2

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I tried partial fraction expansion in this way,

but it's just too cumbersome to solve in a test.

Is there another way?

NK Yu
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  • You can compute the residue of both sides as $s \to \pm i$ to compute $\theta_2$ and $\theta_1$. This lets you compute $\theta_4$ and $\theta_3$ by subtracting off the first term. – Qiaochu Yuan Jul 19 '18 at 07:14
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    @Qiaochu, I'd hazard a guess that someone struggling with partial fractions wouldn't know a residue if it painted itself purple and danced naked on a harpsicord singing happy residues are here again. – Gerry Myerson Jul 19 '18 at 07:23

1 Answers1

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Hint:

$$2s=(a_1+a_2s)(s^2-s+4)+(a_3+a_4s)(s^2+1)$$

Comparing the constant terms $0=4a_1+a_3\iff a_3=?$

Comparing the coefficients of $s^2,0=a_1-a_2+a_3\iff a_2=a_1+a_3=a_1-4a_1=?$

Comparing the coefficients of $s^3, 0=a_2+a_4\iff a_4=-a_2=?$

Comparing the coefficients of $s, 2=-a_1+4a_2+a_4$

Replace the values of $a_2,a_4$ with $a_1$

lab bhattacharjee
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