Definition:Inverse Laplace Transform/Definition 2
Jump to navigation
Jump to search
Definition
Let $\map f s: S \to \C$ be a complex function, where $S \subset \C$.
The inverse Laplace transform of $f$, denoted $\map F t: \R \to S$, is defined as:
This article, or a section of it, needs explaining. In particular: Can the domain of $f$ be clarified -- is it genuinely $\R$? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. 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 {{Explain}} from the code. |
\(\ds \map F t\) | \(=\) | \(\ds \dfrac 1 {2 \pi i} \PV_{c \mathop - i \, \infty}^{c \mathop + i \, \infty} e^{s t} \map f s \rd s\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 \pi i} \lim_{T \mathop \to \infty} \int_{c \mathop - i \, T}^{c \mathop + i \, T} e^{s t} \map f s \rd s\) |
where:
- $\PV$ is the Cauchy principal value of the integral
- $c$ is any real constant such that all the singular points of $\map f s$ lie to the left of the line $\map \Re s = c$ in the complex $s$ plane.
This article, or a section of it, needs explaining. In particular: Establish the fact that $\map \Re s = c$ specifies a line, and define what that line is You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. 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 {{Explain}} from the code. |
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Complex Inversion Formula: $32.2$