Laplace Transform Determination/Miscellaneous Methods
Jump to navigation
Jump to search
Solution Technique for Laplace Transform
To find the Laplace transform of a function $f$, one can evaluate it using one of the following techniques:
Linear Combination of Laplace Transforms
Then:
- $\laptrans {\lambda \, \map f t + \mu \, \map g t} = \lambda \laptrans {\map f t} + \mu \laptrans {\map g t}$
everywhere all the above expressions are defined.
Laplace Transform of Exponential times Function
- $\laptrans {e^{a t} \map f t} = \map F {s - a}$
everywhere that $\laptrans f$ exists, for $\map \Re s > a$
Laplace Transform of Function of t minus a
Let $g$ be the function defined as:
- $\map g t = \begin{cases} \map f {t - a} & : t > a \\ 0 & : t \le a \end{cases}$
Then:
- $\laptrans {\map g t} = e^{-a s} \map F s$
Laplace Transform of Constant Multiple
Let $a \in \C$ or $\R$ be constant.
Then:
- $a \laptrans {\map f {a t} } = \map F {\dfrac s a}$
Laplace Transform of Higher Order Derivatives
\(\ds \laptrans {\map {f^{\paren n} } t}\) | \(=\) | \(\ds s^n \laptrans {\map f t} - \sum_{j \mathop = 1}^n s^{j - 1} \map {f^{\paren {n - j} } } 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds s^n \map F s - s^{n - 1} \, \map f 0 - s^{n - 2} \, \map {f'} 0 - s^{n - 3} \, \map {f} 0 - \ldots - s \, \map {f^{\paren {n - 2} } } 0 - \map {f^{\paren {n - 1} } } 0\) |
Laplace Transform of Integral
- $\ds \laptrans {\int_0^t \map f u \rd u} = \dfrac {\map F s} s$
wherever $\laptrans f$ exists.
Higher Order Derivatives of Laplace Transform
- $\dfrac {\d^n} {\d s^n} \laptrans {\map f t} = \paren {-1}^n \laptrans {t^n \, \map f t}$
Integral of Laplace Transform
- $\ds \laptrans {\dfrac {\map f t} t} = \int_s^{\to \infty} \map F u \rd u$
wherever $\ds \lim_{t \mathop \to 0} \dfrac {\map f t} t$ and $\laptrans f$ exist.
Laplace Transform of Periodic Function
Let $f$ be periodic, that is:
- $\exists T \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + T}$
Then:
- $\laptrans {\map f t} = \dfrac 1 {1 - e^{-s T} } \ds \int_0^T e^{-s t} \map f t \rd t$
where $\laptrans {\map f t}$ denotes the Laplace transform.
Initial Value Theorem of Laplace Transform
Let $\ds \lim_{t \mathop \to 0} \dfrac {\map f t} {\map g t} = 1$.
Then:
- $\ds \lim_{s \mathop \to \infty} \dfrac {\map F s} {\map G s} = 1$
if those limits exist.
Final Value Theorem of Laplace Transform
Let $\ds \lim_{t \mathop \to \infty} \dfrac {\map f t} {\map g t} = 1$.
Then:
- $\ds \lim_{s \mathop \to 0} \dfrac {\map F s} {\map G s} = 1$
if those limits exist.
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Methods of Finding Laplace Transforms: $5$. Method of differential equations