Definition:Improper Integral/Unbounded Open Interval

Definition

Unbounded Above

Let $f$ be a real function which is continuous on the unbounded open interval $\openint a {+\infty}$.

Then the improper integral of $f$ over $\openint a {+\infty}$ is defined as:

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

for some $c \in \openint a {+\infty}$.

Unbounded Below

Let $f$ be a real function which is continuous on the unbounded open interval $\openint {-\infty} b$.

Then the improper integral of $f$ over $\openint {-\infty} b$ is defined as:

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

for some $c \in \openint {-\infty} b$.

A specific and important instance of this occurs when the interval in question is the set of all real numbers:

Unbounded Above and Below

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.