# Partial Fractions: Expanding & Laplace Transforms

• mxpxer7
In summary: I'm not sure what you're asking for, though.In summary, the conversation discusses the expansion of a partial fraction and its relation to the table of Laplace transforms. The solution involves using the properties of Laplace transforms and shifting arguments to simplify the expression.
mxpxer7
I'm sure this is a no brainer to someone, but here it is..

what is does the partial fraction of this look like in expanded form? Or how can I make it fit on the table of laplace transforms?

__(2s+1)__
(s-1)^2 + 1

Last edited:
I think you'd get more replies (and better ones) if you posted this in the algebra forum, as this doesn't really have anything to do with differential equations. Your case would fall under the "irreducible quadratic factor in the denominator", as the denominator has zeroes s=1+i and s=1-i which aren't real:

So what you wrote is the simplest form if you want to use only real numbers.

I apologize for putting this in the wrong section I'm using the answer for la place transforms so maybe this is where the calculus comes in, How can i write this so it fits into the table of laplace transforms?

$$F(s)=\frac{2s+1}{(s-1)^2+1}=2\frac{s-1}{(s-1)^2+1}+3\frac{1}{(s-1)^2+1}=G(s-1)$$

where $$G(s)=2\frac{s}{(s)^2+1}+3\frac{1}{(s)^2+1}$$

The inverse Laplace transform of G(s) is 2*cos(x)+3*sin(x). The inverse Laplace transform of G(s-1) is $$e^x$$[2*cos(x)+3*sin(x)] according to some of the properties of Laplace transforms and a shifted arguments.

RedX said:
$$F(s)=\frac{2s+1}{(s-1)^2+1}=2\frac{s-1}{(s-1)^2+1}+3\frac{1}{(s-1)^2+1}=G(s-1)$$

where $$G(s)=2\frac{s}{(s)^2+1}+3\frac{1}{(s)^2+1}$$

The inverse Laplace transform of G(s) is 2*cos(x)+3*sin(x). The inverse Laplace transform of G(s-1) is $$e^x$$[2*cos(x)+3*sin(x)] according to some of the properties of Laplace transforms and a shifted arguments.

Nice shortcut...

## 1. What are partial fractions and why are they important in expanding and Laplace transforms?

Partial fractions are a method used to break down a complex fraction into simpler fractions. This technique is important in expanding and Laplace transforms because it allows us to simplify complex expressions and make them more manageable for further calculations.

## 2. How do you expand a partial fraction?

To expand a partial fraction, you must first factor the denominator into linear or quadratic factors. Then, use partial fraction decomposition to rewrite the fraction as a sum of simpler fractions. Finally, solve for the unknown coefficients by equating the expanded expression to the original fraction.

## 3. What is the Laplace transform of a partial fraction?

The Laplace transform of a partial fraction is the sum of the Laplace transforms of its individual terms. This means that each term in the expanded expression will have its own Laplace transform, which can be easily calculated using the properties and tables of Laplace transforms.

## 4. Can partial fractions be used for all types of functions?

Partial fractions can only be used for rational functions, which are functions where the numerator and denominator are polynomials. If the function is not rational, other techniques such as trigonometric identities or inverse Laplace transforms must be used.

## 5. How are partial fractions and Laplace transforms related?

Partial fractions are used in expanding expressions before taking the Laplace transform. This is because the Laplace transform of a rational function is easier to calculate when the function is expanded into simpler fractions. Additionally, the Laplace transform can be used to solve integrals involving partial fractions.

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