Definition:Improper Integral/Half Open Interval/Open Below
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Definition
Let $f$ be a real function which is continuous on the half open interval $\hointl a b$.
Then the improper integral of $f$ over $\hointl a b$ is defined as:
- $\ds \int_{\mathop \to a}^b \map f t \rd t := \lim_{\gamma \mathop \to a} \int_\gamma^b \map f t \rd t$
Also presented as
The definition of an improper integral on a half open interval $\hointl a b$ can also be presented as:
- $\ds \int_{\mathop \to a}^b \map f t \rd t := \lim_{\delta \mathop \to 0} \int_{a + \delta}^b \map f t \rd t$
Also denoted as
When presenting an improper integral on a half open interval $\hointl a b$, it is common to abuse notation and write:
- $\ds \int_a^b \map f t \rd t$
which is understood to mean exactly the same thing as $\ds \int_{\mathop \to a}^b \map f t \rd t$.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definition of a Definite Integral: $15.6$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 13.27$