# hwprob1 - In other words plug in f(0 f p(0 and the...

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In class we showed that if f ( z ) is a holomorphic function with simple isolated zeroes located at a 1 , a 2 , · · · , none of the a i zero, then f ( z ) could be written in terms of f (0), f p (0), and the a i as f ( z ) = f (0) exp p z f p (0) f (0) P ± n ² 1 - z a n ³ exp( z/a n ) 1. Suppose f ( z ) is an N th order polynomial: f ( z ) = α ( x - x 1 )( x - x 2 ) · · · ( x - x N ) for α some constant and none of the x i vanishing. Show that the expression above correctly captures the polynomial nature of f ( z
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Unformatted text preview: ). In other words, plug in f (0), f p (0), and the locations of the zeroes, and verify that the abstract formula at top reduces to the N th order polynomial above. 2. The function 1 / Γ( z + 1) can be shown to have simple zeroes at z =-1 ,-2 ,-3 , · · · . Use the formula above to derive the Weierstrass representation of 1 / Γ( z )....
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