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[Solved] Complex number: Decompose inside C

 
Complex Numbers
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June 15, 2021 2:36 pm
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Staff 
(@armor)
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Decompose inside \mathbb{C}:

\frac{z^4+1}{z(z^2+1)}

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June 15, 2021 2:36 pm
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

By a direct computation,

\small \frac{z^4+1}{z(z^2+1)}=\frac{z^4+z^2-z^2+1}{z(z^2+1)}=z+\frac{-z^2+1}{z(z-i)(z+i)}

\small =z+\frac{1}{1}*\frac{1}{z}+\frac{1+1}{i.2i}*\frac{1}{z-i}+\frac{1+1}{-i(-2i)}*\frac{1}{z+i}

\small =z+\frac{1}{z}-\frac{1}{z-i}-\frac{1}{z+i}.

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