How to formally show that f(z) is analytic at z=0? Let z be a complex number. Let f(z)=11z+ln⁡(1z). How to formally show that f(z) is analytic at z=0? I know that for small z we have |1z|&gt;|ln⁡(1z)| and that implies |f(0)|=0. Are there multiple ways to handle this ?

Question

How to formally show that f(z) is analytic at z=0?  Let z be a complex number. Let  f(z)=11z+ln⁡(1z).  How to formally show that f(z) is analytic at z=0? I know that for small z we have  |1z|&amp;gt;|ln⁡(1z)|  and that implies |f(0)|=0. Are there multiple ways to handle this ?

basgrwthej · Accepted Answer

To me, analytic means locally represented by its Taylor series. With this interpretation f is not analytic. Indeed, suppose f(z)=zr∑n=0∞cnzn in a neighborhood of 0, where c0≠0. Then  1z+ln⁡1z=z−r∑{n=0}∞bnzn  in some (possibly smaller) neighborhood of 0. It follows that ln⁡1z has a pole or a removable singularity at 0. If this is not evidently absqrt already, apply the same to ln⁡z=−ln⁡1z and conclude that the logarithm is a rational function.

How to formally show that f(z) is analytic at z=0? Let

Answered question

Answer & Explanation

New Questions in Algebra II