Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose that $P(z)$ and $Q(z)$ are polynomials with no common roots and $\deg(P) < \deg(Q)$. Suppose that $Q(z) = \displaystyle\prod_{k=1}^t (z - z_k)^{n_k}$ then the partial fraction decomposition of $\frac{P(z)}{Q(z)}$ has the form: $$ \frac{P(z)}{Q(z)} = \sum_{k=1}^t \sum_{j=1}^{n_k} \frac{a_{kj}}{(z - z_k)^j}$$ Show that $ a_{kj} = \operatorname{Res}_{z_k} ((z - z_k)^{j-1} \frac{P(z)}{Q(z)})$

I am having a hard time answering this question could someone shed some light?

share|improve this question
1  
Try changing the title of the quesition to something more informative, and less subjective, if complex means complicated and is not related to the question being from complex analysis. –  Pedro Tamaroff Apr 10 '12 at 0:17
    
Shouldn't we write $a_{k,j}$ rather than $a_{k_j}$? –  anon Apr 10 '12 at 0:20
add comment

2 Answers 2

Hint: residue is linear in the function. What would you get for each of the terms?

share|improve this answer
add comment

From the partial fraction decomposition, $(z - z_k)^{j-1} \frac{P(z)}{Q(z)}$ has a holomorphic primitive at all points $z\ne z_k$ except for the single term $\frac{a_{kj}}{z - z_k}$. Hence the residue at $z_k$ is $a_{kj}$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.