How to derive the branch cuts for complex arcsin(z)

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SUMMARY

The branch cuts for the complex arcsin(z) function are established from -∞ to -1 and from 1 to ∞. These specific cuts ensure that the function remains single-valued by preventing any self-intersecting paths in the complex plane. The chosen cuts are widely accepted due to their simplicity and symmetry with respect to the real axis, making them a standard choice in complex analysis.

PREREQUISITES
  • Understanding of complex analysis concepts
  • Familiarity with the properties of the arcsine function
  • Knowledge of branch cuts in complex functions
  • Basic grasp of the complex plane and path connectivity
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  • Research the properties of complex functions and their branch cuts
  • Study the derivation of the arcsine function in complex analysis
  • Explore alternative branch cut configurations for other complex functions
  • Learn about path integrals and their implications in complex analysis
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Mathematicians, students of complex analysis, and anyone interested in understanding the behavior of complex functions and their branch cuts.

monmon_4
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I notice that the branch cuts for arcsin(z) are from -infinity to -1 and 1 to infinity. How do these choices for the branch cuts make the function single-valued? I am having trouble understanding the reasoning here even though these choices for the cuts are widely used/accepted.
 
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I believe that any two paths from -1 and 1 to infinity that don't intersect themselves or each other (i.e. leaves the complex plane connected) will work. The standard ones seem like a good choice. They are simple and symmetric wrt the real axis.
 

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