Definition:Integrable Function/Unbounded
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Definition
Let $f: \R \to \R$ be a real function.
Let $f$ be unbounded on the open interval $\openint a b$.
Let:
- $f^+$ denote the positive part of $f$
- $f^-$ denote the negative part of $f$.
Let $f^+$ and $-f^-$ both be integrable on $\openint a b$.
Then $f$ is integrable on $\openint a b$ and its (definite) integral is understood to be:
- $\displaystyle \int_a^b \map f x \rd x := \int_a^b \map {f^+} x \rd x - \int_a^b \paren {-\map {f^-} x} \rd x$
Sources
- 1970: Arne Broman: Introduction to Partial Differential Equations ... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.3$ Definitions