Partial Fraction Expansion of Repeated Roots by Differentiation

Singly Repeated Roots

Consider the case in which one of the roots is repeated:

In this fraction the denominator polynomial has a repeated root at s=-a.  The remainder of the denominator polynomial is called D'(s); it has no roots as s=-a.  The numerator polynomial is N(s).  If we expand this fraction we get

The term N'(s)/D'(s) represents the expansion of all of the terms except those with roots at s=-a.

We can find A₁ by multiplying by (s+a)² and setting s=-a (i.e., the cover-up  method).

To find A₂ we note that the we can get rid of the A₁ term by differentiating the result above

Now we can solve for A₂ by setting s=-a.

Extension to multiply repeated roots.

Now consider the case of multiply repeated roots (n>2).

The resulting partial fraction expansion is

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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References