Some higher-order trigonometric functions.
Version of Thursday 21 January 2016.
Dave Barber's e-mail and other pages.

In this report are generalized several of the usual trigonometric functions (circular, hyperbolic) defined on the complex numbers.

We start with the familiar imaginary constant i which, in polar form, can be written (1, π / 2). Recall that i is an fourth root of unity. Similarly define j = (1, π / 3), which is a sixth root of unity; and k = (1, π / 4), which is an eighth root of unity. Observe i 2 = j 3 = k 4 = −1. The pattern can be continued, but this is enough to set up an analogy.

First we write some well-known functions, here classified as order two, in a special format:

usual name new name Order twoi = (1, π / 2) z 0 ÷ 0! + z 2 ÷ 2! + z 4 ÷ 4! + z 6 ÷ 6! … 1⁄2 ( i 0 exp (i 0z) + i 0 exp (i 2z) z 0 ÷ 0! − z 2 ÷ 2! + z 4 ÷ 4! − z 6 ÷ 6! … 1⁄2 ( i 0 exp (i 1z) + i 0 exp (i 3z) z 1 ÷ 1! + z 3 ÷ 3! + z 5 ÷ 5! + z 7 ÷ 7! … 1⁄2 ( i 0 exp (i 0z) + i 2 exp (i 2z) z 1 ÷ 1! − z 3 ÷ 3! + z 5 ÷ 5! − z 7 ÷ 7! … 1⁄2 ( i 1 exp (i 3z) + i 3 exp (i 1z)

Here are some new functions:

name Maclaurin series definition Order threej = (1, π / 3) z 0 ÷ 0! + z 3 ÷ 3! + z 6 ÷ 6! + z 9 ÷ 9! … 1⁄3 ( j 0 exp (j 0z) + j 0 exp (j 2z) + j 0 exp (j 4z) ) z 0 ÷ 0! − z 3 ÷ 3! + z 6 ÷ 6! − z 9 ÷ 9! … 1⁄3 ( j 0 exp (j 1z) + j 0 exp (j 3z) + j 0 exp (j 5z) ) z 1 ÷ 1! + z 4 ÷ 4! + z 7 ÷ 7! + z 10 ÷ 10! … 1⁄3 ( j 0 exp (j 0z) + j 2 exp (j 4z) + j 4 exp (j 2z) ) z 1 ÷ 1! − z 4 ÷ 4! + z 7 ÷ 7! − z 10 ÷ 10! … 1⁄3 ( j 1 exp (j 5z) + j 3 exp (j 3z) + j 5 exp (j 1z) ) z 2 ÷ 2! + z 5 ÷ 5! + z 8 ÷ 8! + z 11 ÷ 11! … 1⁄3 ( j 0 exp (j 0z) + j 2 exp (j 2z) + j 4 exp (j 4z) ) z 2 ÷ 2! − z 5 ÷ 5! + z 8 ÷ 8! − z 11 ÷ 11! … 1⁄3 ( j 0 exp (j 3z) + j 2 exp (j 5z) + j 4 exp (j 1z) )

and some more:

name Maclaurin series definition Order fourk = (1, π / 4) z 0 ÷ 0! + z 4 ÷ 4! + z 8 ÷ 8! + z 12 ÷ 12! … 1⁄4 ( k 0 exp (k 0z) + k 0 exp (k 2z) + k 0 exp (k 4z) + k 0 exp (k 6z) ) z 0 ÷ 0! − z 4 ÷ 4! + z 8 ÷ 8! − z 12 ÷ 12! … 1⁄4 ( k 0 exp (k 1z) + k 0 exp (k 3z) + k 0 exp (k 5z) + k 0 exp (k 7z) ) z 1 ÷ 1! + z 5 ÷ 5! + z 9 ÷ 9! + z 13 ÷ 13! … 1⁄4 ( k 0 exp (k 0z) + k 2 exp (k 6z) + k 4 exp (k 4z) + k 6 exp (k 2z) ) z 1 ÷ 1! − z 5 ÷ 5! + z 9 ÷ 9! − z 13 ÷ 13! … 1⁄4 ( k 1 exp (k 7z) + k 3 exp (k 5z) + k 5 exp (k 3z) + k 7 exp (k 1z) ) z 2 ÷ 2! + z 6 ÷ 6! + z 10 ÷ 10! + z 14 ÷ 14! … 1⁄4 ( k 0 exp (k 0z) + k 0 exp (k 4z) + k 4 exp (k 2z) + k 4 exp (k 6z) ) z 2 ÷ 2! − z 6 ÷ 6! + z 10 ÷ 10! − z 14 ÷ 14! … 1⁄4 ( k 2 exp (k 3z) + k 2 exp (k 7z) + k 6 exp (k 1z) + k 6 exp (k 5z) ) z 3 ÷ 3! + z 7 ÷ 7! + z 11 ÷ 11! + z 15 ÷ 15! … 1⁄4 ( k 0 exp (k 0z) + k 2 exp (k 2z) + k 4 exp (k 4z) + k 6 exp (k 6z) ) z 3 ÷ 3! − z 7 ÷ 7! + z 11 ÷ 11! − z 15 ÷ 15! … 1⁄4 ( k 1 exp (k 5z) + k 3 exp (k 7z) + k 5 exp (k 1z) + k 7 exp (k 3z) )

An algorithm to convert a Maclaurin series into a finite formula has is not known to the present author, who simply experimented with the numbers until everything came out right. An open question is whether the finite formulas are unique.

With the Maclaurin series definition, the functions can readily be extended to accept many entities other than complex numbers, such as quaternions or square matrices, as arguments. Whatever the argument type, the Maclaurin series will often converge fast enough to make it the heart of a viable numerical method.

Obvious from the series, but not the finite formula, is that if the input is a real number, so will be the output.

A tauter notation is usually possible, for instance:

namefinite formula more compactly
T4P0 (z) 14 ( exp (z) + exp (iz) + exp (−z) + exp (−iz) )
T4A0 (z) 14 ( exp (kz) + exp (kiz) + exp (−kz) + exp (−kiz) )
T4P1 (z) 14 ( exp (z) + i exp (−iz) − exp (−z) − i exp (iz) )
T4A1 (z) 14 ( k exp (−kiz) + ki exp (−kz) − k exp (kiz) − ki exp (kz) )
T4P2 (z) 14 ( exp (z) + exp (−z) − exp (iz) − exp (−iz) )
T4A2 (z) 14 ( i exp (kiz) + i exp (−kiz) −i exp (kz) − i exp (−kz) )
T4P3 (z) 14 ( exp (z) + i exp (iz) − exp (−z) − i exp (−iz) )
T4A3 (z) 14 ( k exp (−kz) + ki exp (−kiz) − k exp (kz) − ki exp (kiz) )

Here is an explanation of the names:

• the "T" stands for "trigonometry";
• the first digit is the order of the function;
• the "P" means that all the signs in the series are plus, "A" means that they alternate between plus and minus;
• the final digit is the exponent of the first term in the series.

Although the signs alternate in an "A" series, we cannot describe it as an alternating series, because in a genuine alternating series the terms must be real, and further must be positive. Here by contrast the terms within the series are unrestricted complex numbers.

Orders five and six are displayed on a separate page.

Infrapolating, define h = (1, π) = −1, which is a second root of unity. Then:

usual name new name Order oneh = (1, π / 1) z 0 ÷ 0! + z 1 ÷ 1! + z 2 ÷ 2! + z 3 ÷ 3! … 1⁄1 ( h 0 exp (h 0z) ) z 0 ÷ 0! − z 1 ÷ 1! + z 2 ÷ 2! − z 3 ÷ 3! … 1⁄1 ( h 0 exp (h 1z) )

Because the quantity of formulas is twice the order, we get nothing when attempting order zero.

Finding derivatives is not difficult. One might differentiate the finite formula directly, or the Maclaurin series term by term. For instance, d T4P0(z) = T4P3(z) dz.

In order four, two differentiation cycles emerge, with similar results in other orders:

• T4P0 → T4P3 → T4P2 → T4P1 → T4P0
• +T4A0 → −T4A3 → −T4A2 → −T4A1 → −T4A0 → +T4A3 → +T4P2 → +T4A1 → +T4A0

The higher orders offer many identities resembling those of order two, i.e., ordinary trigonometry. Here for example are a few identities from order three:

3 · T3A1 (z) · T3A2 (z) = T3P0 (z) − T3A0 (2 · z)

3 · T3A2 (z) · T3A0 (z) = T3P2 (2 · j 3 · z) + T3P2 ( j 2 · · z) + T3P2 ( j 4 · · z)

 T3P0 (j 1 · z) = T3A0 (z) T3P1 (j 5 · z) = j 5 · T3A1 (z) T3P2 (− z) = T3A2 (z)

 3 · T3P0(z) · T3P0(z) = + 2 · T3P0(2z) + T3A0(z) 3 · T3A0(z) · T3A0(z) = + 2 · T3A0(2z) + T3P0(z) 3 · T3P1(z) · T3P1(z) = + 2 · T3P2(2z) + T3A2(z) 3 · T3A1(z) · T3A1(z) = + 2 · T3A2(2z) + T3P2(z) 3 · T3P2(z) · T3P2(z) = + 2 · T3P1(2z) − T3A1(z) 3 · T3A2(z) · T3A2(z) = − 2 · T3A1(2z) + T3P1(z)

 3 · T3P0 (z) · T3P0 (w) = T3P0 (z + j 0 · w) + T3P0 (z + j 2 · w) + T3P0 (z + j 4 · w) = T3P0 (j 0 · z + w) + T3P0 (j 2 · z + w) + T3P0 (j 4 · z + w) 3 · T3P0 (z) · T3A0 (w) = T3P0 (z + j 1 · w) + T3P0 (z + j 3 · w) + T3P0 (z + j 5 · w) = T3A0 (j 1 · z + w) + T3A0 (j 3 · z + w) + T3A0 (J5 · z + w) 3 · T3A0 (z) · T3A0 (w) = T3A0 (z + j 0 · w) + T3A0 (z + j 2 · w) + T3A0 (z + j 4 · w) = T3A0 (j 0 · z + w) + T3A0 (j 2 · z + w) + T3A0 (j 4 · z + w)

 3 · T3P0 (z) · T3P1 (w) = T3P1 (j 0 · z + w) + T3P1 (j 2 · z + w) + T3P1 (j 4 · z + w) 3 · T3P0 (z) · T3A1 (w) = T3A1 (j 1 · z + w) + T3A1 (j 5 · z + w) + T3A1 (j 3 · z + w)

A further generalization arises from observing that half the series above have alternating signs; this can be extended into a rotation of "complex signs". The following are some roots of unity mentioned above, and some new ones, in polar notation:

 g = (1, 0) h = (1, π / 1) i = (1, π / 2) j = (1, π / 3) k = (1, π / 4) l = (1, π / 5) m = (1, π / 6) n = (1, π / 7) o = (1, π / 8)

Recall:

T3A0 (z) = z 0 ÷ 0! − z 3 ÷ 3! + z 6 ÷ 6! − z 9 ÷ 9! + z12 ÷ 12! − z15 ÷ 15! …

The sign alternation can instead be effected with powers of m:

 T3A0 (z) = m 0z 0 ÷ 0! + m 3z 3 ÷ 3! + m 6z 6 ÷ 6! + m 9z 9 ÷ 9! + m12z12 ÷ 12! + m15z15 ÷ 15! … = (mz) 0 ÷ 0! + (mz) 3 ÷ 3! + (mz) 6 ÷ 6! + (mz) 9 ÷ 9! + (mz)12 ÷ 12! + (mz)15 ÷ 15! …

This leads to a wide family of identities of which one example is this:

T3P0 (mz) = T3A0 (z)

In place of m can be substituted any complex number whose magnitude is unity (a rotator), and a different function is generated. With o as the rotator:

 o 0z 0 ÷ 0! + o 4z 4 ÷ 4! + o 8z 8 ÷ 8! + o12z12 ÷ 12! + o16z16 ÷ 16! + o20z20 ÷ 20! … = z 0 ÷ 0! + i z 4 ÷ 4! − z 8 ÷ 8! − i z12 ÷ 12! + z16 ÷ 16! + i z20 ÷ 20! … = T8A0 (z) + i · T8A4 (z)

Another example:

T3P0 (z) = g 0z 0 ÷ 0! + g 3z 3 ÷ 3! + g 6z 6 ÷ 6! + g 9z 9 ÷ 9! + g12z12 ÷ 12! + g15z15 ÷ 15! …

The rotator can be written (1, θ) for some real θ. If π ÷ θ is a rational number, then the rotator is a root of unity and the "complex signs" will eventually repeat. If on the other hand π ÷ θ is irrational, the rotator is not a root of unity and the "complex signs" will never repeat.