Building complex functions out of real parts

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A couple months ago I wrote about how to compute the sine and cosine of a complex number using only real functions of real variables using the equations

You can do something analogous for all the elementary functions, though some of the equations are quite a bit more complicated than the ones above. See the equations here.

The equations come from a paper by Henry G. Baker, cited in the linked page. I wrote up Baker’s equations in LaTeX, then used ChatGPT to generate Python code from the LaTeX to numerically verify the equations and my typesetting of them. This caught a few typos on my part.

The test code evaluated the equations at points from each quadrant. All matched NumPy, implying that Baker and NumPy use the same branch cuts on inverse functions.

This post is part of a thread that has gone on for a few days. Maybe it’s the last post in the thread; we’ll see.

It all started with a post on Markov’s equation

x² + y² + z² = 3 xyz

and an approximation to the equation that has a closed-form solution. That led to the identity

cosh( arccosh(a) + arccosh(b) ) = ab + √(a² − 1) √(b² − 1).

The approximation to Markov’s equation only needed the identity to be valid for real a and b greater than 1. But when I looked closer at the identity I found several complications with branch cuts. The identity doesn’t hold everywhere using the principle branch of the square root function. But if you define √(z² − 1) to have a branch cut along [−1, 1] then the equation holds everywhere in the complex plane. And that led to my writing up some notes on how to define all the elementary inverse functions in terms of log.

Someone reading these posts suggested I look at a paper that mentioned "couth" and "uncouth" function pairs, which led to this post and its warm up.

I find all this interesting because it’s an advanced perspective on a questions that are latent in a intro calculus class. What exactly do functions like arccos mean and why where they defined as they were? These are fairly deep and interesting questions that are swept under the rug, and swept there for good reason. A calculus class has to cover an enormous amount of material and there’s no time to dwell on fine points. Some of my favorite posts look back leisurely on things that go by in a blur when you’re a student.

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John D. Cook, PhD

My colleagues and I have decades of consulting experience helping companies solve complex problems involving data privacy, applied math, and statistics.

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