Image of upper half-plane under the inverse sine

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SUMMARY

The discussion focuses on determining the image of the upper half-plane under the inverse sine function, defined as arcsin(w) = π/2 - arccos(w), where arccos(w) is expressed using the principal branches of the logarithm and square root. The user has attempted to analyze the function by composing it with elementary functions but has struggled to find a suitable breakdown for identifying the images. The need for guidance on approaching this problem is emphasized, indicating a gap in understanding the application of these mathematical concepts.

PREREQUISITES
  • Understanding of complex analysis, specifically the properties of inverse trigonometric functions.
  • Familiarity with logarithmic functions and their branches in complex analysis.
  • Knowledge of the square root function and its principal branch in the complex plane.
  • Basic skills in function composition and mapping in complex functions.
NEXT STEPS
  • Study the properties of the inverse sine function in complex analysis.
  • Learn about the principal branches of logarithmic and square root functions.
  • Explore complex mappings and transformations of the upper half-plane.
  • Investigate the relationship between arcsin and arccos functions in the context of complex variables.
USEFUL FOR

Students and mathematicians interested in complex analysis, particularly those studying inverse trigonometric functions and their applications in mapping complex planes.

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Homework Statement



The problem is simply to find the image of the upper half-plane under the inverse sine function.

Homework Equations



The textbook defines the inverse sine in the following way. First, it defines arccos(w) = -i * log(w +/- sqrt(w^2 - 1)) and then it defines arcsin(w) = pi/2 - arccos(w).

The problem doesn't specify what branches of the logarithm and square root should be used. I'm going to assume the principal branches.

The Attempt at a Solution



I've attempted to just break the inverse sine into a sequence of compositions and to, so to speak, push the upper half-plane through this sequence, but I've not been able to identify a breakdown into suitably elementary functions allowing me to determine the images.

As for doing it directly, I'm just now seeing how to begin.
 
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I really hate to bump this, but does anybody have any ideas on how to approach the problem? I would significantly appreciate any advice!
 

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