Translation theorems

Section 3.3 Translation theorems

It is not always convenient to use Definition 3.1.2 to calculate a Laplace transform.

Theorem 3.3.1. First translation theorem (along \(s\)-axis).

Let \(f\) be piecewise continuous on \(t\geq 0\) and of exponential order \(c\text{.}\) If \(F\) is the Laplace transform of \(f\) and \(h(t)=e^{at}f(t)\text{,}\) then

\begin{equation*} \mathcal{L}\{h\}(s)=F(s-a), s>c+a, \end{equation*}

and

\begin{equation*} \mathcal{L}^{-1}\{F(s-a)\}(t)=e^{at}f(t), t>0. \end{equation*}

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Proof.

later!

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Example 3.3.2.

Find the Laplace transform of...

\(\displaystyle f(t)=t^2e^{4t}\)

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\(\displaystyle g(t)=e^{-t}\sin(3t)\)

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Example 3.3.3.

Find the invers Laplace transform of...

\(\displaystyle F(s)=\dfrac{s}{s^2+4s+13}\)

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\(\displaystyle G(s)=\dfrac{s}{(s+3)^2}\)

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Example 3.3.4.

Use the Laplace transform to solve the initial value problem

\begin{equation*} y''+6y'+25y=0, y(0)=2, y'(0)=3. \end{equation*}

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Example 3.3.5.

Use the Laplace transform to solve the initial value problem

\begin{equation*} y'+4y=e^{-4t}, y(0)=1 \end{equation*}

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Example 3.3.6.

Sketch the following piecewise function. Use Theorem 3.1.13 to rewrite it a more compact form.

\begin{equation*} f(t)=\left\{ \begin{array}{ll} \sin(t), & \quad 0\leq t < \pi \\ \cos(t), & \quad t\geq \pi. \end{array}\right. \end{equation*}

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Theorem 3.3.7. Second translation theorem.

Let \(f\) be piecewise continuous on \(t \geq 0\) and of exponential order \(c\text{.}\) If \(a>0\text{,}\) then

\begin{equation*} \mathcal{L}\{f(t-a)\mathcal{U}(t-a)\}(s)=e^{-as}F(s). \end{equation*}

An alternative way to express the same formula is

\begin{equation*} \mathcal{L}\{g(t)\mathcal{U}(t-a)\}(s)=e^{-as}\mathcal{L}\{g(t+a)\}(s). \end{equation*}

The inverse form is

\begin{equation*} \mathcal{L}^{-1}\{e^{-as}F(s)\}(t) = f(t-a)\mathcal{U}(t-a). \end{equation*}

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Example 3.3.8.

Find the inverse Laplace transform of...

\(\displaystyle F(s)=\dfrac{e^{-\frac{3\pi}{2}s}}{s^2+1}\)

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\(\displaystyle G(s)=\dfrac{e^{-s}}{(s-1)^2}\)

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Example 3.3.9.

Solve the initial value problem

\begin{equation*} y'+y=\mathcal{U}(t-s), y(0)=1. \end{equation*}

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Theorem 3.3.10. Laplace transform of periodic functions.

Let \(f\) be piecewise continuous on \(t \geq 0\) and of exponential order. If \(f\) is periodic with period \(T\text{,}\) i.e. \(f(t+T)=f(t)\) for all \(t\text{,}\) then

\begin{equation*} \mathcal{L}\{f\}(s)=\dfrac{1}{1-e^{-sT}} \displaystyle\int_0^T e^{-st}f(t)\mathrm{d}t. \end{equation*}

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Proof.

later!

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Example 3.3.11.

Find the Laplace transform of the \(T=1\)-periodic square wave function \(f(t)=\left\{ \begin{array}{ll} 1, & \quad t \in (2n,2n+1), n=0,1,2,\ldots \\ -1, & \quad t \in (2n+1,2n+2), n=0,1,2,\ldots \end{array}\right.\)

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Example 3.3.12.

Find the Laplace transform of the \(T=a\)-periodic extension of the line \(at\) for \(0 < t < a\text{.}\)

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