of infinite singularity) for $f$, but $f$ has no interior infinite singularities, then $\int_a^b f(x)\,dx$ can be defined as the sum of the one-step improper integrals $\int_a^c f(x)\,dx$ and $\int_c^b f(x)\,dx$, for any choice of interior point $c\in (a,b)$.