How Can find the Laurent series for this function valid for 0<|z−i|<2 f(z)=sin(2πz)z(z2+1)
Let g(z)=sin(πz)
sin(πz)=sin(2π(z−i))cos(2πi)+cos(2π(z−i))sin(2πi)
And Let h(z)=1z2+1
1z(z2+1)=1i(1−(−(z−i))[1/2iz−i+−1/2i2i(1−(−z−i2i))]
So it's easy to find expansion for g(z) and h(z) and then multiply the two expansions
We notice that f has simple pole at z=i So, we can get the principal part easily Or using this
2πia1=∫|z−i|=1f(z)dz
Is there a trick to find the Laurent series quickly ?
This question was in my exam .I Calculated the principal part , but I didn't have enough time to calculate the exact form for the analytic part .
Thank you
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