Definition:Improper Integral/Unbounded Open Interval/Unbounded Above and Below
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Definition
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 $\ds \int_{\mathop \to -\infty}^{\mathop \to + \infty} \map f t \rd t$.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definition of a Definite Integral: $15.4$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 13.27$