Dgdrxrt

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Avoiding the "not like other girls" trope?

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Is the function $f(z)=log z +z$ injective in the neighborhood of the infinity?

Understanding the definition of the order of an entire function in Ahlfors's Complex AnalysisWhy the function $g(q)=f(log(q)/(2pi i))$ is well defined?Adding infinity to the upper half planeconformal mapping, regions of the complex plane marked +/-, find the function f,Injectivity of the extension of a injective functionAlternative definitions of residue at infinityShow holomorphic branch of $log z$ on $Omega=mathbbCbackslash ztext real : zleq 0$ satisfying $log(1)=0$.Prove that the principal logarithm ($Log$) is a bijection between $mathbbC-[0,infty)$ and $Omega=z in mathbbC $The complex cosine is injective in the range $0<Re(z)<pi/2$Find a branch of the logarithm such that the function is holomorphic everywhere except at the given half line

0

$begingroup$

Let $U_R=$ be a neighborhood of the infinity. I know $f(z)=log z+z$ is well-defined in $U_R$ with image in $V_R$ equal the $U_R$ minus the $pi$-neighborhood of real half line $mathbbR^-$.

I saw for $R> 1$ that $f$ is locally injective. Is it globally injective, for $R$ sufficiently large?

complex-analysis complex-numbers complex-geometry

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edited Mar 28 at 19:58

Alan Muniz

2,61311030

asked Mar 5 at 16:50

genasgenas

42

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  The function $zmapsto rm Log,z$, or any other branch of $log$, is not well defined in $U_R$.
  $endgroup$
  – Christian Blatter
  Mar 5 at 19:02
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax.
  $endgroup$
  – dantopa
  Mar 7 at 0:57
  

  
  
  
  

add a comment | 

0

$begingroup$

Let $U_R=$ be a neighborhood of the infinity. I know $f(z)=log z+z$ is well-defined in $U_R$ with image in $V_R$ equal the $U_R$ minus the $pi$-neighborhood of real half line $mathbbR^-$.

I saw for $R> 1$ that $f$ is locally injective. Is it globally injective, for $R$ sufficiently large?

complex-analysis complex-numbers complex-geometry

share|cite|improve this question

edited Mar 28 at 19:58

Alan Muniz

2,61311030

asked Mar 5 at 16:50

genasgenas

42

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  The function $zmapsto rm Log,z$, or any other branch of $log$, is not well defined in $U_R$.
  $endgroup$
  – Christian Blatter
  Mar 5 at 19:02
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax.
  $endgroup$
  – dantopa
  Mar 7 at 0:57
  

  
  
  
  

add a comment | 

$begingroup$

Let $U_R=$ be a neighborhood of the infinity. I know $f(z)=log z+z$ is well-defined in $U_R$ with image in $V_R$ equal the $U_R$ minus the $pi$-neighborhood of real half line $mathbbR^-$.

I saw for $R> 1$ that $f$ is locally injective. Is it globally injective, for $R$ sufficiently large?

complex-analysis complex-numbers complex-geometry

share|cite|improve this question

edited Mar 28 at 19:58

Alan Muniz

2,61311030

asked Mar 5 at 16:50

genasgenas

42

$endgroup$

share|cite|improve this question

edited Mar 28 at 19:58

Alan Muniz

2,61311030

asked Mar 5 at 16:50

genasgenas

42

share|cite|improve this question

edited Mar 28 at 19:58

Alan Muniz

2,61311030

asked Mar 5 at 16:50

genasgenas

42

edited Mar 28 at 19:58

Alan Muniz

2,61311030

edited Mar 28 at 19:58

Alan Muniz

2,61311030

asked Mar 5 at 16:50

genasgenas

42

asked Mar 5 at 16:50

genasgenas

42