The expression you’ve given is $\frac{1}{(s+4)^{3/2}}$, and it looks like you want to find its inverse Laplace transform using partial fractions. However, this form is not directly suitable for partial fraction decomposition because the power of $(s+4)$ is fractional.

Instead, we approach this using the inverse Laplace transform formulas associated with functions of the form $\frac{1}{(s+a)^{n}}$, typically handled with gamma functions or known tables of inverse Laplace transforms.

Here is the detailed approach:

Step-by-Step Solution:

The inverse Laplace transform of $\frac{1}{(s+a)^{n}}$ is known to be: $\mathcal{L}^{-1} \left\{ \frac{1}{(s+a)^{n}} \right\} = \frac{t^{n-1}}{\Gamma(n)} e^{-at}, \quad n > 0$ where $\Gamma(n)$ is the Gamma function.

1. Compare $\frac{1}{(s+4)^{3/2}}$ with $\frac{1}{(s+a)^n}$. Here, $a = 4$ and $n = \frac{3}{2}$.

2. Using the known inverse Laplace transform formula: $\mathcal{L}^{-1} \left\{ \frac{1}{(s+4)^{3/2}} \right\} = \frac{t^{\frac{3}{2}-1}}{\Gamma\left(\frac{3}{2}\right)} e^{-4t}$

3. Simplify the exponent of $t$: $t^{\frac{1}{2}} = \sqrt{t}$

4. The value of $\Gamma\left(\frac{3}{2}\right)$ is known and can be computed as: $\Gamma\left(\frac{3}{2}\right) = \frac{1}{2} \sqrt{\pi}$

Thus, the inverse Laplace transform is: $\mathcal{L}^{-1} \left\{ \frac{1}{(s+4)^{3/2}} \right\} = \frac{2}{\sqrt{\pi}} \cdot \sqrt{t} \cdot e^{-4t}$

Final Answer:

The inverse Laplace transform of $\frac{1}{(s+4)^{3/2}}$ is: $f(t) = \frac{2}{\sqrt{\pi}} \sqrt{t} e^{-4t}$

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Would you like further details on the Gamma function or more about the properties of Laplace transforms?

Related Questions:

1. What is the inverse Laplace transform of $\frac{1}{(s+a)^n}$ for different $n$?
2. How is the Gamma function $\Gamma(n)$ defined and evaluated?
3. Can the inverse Laplace transform be applied to complex functions?
4. What are some applications of inverse Laplace transforms in differential equations?
5. How can fractional powers of $s$ affect the inverse Laplace transform?

Tip:

The Gamma function $\Gamma(n)$ generalizes the factorial function for non-integer values, and is widely used in Laplace transforms when handling fractional exponents.