Tangential Identities

 

To the first order, the expressions (1+x)/(1−x) and e2x both equal 1+2x, although they differ at higher orders. This suggests that an interesting function might be definable in terms of these expressions. Indeed, the hyperbolic tangent can be defined implicitly by the relation

 

 

Solving for tanh(θ) yields the more usual definition

 

 

This is just the hyperbolic sine divided by the hyperbolic cosine. Putting t = tanh(θ) we have θ = inv tanh(t), and taking the natural log of both sides of equation (1) and dividing through by 2 we get the identity

 

 

Several other useful identities follow from these expressions. For example, given any three quantities θ1, θ2, θ3, we see from equation (1) that

 

 

Taking the natural log of both sides and dividing through by 2, we have

 

 

where tj denotes tanh(θj). If θ1 + θ2 + θ3 = 0 then the right side of equation (4) equals 1, and we can clear the denominator and cancel terms to arrive at

 

 

More generally, given any n quantities θ1, θ2, ..., θn and the corresponding values tj = tanh(θj), we have

 

 

Consequently, we can clear the denominator and re-arrange terms to give the identity

 

 

where σm is the mth elementary symmetric function of the quantities tj. For example, with n = 3 the elementary symmetric functions are

 

 

We stipulate that σm = 0 for m > n. In terms of these quantities we have

 

 

In other words, the hyperbolic tangent of any sum of quantities equals the sum of the odd elementary symmetric functions of the hyperbolic tangents of those individual quantities divided by the sum of the even elementary symmetric functions. If we consider the n individual quantities as orthogonal coordinates, its interesting that this complicated function of the n coordinates in invariant throughout any (n-1)-dimensional subspace with a given sum, such as the diagonal planes in 3-dimensional space.

 

With n = 2 we have just two quantities θ1, θ2 and the corresponding quantities tj = tanh(θj), from which we get σ0 = 1, σ1 = t1 + t2, and σ2 = t1t2. The above relation then gives the familiar addition rule for the hyperbolic tangent

 

 

Analogously to equation (1), the usual tangent function is defined implicitly by the relation

 

 

Solving for tan(θ) we get the more usual definition

 

 

Putting t = tan(θ), we have θ = inv tan(t), and we can solve equation (5) for θ to give the identity

 

 

Given any three quantities θ1, θ2, θ3, we see from equation (4) that

 

 

Taking the natural log of both sides and dividing through by 2i, we have

 

 

where tj denotes tanh(θj). If θ1 + θ2 + θ3 = kπ for any integer k, then the right side of (8) equals 1, and we can clear the denominator and cancel terms to arrive at

 

 

where tj denotes tan(θj). On the other hand, if θ1 + θ2 + θ3 = (k + 1/2)π for any integer k, then the right side of the relation equals −1, and we get

 

 

More generally, given any n quantities θ1, θ2, ..., θn and the corresponding values tj = tan(θj), we have

 

 

Consequently, we can clear the denominator and re-arrange terms to give the identity

 

 

where σm is the mth elementary symmetric function of the quantities tj. In terms of these quantities we have

 

 

This can also be written as

 

 

which shows that if s = kπ for any integer k, then the sum of the odd symmetric functions with alternating signs vanishes, whereas if s = (k + 1/2)π then the sum of the even symmetric functions with alternating signs vanishes.

 

Now consider three arbitrary quantities q1, q2, q3 and let uij = qi qj. It follows that

 

 

Putting θ1 = u12, θ2 = u23, θ3 = u31, and letting vij = tanh(uij) we see from equation (4) that we have the relation

 

 

Identifying the quantities vij as the speed of the spatial origin of inertial coordinate system i in terms of inertial coordinate system j, the quantities uij are called the rapidities, and the above expression gives the rule for velocity composition in special relativity.

 

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