<s>[INST] partial fraction for complex roots While solving Laplace transform using Partial fraction expansion. I have confusion in solving partial fraction for complex roots.
I have this equation $$ \frac {2s^2+5s+12} {(s^2+2s+10)(s+2)}$$
Please anyone help to tell me to understand the steps for solving partial fraction for complex roots
 [/INST] It is the same story as with real roots.
Consider $$A=\frac {2s^2+5s+12} {(s^2+2s+10)(s+2)}$$ the first partial fraction decomposition gives $$A=\frac{s+1}{s^2+2 s+10}+\frac{1}{s+2}$$ So, now, just focus on $$B=\frac{s+1}{s^2+2 s+10}=\frac{s+1}{(s+1+3i)(s+1-3i)}=\frac \alpha {(s+1+3i)}+\frac \beta {(s+1-3i)} $$ Reducing to same denominator $$s+1=\alpha(s+1+3i)+\beta (s+1-3i)=(\alpha+\beta)(s+1)+3(\alpha-\beta)i$$ Idenitfyng the real and imaginary parts than gives $$\alpha+\beta=1 \, \, \, \quad \alpha-\beta=0$$ that is to say $\alpha=\beta=\frac 12$.
Edit
To clarify the first partial fraction decomposition $$\frac {2s^2+5s+12} {(s^2+2s+10)(s+2)}=\frac a{s+2}+\frac {b+cs}{s^2+2s+10}$$ Reducing to same denominator and removing the common denominator $$2s^2+5s+12=a(s^2+2s+10)+(b+cs)(s+2)$$ Expanding the rhs and grouping terms $$2s^2+5s+12=(a+c)s^2+ (2 a+b+2 c) s+(10 a+2 b)$$ Comparing the coefficients $$a+c=2 \quad, 2a+b+2c=5 \quad, 10a+2b=12$$ Solving the three equations for the three unknowns leads to $a=b=c=1$.
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