Hyperbolic tangent sum

In the previous post I said I was trying remember where I’d seen the tangent sum applied. I mentioned a couple near misses, and it turns out that what I was trying to remember was another near miss. What I’d seen before was not the tangent sum but the hyperbolic tangent sum. Several people suggested that this was I probably had in mind.

The tangent sum of two numbers is defined as the tangent of the sum of their arc tangents:

tan(arctan a + arctan b).

The hyperbolic arctangent sum sticks an h on the end of the tans:

tanh(arctanh a + arctanh b).

We used ⊕ to denote tangent sum in the previous post; here we’ll use it to denote hyperbolic tangent sum:

a ⊕ b = tanh(arctanh a + arctanh b).

The hyperbolic tangent sum is defined on (−1, 1) because that is the range of hyperbolic tangent.

The hyperbolic tangent sum can be simplified to

a ⊕ b = (a + b) / (1 + ab).

Note the sign change in the denominator compared to the tangent sum. This means the hyperbolic tangent sum is better behaved than the tangent sum. If a and b are positive, then a ⊕ b is positive, and if a and b are in the domain of arctanh, then a ⊕ b is finite. The analogous statements weren’t true for the tangent sum.

I wrote about hyperbolic tangent sum three years ago, using the simplified form above and not mentioning hyperbolic tangents. At the time I called it relativistic addition because of the connection with special relativity.

Hyperbolic tangent sum is associative, and the expressions for a ⊕ b ⊕ c ⊕ … have the same form as for the tangent sum, except all the negative signs go away: the sums in the numerator and denominator do not alternate. See this post for more details and a proof.