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SC2  – partial fractional exspansion

Why perform partial fraction expansion?

Partial fraction expansion is performed

whenever we want to represent a complicatedfraction as a sum of simpler fractions.

We can represent this as a sum of simple fractions:

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SC2  – partial fractional exspansion

o find A2, multiply F(s) by s+2 and set s=-2.

.

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SC2  – partial fractional exspansion

Likewise, for A3 multiply by s+5 and set s=-5

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SC2  – partial fractional exspansion

To find A1, multiply F(s) by s,

and then set s=0.

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SC2  – partial fractional exspansion

Example: Distinct Real RootsConsider the example from above:

Find A1 Find A2

Find A3

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SC2  – partial fractional exspansion

Example: Distinct Real Roots; the cover-up method

Consider the example from above:

Find A1 by first "covering-up" the first term in the denominator (i.e., the term that is

associated with A1) with your finger (shown as a gray ellipse), and then letting s=0.

Likewise for A2

and A3

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SC2  – partial fractional exspansion

Repeated Real Roots.Situations in which there exists a repeated root in the

denominator polynomial are more complicated. Consider thefraction:

The expansion for this case is:

Most of the unknown coefficients can be found using the "cover-up" method.

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SC2  – partial fractional exspansion

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SC2  – partial fractional exspansion

Extens ion to mult ip ly repeated 

roots .Now consider the case of multiply repeated roots

(n>2).

 As before we can easily find A1 and A2

If we continue to differentiate we can find

the other coefficients in a similar manner 

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SC2  – partial fractional exspansion

Complex roots. Consider the fraction:

The second term in the denominator cannot be factored into real terms. This

leaves us with two possibilities - either accept the complex roots, or find a way

to include the second order term. Method 1 - Using the complex (first order) roots

If we use complex roots, we can expand the fraction as we did before. This is

not typically the way you want to proceed if you are working by hand, but may

be easier for computer solutions (where complex numbers are handled as

easily as real numbers). To perform the expansion, continue as before.

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SC2  – partial fractional exspansion

where

Note that A2 and A3 must be complex conjugates of each other since

they are equivalent except for the sign on the imaginary part. Performing

the required calculations:

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SC2  – partial fractional exspansion

SC2 i l f i l i

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SC2  – partial fractional exspansion

SC2 ti l f ti l i

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SC2  – partial fractional exspansion

SC2 ti l f ti l i

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SC2  – partial fractional exspansion