### The Riemann Surfaces of the Arc* Functions

Because of their practical importance, in this section we will construct pictures of the real and imaginary part as well as the absolute value of the inverse trigonometric and hyperbolic functions. Here is a table of some properties of these functions.

Table 1. Properties of inverse trigonometric and hyperbolic functions.

The second column gives the representation of the function using logarithms and square roots (which can be obtained using `TrigToExp`). Such a representation is particularly useful here because it is easy to continue the logarithm and the square root to the other sheet(s). The third column gives the branch points. All finite branch points are at the origin or at distance 1 at the real or imaginary axes. The fourth column gives the location of the branch cuts as intervals. This suggests subdividing the complex -plane into sectors  such that we do not encounter a branch cut inside any given sector. The last two columns give the real and imaginary part of the jump across a branch cut.

The function `rsfPicture` generates a picture of the real part, the imaginary part, and the absolute value of the sheets of the Riemann surface of the list `functions`.

Using the logarithmic representations given, we get the following pictures of parts of the Riemann surface of the `Arc*` functions. We analytically solve the square roots by taking into account the negative branch and the logarithms by . Some of the sheets will coincide and so either the real or imaginary part will give a faithful representation of the Riemann surface.

A function which would produce the first arguments of the inputs below would be as follows.

Here are two examples of continued functions.

The function `arcTable` will generate a table of some sheets of the Riemann surface of the function `f`.

Here are pictures of the real and imaginary parts of all inverse trigonometric and hyperbolic functions.

``rsfPicture[arcTable[ArcSin,z],z];``

``rsfPicture[arcTable[ArcCos,z],z];``

``rsfPicture[arcTable[ArcTan,z],z];``

``rsfPicture[arcTable[ArcCot,z],z];``

``rsfPicture[arcTable[ArcSec,z],z];``

``rsfPicture[arcTable[ArcCsc,z],z];``

``rsfPicture[arcTable[ArcSinh,z],z];``

``rsfPicture[arcTable[ArcCosh,z],z];``

``rsfPicture[arcTable[ArcTanh,z],z];``

``rsfPicture[arcTable[ArcCoth,z],z];``

``rsfPicture[arcTable[ArcSech,z],z];``

``rsfPicture[arcTable[ArcCsch,z],z];``

Converted by Mathematica

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