Definition:Improper Integral/Unbounded Open Interval/Unbounded Above and Below

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Let $f$ be a real function which is continuous everywhere.

Then the improper integral of $f$ over $\R$ is defined as:

$\ds \int_{\mathop \to -\infty}^{\mathop \to +\infty} \map f t \rd t := \lim_{\gamma \mathop \to -\infty} \int_\gamma^c \map f t \rd t + \lim_{\gamma \mathop \to +\infty} \int_c^\gamma \map f t \rd t$

for some $c \in \R$.

Usually $c$ is taken to be $0$ as this usually simplifies the evaluation of the expressions.

Also defined as

This concept can also be seen defined as:

$\ds \int_{\mathop \to -\infty}^{\mathop \to +\infty} \map f t \rd t := \lim_{\substack {b \mathop \to \infty \\ a \mathop \to -\infty} } \int_a^b \map f t \rd t$

but this can be argued as being more opaque and hence less intuitively easy to grasp accurately.

Also denoted as

It is common to abuse notation and write:

$\ds \int_{-\infty}^\infty \map f t \rd t$

which is understood to mean exactly the same thing as $\ds \int_{\mathop \to -\infty}^{\mathop \to + \infty} \map f t \rd t$.