Monday, 30 March 2015

complex analysis - Laurent series for f(z)=fracsin(2piz)z(z2+1)

How Can find the Laurent series for this function valid for 0<|zi|<2 f(z)=sin(2πz)z(z2+1)




Let g(z)=sin(πz)



sin(πz)=sin(2π(zi))cos(2πi)+cos(2π(zi))sin(2πi)



And Let h(z)=1z2+1



1z(z2+1)=1i(1((zi))[1/2izi+1/2i2i(1(zi2i))]



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=|zi|=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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