Category:Definitions/Cauchy Principal Value
This category contains definitions related to Cauchy Principal Value.
The Cauchy principal value is an extension of the concept of an improper integral when the latter might not exist.
This article is complete as far as it goes, but it could do with expansion. In particular: Needs to be include the situation where there is an undefined point at some $b \in \R$ where $\map f b$ is undefined, for example $\ds \int_{-\infty}^x \dfrac {e^u} u \rd u$ at $u = 0$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Real Integral
Let $f: \R \to \R$ be a real function which is piecewise continuous everywhere.
Then the Cauchy principal value of $\ds \int f$ is defined as:
- $\PV_{-\infty}^{+\infty} \map f t \rd t := \lim_{R \mathop \to +\infty} \int_{-R}^R \map f t \rd t$
where $\ds \int_{-R}^R \map f t \rd t$ is a Riemann integral.
Complex Integral
Let $f: \R \to \C$ be a bounded complex function.
Then the Cauchy principal value of $\ds \int f$ is defined as:
- $\PV_{-\infty}^{+\infty} \map f t \rd t := \lim_{R \mathop \to +\infty} \int_{-R}^R \map f t \rd t$
where $\ds \int_{-R}^R \map f t \rd t$ is a complex Riemann integral.
Contour Integral
Let $C$ be a contour defined by a directed smooth curve.
Let $C$ be parameterized by the smooth path $\phi: \closedint {-R} R \to \C$, where $R > 0$.
Let $f: \Img C \to \C$ be a continuous complex function, where $\Img C$ denotes the image of $C$.
Then the Cauchy principal value of $\ds \int f$ is defined as:
- $\PV_C \map f z \rd z = \PV_{\map \phi {-\infty} }^{\map \phi {+\infty} } \map f z \rd z := \lim_{R \mathop \to +\infty} \int_{-R}^R \map f {\map \phi t} \map {\phi'} t \rd t$
where $\ds \int_{-R}^R \map f {\map \phi t} \map {\phi'} t \rd t$ is a complex Riemann integral defining a contour integral.
Pages in category "Definitions/Cauchy Principal Value"
The following 6 pages are in this category, out of 6 total.