A meromorphic function $f$ with poles of order $2$ at $\sqrt { n }$ ($n=1,2,3,...$),

Asked Nov 23 '19 at 10:42

Active Nov 07 '22 at 08:05

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find a meromorphic function $f$ with poles of order 2 at $\sqrt { n }$ ($n=1,2,3,...$), the Residue at each pole is $2$ , and $\lim _ { z \rightarrow \sqrt { n } } ( z - \sqrt { n } ) ^ { 2 } f ( z ) = 1$ for all $n \in \mathbb N$.

I tried with $f ( z ) = \frac { 1 } { \left( \sin \pi z^ { 2 } \right) ^ { 2 } }$ , but conditions are not fully conciding

edited Nov 07 '22 at 08:05

asked Nov 23 '19 at 10:42

Riaz

2,174

2

Yep there will be issues, because it will have zeroes at every negative integer too. But it is a good start. – mathreadler Nov 23 '19 at 11:10
1

https://en.wikipedia.org/wiki/Mittag-Leffler%27s_theorem – metamorphy Nov 07 '22 at 09:34

A meromorphic function $f$ with poles of order $2$ at $\sqrt { n }$ ($n=1,2,3,...$),

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