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Laplace Transformation

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- Laplace transform
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Laplace Transform

Laplace transform is a mathematical operation that is used to “transform” a variable (such as $x$ , or $y$ , or $z$ in space, or at the time $t$ ) to a parameter $(s) - a$ “constant” under certain conditions. It transforms ONE variable at a time.

Let $f (t)$ be a function of ‘ $t$ ’ defined for all positive values of $t$ . Then Laplace transforms of $f (t)$ is denoted by $L \{f (t)\}$ is defined by:

$L \{f (t)\} = \int_{0}^{\infty} e^{- st} f (t) dt = \overline{f} (s)$

provided that the integral exists. Here the parameter $s$ is a real (or) complex number.
The relation can also be written as $f (t) = L^{- 1} \{\overline{f} (s)\}$

Example 1:

Express the Laplace transforms for $f (t) = a t^{2}$ with $0 \leq t < \infty$ .

Solution:

$L [f (t)]$

$= \int_{0}^{\infty} e^{- st} (t^{2}) dt$

$= e^{- st} [- \frac{2 t^{2}}{2 s} - \frac{2 t}{s^{2}} - \frac{2}{s^{3}}]$ $\Biggr|_{0}^{ \infty} $

$= \frac{2}{s^{3}}$

Practice question 1:

Express the Laplace transforms for $f (t) = e^{a t}$ with $0 \leq t < \infty$ .

$L [f (t)] =$

How Did I Do?

Properties of Laplace transform.

Linearity: The Laplace transform of a linear combination of signals is equal to the sum of their individual Laplace transforms.

Time shifting: The Laplace transform of a time-shifted signal is related to the Laplace transform of the original signal through a simple scaling factor.

Multiplication: The Laplace transform of the convolution of two signals is equal to the product of their individual Laplace transforms.

Transforms of Integral: Laplace transform states that for a signal and it's Laplace transform, the Laplace transform of the antiderivative of signal with respect to time is related to it's Laplace transform through a simple scaling factor:

Initial value theorem: The Laplace transform provides a relationship between the initial value of a signal and its Laplace transform.

Final value theorem: The Laplace transform provides a relationship between the final value of a signal and its Laplace transform.

Practice question 2:

Match the correct property name of the following Laplace transform

	$L (\int_{0}^{\infty} f (t) dt) = F (s - a)$
	$L (t^{n} f (t)) = (- 1) \frac{d^{n}}{{ds}^{n}} F (s)$
	$L (e^{at} f (t)) = F (s - a)$
	$L (a (f (t)) + b (g (t))) = a \cdot F (s) + b \cdot G (s)$