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Ok, the quick and dirty: The given Theorem: If f is analytic everywhere in the finite complex plane except for a finite number of singular points interior to a positively oriented simple closed contour C, then
\int_{C}f(z)dz=2\pi i \mbox{Res}\left[ \frac{1}{z^2}f\left( \frac{1}{z}\right) ;z=0\right]
The problem: If P(z)=a_0+a_1z+\cdots +a_nz^n and Q(z)=b_0+b_1z+\cdots+b_mz^m where m\geq n+2,a_0\neq 0, b_0\neq 0 and all the zeros of Q(z) lie interior to C, prove that
\int_{C}\frac{P(z)}{Q(z)}dz=0
Apply the theorem to get
\int_{C}\frac{P(z)}{Q(z)}dz=2\pi i \mbox{Res}\left[ \frac{1}{z^2}\frac{P\left( \frac{1}{z}\right) }{Q\left( \frac{1}{z}\right) } ;z=0\right]
now what? Pratial fractions? That is somewhat brutal, isn't there some simple way to do this. Thanks. --Ben
Papers due in a few hours, please hurry...
\int_{C}f(z)dz=2\pi i \mbox{Res}\left[ \frac{1}{z^2}f\left( \frac{1}{z}\right) ;z=0\right]
The problem: If P(z)=a_0+a_1z+\cdots +a_nz^n and Q(z)=b_0+b_1z+\cdots+b_mz^m where m\geq n+2,a_0\neq 0, b_0\neq 0 and all the zeros of Q(z) lie interior to C, prove that
\int_{C}\frac{P(z)}{Q(z)}dz=0
Apply the theorem to get
\int_{C}\frac{P(z)}{Q(z)}dz=2\pi i \mbox{Res}\left[ \frac{1}{z^2}\frac{P\left( \frac{1}{z}\right) }{Q\left( \frac{1}{z}\right) } ;z=0\right]
now what? Pratial fractions? That is somewhat brutal, isn't there some simple way to do this. Thanks. --Ben
Papers due in a few hours, please hurry...
