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https://texercises.com/exercise/polynomial-proof/
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Exercise:
Let fz be an entire function such that lim_|z|rightarrow infty|fz| +infty Show that fz is a polynomial.

Solution:
lim_|z|rightarrow infty|fz| +infty means that fz has a pole at infinity fz has pole at infinity rightarrow f/z has a pole at then we can expand in Laurent's series fleftfraczright _n-d^+inftya_nz^n for some dgeq . Hence fz _n-infty^d a_-nz^n but f is holomorphic in this means that a_-n for all n so fz _n^d a_-nz^n a polynomial.
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Exercise:
Let fz be an entire function such that lim_|z|rightarrow infty|fz| +infty Show that fz is a polynomial.

Solution:
lim_|z|rightarrow infty|fz| +infty means that fz has a pole at infinity fz has pole at infinity rightarrow f/z has a pole at then we can expand in Laurent's series fleftfraczright _n-d^+inftya_nz^n for some dgeq . Hence fz _n-infty^d a_-nz^n but f is holomorphic in this means that a_-n for all n so fz _n^d a_-nz^n a polynomial.
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ca, entire, eth, holomorphic, polynomial
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ENG (English)
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Creator rk
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