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Find the residues of $f(z) = \frac{sin \pi z^2 + cos \pi z^2}{(z - 1)(z - 2)^2}$ at its pole.

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Find the residues of $f(z) = \frac{sin \pi z^2 + cos \pi z^2}{(z - 1)(z - 2)^2}$ at its pole.

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written 3.8 years ago by | modified 3.6 years ago by |

Poles are given by $(z-1)(z-2)^2=0$

i.e. z=1 is the pole of order 1 & z=2 is the pole of order 2.

Residue at the pole z=1 is given as

$R_1 = lim_{z→1} \,\ (z-1). \frac{sinπz^2+cosπz^2}{(z-1)(z-2)^2}$

$= lim_{z→1} \ \frac{sinπz^2+cosπz^2}{(z-2)^2} = \frac{(sinπ+cosπ)}{(1-2)^2} = -1$

Also Residue at the pole z=1 is given as

$R_2 = lim_{z→2} \,\ \ \frac{1}{1!} \frac{d}{dz} (z-2)^2 . \frac{sinπz^2+cosπz^2}{(z-1)(z-2)^2}$

= $ lim_{z→2} \,\ \frac{d}{dz} (\frac{sinπz^2+cosπz^2}{z-1}) = 4π- 1$

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