No Branch Cut Needed for cos(sqrt(z))

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SUMMARY

The discussion clarifies that no branch cut is necessary for the function cos(sqrt(z)). This conclusion stems from the even nature of the cosine function, which allows for the simplification of multivalued functions like sqrt(z). Participants emphasize the importance of expressing z in polar form, specifically as r*exp(i(t+2*k*pi)), to understand the behavior of sqrt(z) and its implications for cos(sqrt(z)).

PREREQUISITES
  • Understanding of complex functions, particularly multivalued functions.
  • Familiarity with the properties of the cosine function, especially its evenness.
  • Knowledge of polar coordinates and their application in complex analysis.
  • Experience with the principal branch of the square root function, Sqrt(z).
NEXT STEPS
  • Study the implications of even functions in complex analysis.
  • Learn about the principal branch of multivalued functions and their representations.
  • Explore the properties of polar coordinates in complex number calculations.
  • Investigate other trigonometric functions and their behavior with multivalued inputs.
USEFUL FOR

Students and professionals in mathematics, particularly those focusing on complex analysis, as well as educators looking to clarify concepts related to multivalued functions and branch cuts.

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Homework Statement

Branch cut for cos(sqrt(z)).



Homework Equations





The Attempt at a Solution

Apparently there is no need for a branch cut for this function, but I am not sure why - I heard it has something to do with cos being an even function. Any clarification would be appreciated.
 
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I'm sure you've done an exercise like writing down all of the values of a multivalued function like sqrt(z) in terms of ordinary functions. (e.g. in terms of the principal branch Sqrt(z))

That seems like an obvious place to start to me.
 
Hurkyl said:
I'm sure you've done an exercise like writing down all of the values of a multivalued function like sqrt(z) in terms of ordinary functions. (e.g. in terms of the principal branch Sqrt(z))

That seems like an obvious place to start to me.

You mean write z as r*exp(i(t+2*k*pi)), so sqrt(z) = (r^1/2)*exp(it/2) or (r^1/2)*exp(i(t/2+pi)? = -(r^1/2)*exp(it/2) ?
 

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