# Dharshan's question via email about a Laplace Transform

• MHB
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In summary, the Laplace transform of $\displaystyle 5\sin{(11\,t)}\sinh{(11\,t)}$ is $\displaystyle \frac{55}{2} \left[ \frac{1}{\left( s - 11 \right) ^2 + 121} - \frac{1}{\left( s + 11 \right) ^2 + 121} \right]$.
Prove It
Gold Member
MHB
Evaluate $\displaystyle \mathcal{L}\left\{ 5\sin{ \left( 11 \, t \right) } \sinh{ \left( 11\,t \right) } \right\}$.

\displaystyle \begin{align*} \mathcal{L} \left\{ 5\sin{ \left( 11\,t \right) } \sinh{ \left( 11\,t \right) } \right\} &= \mathcal{L} \left\{ 5\sin{ \left( 11\,t \right) } \cdot \frac{1}{2} \left( \mathrm{e}^{11\,t} - \mathrm{e}^{-11\,t} \right) \right\} \\ &= \frac{5}{2} \,\mathcal{L} \left\{ \mathrm{e}^{11\,t} \sin{ \left( 11\,t \right) } - \mathrm{e}^{-11\,t} \sin{ \left( 11\,t \right) } \right\} \\ &= \frac{5}{2} \left[ \mathcal{L}\left\{ \mathrm{e}^{11\,t}\sin{ \left( 11\,t \right) } \right\} - \mathcal{L}\left\{ \mathrm{e}^{-11\,t} \sin{ \left( 11\,t \right) } \right\} \right] \\ &= \frac{5}{2} \left\{ \left[ \frac{11}{s^2 + 11^2} \right]_{s \to s - 11} - \left[ \frac{11}{s^2 + 11^2} \right]_{s \to s + 11} \right\} \\ &= \frac{55}{2} \left[ \frac{1}{\left( s - 11 \right) ^2 + 121} - \frac{1}{\left( s + 11 \right) ^2 + 121} \right] \end{align*}

It would be fine to leave your answer in this form, but if you get a common denominator and simplify, you could write the answer as $\displaystyle \frac{1210\,s}{s^4 + 58564}$.

benorin
the transforms are correct, I didn’t check the algebra on your last simplification.

## 1. What is a Laplace Transform?

A Laplace Transform is a mathematical tool used to convert a function of time into a function of complex frequency. It is commonly used in engineering and physics to solve differential equations and analyze systems in the frequency domain.

## 2. How is a Laplace Transform calculated?

The Laplace Transform is calculated using an integral formula that involves the function of time and a complex exponential term. The integral is evaluated from 0 to infinity, and the resulting function is a complex function of frequency.

## 3. What are the applications of Laplace Transform?

Laplace Transform has many applications in various fields such as control systems, signal processing, circuit analysis, and differential equations. It is used to solve problems that involve time-dependent systems and to analyze the behavior of these systems in the frequency domain.

## 4. How is the Laplace Transform related to the Fourier Transform?

The Laplace Transform is a generalization of the Fourier Transform. While the Fourier Transform is used for functions defined on the real line, the Laplace Transform can be used for functions defined on the entire complex plane. Additionally, the Laplace Transform can handle more general functions than the Fourier Transform.

## 5. Are there any limitations of using Laplace Transform?

One limitation of Laplace Transform is that it cannot be applied to functions that grow faster than exponential. It also requires the function to be defined for all positive time values. Another limitation is that the inverse Laplace Transform may not exist for all functions, making it difficult to retrieve the original function from its transformed form.

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