Image of upper half-plane under the inverse sine

In summary, the problem is to find the image of the upper half-plane under the inverse sine function, defined as arccos(w) = -i * log(w +/- sqrt(w^2 - 1)) and arcsin(w) = pi/2 - arccos(w). The problem does not specify which branches of the logarithm and square root to use, so the principal branches are assumed. The attempt at a solution involves breaking the inverse sine into a sequence of compositions, but it has not been successful in determining the images. Directly approaching the problem is also being considered.
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zpconn
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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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  • #2
I really hate to bump this, but does anybody have any ideas on how to approach the problem? I would significantly appreciate any advice!
 

FAQ: Image of upper half-plane under the inverse sine

What is the image of the upper half-plane under the inverse sine function?

The image of the upper half-plane under the inverse sine function is the set of all complex numbers that can be obtained by taking the inverse sine of any complex number in the upper half-plane.

How is the image of the upper half-plane under the inverse sine related to the unit circle?

The image of the upper half-plane under the inverse sine function is closely related to the unit circle. In fact, the points on the unit circle correspond to the boundary of the image, with the rest of the image being filled in by points in the interior of the circle.

Can the inverse sine function map any point in the upper half-plane to a point on the unit circle?

No, the inverse sine function cannot map any point in the upper half-plane to a point on the unit circle. The inverse sine function is only defined for complex numbers within a certain range, and there are some points in the upper half-plane that cannot be mapped to a point on the unit circle.

How does the image of the upper half-plane under the inverse sine change as the input points move further away from the origin?

The image of the upper half-plane under the inverse sine expands and becomes more spread out as the input points move further away from the origin. This is because the inverse sine function has a larger range for larger input values, resulting in a larger image.

What is the significance of studying the image of the upper half-plane under the inverse sine function?

The image of the upper half-plane under the inverse sine function is important in understanding the behavior of the inverse sine function and its relationship with the unit circle. It also has applications in fields such as complex analysis, geometry, and physics.

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