Inverse circular and hyperbolic functions

What is it about?

The inverse trigonometric and inverse hyperbolic functions are multivalued when considered as functions of a complex variable, meaning they can take on multiple values for a single input. To resolve this ambiguity, the NIST Digital Library of Mathematical Functions (DLMF) defines principal branches, single-valued versions of these functions, along with the corresponding cut planes that establish where these functions are valid. The DLMF provides complex-valued formulas for evaluating these principal branches. In two cases—the inverse sine and inverse cosine—it also includes explicit formulas that depend only on the real and imaginary parts of the complex input, making evaluation more direct. We refer to these as concrete expressions, which are particularly useful because they eliminate the need for deeper knowledge of complex analysis. In this article, we extend these results by deriving concrete expressions for four remaining principal branches. Our findings offer a computationally efficient resource for computer algebra systems, programming languages, and individual users, making it easier to evaluate these functions accurately and efficiently.

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Why is it important?

This work completes work that was started at Bell Labs over a century ago. It also collects in one place concrete expressions for the principal branches that are missing from the Digital Library of Mathematical Functions (DLMF).

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