Principal Values of Inverse Cosine and Related Functions

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I’ve recently been working, with Mary Aprahamian, on theory and algorithms for the matrix inverse sine and cosine and their hyperbolic counterparts. Of course, in order to treat the matrix functions we first need a good understanding of the scalar case. We found that, as regards practical computation, the literature is rather confusing. The reason can be illustrated with the logarithm of a complex number.

Consider the question of whether the equation

\log(z_1 z_2) = \log z_1 + \log z_2

is valid. In many textbooks this equation is stated as is, but with the (often easily overlooked) proviso that each occurrence of \log denotes a particular branch of the logarithm—possibly different in each case. In other words, the equation is true for the multivalued function that includes all branches.

In practice, however, we are usually interested in the principal logarithm, defined as the one for which the complex argument of \log z lies in the interval (-\pi,\pi] (or possibly some other specific branch). Now the equation is not always true. A correction term that makes the equation valid can be expressed in terms of the unwinding number introduced by Corless, Hare, and Jeffrey in 1996, which is discussed in my earlier post Making Sense of Multivalued Matrix Functions with the Matrix Unwinding Function.

The definition of principal logarithm given in the previous paragraph is standard. But for the inverse (hyperbolic) cosine and sine it is difficult to find clear definitions of principal values, especially over the complex plane. Some authors define these inverse functions in terms of the principal logarithm. Care is required here, since seemingly equivalent formulas can yield different results (one reason is that (z^2-1)^{1/2} is not equivalent to (z-1)^{1/2}(z+1)^{1/2} for complex z). This is a good way to proceed, but working out the ranges of the principal functions from these definitions is not trivial.

In our paper we give diagrams that summarize four kinds of information about the principal inverse functions acos, asin, acosh, and asinh.

  • The branch points.
  • The branch cuts, marked by solid lines.
  • The domain and range, shaded gray and extending to infinity in the obvious directions).
  • The values attained on the branch cuts: the value on the cut is the limit of the values of the function as z approaches the cut from the side without the hashes.

The figures are below. Once we know the principal values we can address questions analogous to the log question, but now for identities relevant to the four inverse functions.

For more, including an explanation of the figures in words and all the details of the matrix case—including answers to questions such as “when is \mathrm{acos}(\cos A) equal to A?”—see our recent EPrint Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms.

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