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A trick to eliminate 2p2\pi2p (sometimes)
Published on 2023-06-28 (Tau Day)
As a physicist, I've learned that the presence of 2p2\pi2p^[V][ ] [Or
rather t\taut, as it should actually be called.] in a formula usually
spells trouble. It's kind of annoying to write by hand, easy to
forget, and it's dimensionless, so you can't easily check if you
forgot to divide or multiply by it. Problems appear especially often
when translating between theory and experiments.^[V][ ] [I remember
one of the very first lectures of my "Introduction to University
Physics" course, where the lecturer wrote a long and complicated
equation on the blackboard, and then went "Oh, and maybe there should
be a 2p2\pi2p here, I'm not exactly sure now. But you get the idea."
I was absolutely shocked. How could anyone do physics with such
imprecision and carelessness?! Later of course I've learned that
sometimes it's OK to be off by an order of magnitude here or there.
But I think this is where my frustration with 2p2\pi2p started.]
Recently, I came up with a trick that can get rid of 2p2\pi2p in many
cases. It's pretty simple, but it has some interesting implications.
This is the trick: I just define a new derivative operator, like so:
ddxf(x)[?]12p[?]ddxf(x). \frac{\mathrm{d}}{\mathrm{d} x} f(x) \equiv \
frac{1}{2\pi} \cdot \frac{\mathrm{d}}{\mathrm{d} x} f(x). dxd f(x)[?]2p
1 [?]dxd f(x). That's all. You just take the derivative and then divide
by 2p2\pi2p. I call this derivative operator with the bar on the
upper d\mathrm{d}d the reduced derivative.^[V][ ] [I discuss this
name and notation below.]
Now, why is this interesting? To start off, we'll note that the
unique function f(x)f(x)f(x) which satisfies ddxf(x)=f(x),f(0)=1 \
frac{\mathrm{d}}{\mathrm{d} x} f(x) = f(x), \quad f(0) = 1 dxd f(x)=f
(x),f(0)=1 is f(x)=e2px=Thx, f(x) = e^{2\pi x} = \Theta^x, f(x)=e2px=
Thx, where Th\ThetaTh (capital Theta) is defined as Th[?]e2p=535.491...\Theta
\equiv e^{2\pi} = 535.491\dotsTh[?]e2p=535.491....^[V][ ] [I have a few
words about the choice of symbol below.] Let me show you why Thx\Theta
^xThx is actually a pretty nice function.
Probably the nicest thing about Thx\Theta^xThx has to do with
trigonometric functions. You may or may not know that sine and cosine
can be defined using complex exponentials like this: sin[?](x)=
eix-e-ix2i,cos[?](x)=eix+e-ix2, \begin{align*} \sin(x) &= \frac{e^{ix}
- e^{-ix}}{2i}, \\ \cos(x) &= \frac{e^{ix} + e^{-ix}}{2}, \end
{align*} sin(x)cos(x) =2ieix-e-ix ,=2eix+e-ix , but using Th\ThetaTh,
we can define new "trigonometric functions": usin[?](x)=
Thix-Th-ix2i,ucos[?](x)=Thix+Th-ix2. \begin{align*} \operatorname{usin}(x)
&= \frac{\Theta^{ix} - \Theta^{-ix}}{2i}, \\ \operatorname{ucos}(x) &
= \frac{\Theta^{ix} + \Theta^{-ix}}{2}. \end{align*} usin(x)ucos(x) =
2iThix-Th-ix ,=2Thix+Th-ix . These are some names that I just came up
with, but the "u\mathrm{u}u" stands for "unit" -- these functions have
the nice property that their period is 111 instead of 2p2\pi2p: usin[?]
(x+1)=usin[?](x),ucos[?](x+1)=ucos[?](x). \begin{align*} \operatorname
{usin}(x + 1) &= \operatorname{usin}(x), \\ \operatorname{ucos}(x +
1) &= \operatorname{ucos}(x). \end{align*} usin(x+1)ucos(x+1) =usin(x
),=ucos(x). I like this, because it kind of eliminates the need for
radians: the xxx in usin[?](x)\operatorname{usin}(x)usin(x) has the
unit of "turns". I think this is conceptually much simpler.^[V][ ] [
These Th\ThetaTh-based unit-period trigonometric functions are also
nicer to calculate on a computer. For example, MATLAB and Julia have
functions sinpi(x) and cospi(x) for calculating sin(pi*x) and cos
(pi*x) with better precision (they are exactly zero at integer values
of x), although they miss the mark by using p\pip instead of 2p2\pi2p
. I'm no floating point expert, but I feel that in addition to
precision, using such functions should be more efficient as well. The
range 0...10\dots 10...1 can be represented exactly in binary (and range
reduction amounts to just discarding the integer part!), so it's much
simpler to deal with than 0...2p0\dots 2\pi0...2p. This point is also
brought up by Casey Muratori. It's also interesting to note that
dealing with radians is still a topic of research.] For the
physicists out there, this is pretty cool because usin[?](ft)\
operatorname{usin}(f t)usin(ft) is a periodic function with frequency
fff, without the need for angular frequency. Instead of mushing 2pf2\
pi f2pf into the single symbol o\omegao, we meld 2p2\pi2p into the
trig function. Yes, this is just an alternative way of sweeping 2p2\
pi2p under the rug, but I claim it makes it easier to avoid mistakes.
These unit trigonometric functions satisfy similar differential
equations as their more well-known siblings, except in terms of the
funny reduced derivative operator we defined above: ddxusin[?](x)=
-ucos[?](x),ddxucos[?](x)=-usin[?](x). \begin{align*} \frac{\mathrm{d}}{\
mathrm{d} x} \operatorname{usin}(x) & = \hphantom{-} \operatorname
{ucos}(x), \\ \frac{\mathrm{d}}{\mathrm{d} x} \operatorname{ucos}(x)
& = -\operatorname{usin}(x). \end{align*} dxd usin(x)dxd ucos(x) =-
ucos(x),=-usin(x). Nice!
You might be aware that eee, 2p2\pi2p and trigonometric functions are
pretty widespread in math. Because of this, there are actually quite
many situations where Thx\Theta^xThx pops up and makes that pesky 2p2\
pi2p go away. Here are some examples I came up with off the top of my
head and with a quick skim through Wikipedia:
The Fourier transform f^\hat ff^ of a function fff: f^(s)=
[?]-[?][?]e-2pisxf(x)dx=[?]-[?][?]Th-isxf(x)dx. \begin{align*} \hat f(s) &= \int_
{-\infty}^\infty e^{-2\pi i s x} f(x) \mathrm{d}x \\ &= \int_{-\
infty}^\infty \Theta^{-i s x} f(x) \mathrm{d}x. \end{align*} f^ (s) =
[?]-[?][?] e-2pisxf(x)dx=[?]-[?][?] Th-isxf(x)dx.
The inverse Fourier transform: f(x)=[?]-[?][?]e2pisxf^(s)ds=[?]-[?][?]Thisxf^(s)
ds. \begin{align*} f(x) &= \int_{-\infty}^\infty e^{2\pi i s x} \hat
f(s) \mathrm{d}s \\ &= \int_{-\infty}^\infty \Theta^{i s x} \hat f(s)
\mathrm{d}s. \end{align*} f(x) =[?]-[?][?] e2pisxf^ (s)ds=[?]-[?][?] Thisxf^ (s)ds
. Note how the forward and inverse transforms are symmetric without
any prefactors.
Dirac delta function: d(x-a)=[?]e2pis(x-a)ds=[?]This(x-a)ds. \begin
{align*} \delta(x-a) &= \int e^{2\pi i s (x-a)} \mathrm{d} s \\ &= \
int \Theta^{i s (x-a)} \mathrm{d} s. \end{align*} d(x-a) =[?]e2pis(x-a)
ds=[?]This(x-a)ds.
Kronecker delta: dnm=1N[?]k=1Ne2pikN(n-m)=1N[?]k=1NThikN(n-m). \begin
{align*} \delta_{nm} &= \frac{1}{N} \sum_{k=1}^N e^{2\pi i \frac{k}
{N} (n-m)} \\ &= \frac{1}{N} \sum_{k=1}^N \Theta^{i \frac{k}{N}
(n-m)} . \end{align*} dnm =N1 k=1[?]N e2piNk (n-m)=N1 k=1[?]N ThiNk (n-m
).
Euler's identity: e2pi=Thi=1. e^{2\pi i} = \Theta^i = 1. e2pi=Thi=1.
Roots of unity: zn=1=z=e2pik/n=Thik/n, \begin{align*} z^n = 1 \
Leftrightarrow z &= e^{2\pi i k / n} = \Theta^{ik/n} , \end{align*}
zn=1=z =e2pik/n=Thik/n, where n[?]N, k=0,1,...,n-1n \in \mathbb{N}, \, k
= 0, 1, \dots, n-1n[?]N,k=0,1,...,n-1.
Square of the nome: q2=(eipt)2=Thit, q^2 = \left(e^{i\pi \tau}\right)^
2 = \Theta^{i\tau} , q2=(eipt)2=Thit, where t\taut is the half-period
ratio of an elliptic function.
Gaussian-like integrals:^[V][ ] [This result is easy to see from
[?]-[?][?]e-(x-b)2/(2c2)dx=|c|2p\int_{-\infty}^\infty e^{-(x-b)^2/(2c^2)} \
mathrm{d} x = |c| \sqrt{2\pi}[?]-[?][?] e-(x-b)2/(2c2)dx=|c|2p , by
substituting c=1/2pc = 1/\sqrt{2\pi}c=1/2p .] [?]-[?][?]Th-x2/2dx=1, \int_{-
\infty}^\infty \Theta^{-x^2/2} \mathrm{d}x = 1 , [?]-[?][?] Th-x2/2dx=1, or
more generally, [?]-[?][?]Th-(x-b)2/(2c2)dx=|c|. \int_{-\infty}^\infty \
Theta^{-(x-b)^2/(2c^2)} \mathrm{d}x = |c| . [?]-[?][?] Th-(x-b)2/(2c2)dx=|c
|. I admit that this may not be so useful for probability theory: the
distribution with probability density function Th-x2/2\Theta^{-x^2/2}Th
-x2/2 has a standard deviation of 1/2p1/\sqrt{2\pi}1/2p instead of
111, so the ugliness has just been pushed elsewhere.
Apart from Thx\Theta^xThx, it's interesting to see whether the reduced
derivative could simplify some equations. I think it makes sense to
try to rewrite differential equations in terms of ddx\frac{\mathrm
{d}}{\mathrm{d} x}dxd whenever we expect some sort of periodic
solution. For example, the equation of a mass mmm attached to a
damped harmonic oscillator is^[V][ ] [This can be obtained by
dividing Newton's second law through by (2p)2m(2\pi)^2 m(2p)2m.]
d2dt2x(t)+2zf0ddtx(t)+f02x(t)=0, \frac{\mathrm{d}^2}{\mathrm{d} t^2}
x(t) + 2 \zeta f_0 \frac{\mathrm{d}}{\mathrm{d} t} x(t) + f_0^2 x(t)
= 0, dt2d2 x(t)+2zf0 dtd x(t)+f02 x(t)=0, where f0=k(2p)2mf_0 = \sqrt
{\frac{k}{(2\pi)^2 m}}f0 =(2p)2mk is the oscillation frequency, z=c
/(2mk)\zeta = c / (2\sqrt{mk})z=c/(2mk ) is the damping ratio, kkk is
the spring constant and ccc is the damping coefficient. This has the
solution x(t)=ATh-zf0tusin[?](1-z2f0t+ph), x(t) = A \Theta^{-\zeta f_0 t}
\operatorname{usin}\left(\sqrt{1 - \zeta^2} f_0 t + \varphi\right), x
(t)=ATh-zf0 tusin(1-z2 f0 t+ph), which is exactly the same as the
regular solution, except that we have linear frequency f0f_0f0
instead of angular frequency o0\omega_0o0 , and the phase offset ph\
varphiph is in units of turns.^[V][ ] [I actually like how the time
constant of the decay 1/(f0z)1/(f_0 \zeta)1/(f0 z) is defined in
terms of the linear frequency f0f_0f0 . With this definition, if the
oscillation has decayed for "one time constant", the amplitude is
Th-1[?]0.2%\Theta^{-1} \approx 0.2\%Th-1[?]0.2% of the original. Normally,
at least I use "three time constants" as some sort of cutoff for "the
oscillation has mostly decayed", because e-3[?]5%e^{-3} \approx 5\%e-3[?]
5%, but that's still pretty far from zero.] Admittedly, 2p2\pi2p
shows up in the definition of f0f_0f0 ,^[V][ ] [But notably not in
that of z\zetaz!] so we haven't completely gotten rid of it.^[V][ ] [
Maybe it could be hidden by defining an "angular spring constant" k~=
k/(2p)2\tilde k = k / (2\pi)^2k~=k/(2p)2 or something. Or, could we
interpret k/m\sqrt{k/m}k/m as the circumference of a circle with
radius f0f_0f0 ? I'll concede that I'm grasping at straws here.]
Similarly, I dare to propose writing the Schrodinger equation like
this: ihddt|Ps> =H^|Ps> . ih \frac{\mathrm{d}}{\mathrm{d} t} \ket{\Psi} =
\hat H\ket{\Psi}. ihdtd |Ps> =H^|Ps> . Note the use of the regular hhh
(which is really the fundmnental constant of nature) instead of \
hbar. With this equation, some basic results of quantum mechanics
can be rewritten without 2p2\pi2p and using just plain hhh, and sure
enough, Thx\Theta^xThx, usin[?](x)\operatorname{usin}(x)usin(x) and ucos[?]
(x)\operatorname{ucos}(x)ucos(x) show up in there. These are a bit
too much to include here, so I'll leave that [DEL:as an exercise to
the reader:DEL] for a separate blog post.
This notation could be abused even further by denoting dx=12pdx\
mathrm{d}x = \frac{1}{2\pi}\mathrm{d}xdx=2p1 dx, which can then
simplify some integral formulae, such as the Gauss-Bonnet theorem:
[?]MKdA+[?][?]Mkgds=2pkh(M)=12p([?]MKdA+[?][?]Mkgds)=kh(M)=[?]MKdA+[?][?]Mkgds=kh(M) \
begin{align*} \int_M K \mathrm{d}A + \int_{\partial M} k_g \mathrm{d}
s &= 2 \pi \chi(M) \\ \Rightarrow \frac{1}{2\pi}\left(\int_M K \
mathrm{d}A + \int_{\partial M} k_g \mathrm{d}s\right) &= \chi(M) \\ \
Rightarrow \int_M K \mathrm{d}A + \int_{\partial M} k_g \mathrm{d}s &
= \chi(M) \end{align*} [?]M KdA+[?][?]M kg ds=2p1 ([?]M KdA+[?][?]M kg ds)=[?]M KdA
+[?][?]M kg ds =2pkh(M)=kh(M)=kh(M) or Cauchy's integral formula: f(a)=
12pi[?]Cf(z)z-adz=1i[?]Cf(z)z-adz=i[?]Cf(z)a-zdz. \begin{align*} f(a) &= \
frac{1}{2\pi i} \oint_C \frac{f(z)}{z-a} \mathrm{d} z \\ &= \frac{1}{
i} \oint_C \frac{f(z)}{z-a} \mathrm{d} z \\ &= i \oint_C \frac{f(z)}
{a-z} \mathrm{d} z . \end{align*} f(a) =2pi1 [?]C z-af(z) dz=i1 [?]C z-af
(z) dz=i[?]C a-zf(z) dz. But this is perhaps stretching it a bit. I'm
a meager experimental physicist who never even took a course on
measure theory, so maybe this is total nonsense.
What is the meaning of all this?
So that's quite a few occurrences of Th\ThetaTh in the wild. I hope
that I've convinced you at least a little bit that treating e2pe^{2\
pi}e2p as a single symbol might make sense. But is there some deeper
meaning to this? Should we declare Th\ThetaTh as the new circle
constant, and throw 2p2\pi2p and the teachings of The Tau Manifesto
in the garbage? Or is this all just sweeping 2p2\pi2p under the rug
to make some formulae superficially prettier?
Physicists use angular frequency quite often, because 2p2\pi2p is
genuinely annoying to deal with. It's a major pain when numerical
simulations don't quite give the expected results, and you have to
hunt for a possibly missing factor of 2p2\pi2p or 1/(2p)1/(2\pi)1/(2p
) or 1/2p1/\sqrt{2\pi}1/2p (shudder). I do believe that using Thix\
Theta^{ix}Thix, usin[?]\operatorname{usin}usin, and ucos[?]\operatorname
{ucos}ucos really presents a new concept that lies somewhere between
angular and linear frequency, and I think it can reduce mistakes
quite a lot. It could be argued that remembering to put the bar on d\
mathrm{d}d is as much mental effort as remembering to keep 2p2\pi2p
around, but I say this is not the case: differentials appear less
often than e2pe^{2\pi}e2p, and they are more of a special object, so
they are handled with more care by default.^[V][ ] [Except when
physicists treat dydx\frac{\mathrm{d}y}{\mathrm{d}x}dxdy as a
fraction.] Overall, I've had some moderate success with simplifying
code by using turns instead of radians or degrees.
I'm not entirely sure about the intuitive meaning of "taking the
derivative and dividing by 2p2\pi2p". Is there some sort of
fundamental connection to periodic functions? Maybe it helps with
some dimensional analysis of functions on a circle, if we give angles
units like in this paper. But I have yet to do any serious math using
the reduced derivative, so this needs some more fleshing out.
Originally, I only thought about Th\ThetaTh, usin[?]\operatorname{usin}
usin, and ucos[?]\operatorname{ucos}ucos, and the reduced derivative
came up as a post-hoc argument against sin[?]\sinsin and cos[?]\coscos
being fundamental functions that satisfy ddxf(x)=g(x)\frac{\mathrm
{d}}{\mathrm{d} x} f(x) = g(x)dxd f(x)=g(x) and ddxg(x)=-f(x)\frac{\
mathrm{d}}{\mathrm{d} x} g(x) = -f(x)dxd g(x)=-f(x).
One thing I can say for sure is that this trick doesn't eliminate 2p2
\pi2p from everywhere for all eternity. You could express the
circumference of a circle as ln[?](Th)r\ln(\Theta) rln(Th)r, but I don't
expect this to catch on. Euler's formula is also a fairly advanced
concept, so the right place for Thix\Theta^{i x}Thix might be an
undergraduate physics or complex analysis course rather than in every
textbook. But I'm pretty sure that working with just fractions of a
turn instead of degrees makes trigonometry easier to grasp even at
the high-school level, as The Tau Manifesto argues.
In general, I think that even though this kind of fiddling with
notation may seem superficial, it's actually important. Not having to
worry about 2p2\pi2p reduces friction, so that you can focus on the
problem you are trying to solve. If you reduce friction enough, even
by small steps like this one, some untractable problems can become
tractable.
So, even if it's a bit unclear if there really is something to this
idea, for now I plan to try and see if Thx\Theta^xThx makes my life
easier. I encourage you to do the same. If you find something
interesting while doing so, feel free to drop me an email!
---------------------------------------------------------------------
Some additional notes
Who came up with this?
I could not really find any references online or in the literature
discussing a concept like the reduced derivative I've introduced
here. Searching for "derivative divided by 2pi" brings up mostly
high-school level homework help of course, but that's basically all I
could find with my search-engine-fu. As far as I can tell, e2pe^{2\
pi}e2p as a single, explicit constant worth considering has not been
discussed anywhere either. The closest I could find was Gelfond's
constant, which is ep=The^{\pi} = \sqrt{\Theta}ep=Th . For example,
OEIS describes e2pe^{2\pi}e2p as just "Gelfond's constant squared".
If you know of any reference where this is discussed before, please
let me know! I'd hate to take undeserved credit.
Naming and notation
I'm not sure what "the derivative divided by 2p2\pi2p" should really
be called. I picked "reduced derivative", like how the Planck
constant divided by 2p2\pi2p is called the reduced Planck constant.
This is also the inspiration for the notation d\mathrm{d}d (\text{\
dj} in LaTeX\LaTeXLATE X), which I propose to be pronounced as
"d-bar".^[V][ ] [I put the bar only on the d\mathrm{d}d in the
numerator of ddx\frac{\mathrm{d}}{\mathrm{d}x}dxd , so that the
division works out if you think of it as "d=d/(2p)\mathrm{d} = \
mathrm{d} / (2\pi)d=d/(2p)". Of course someone who is more versed in
differentials (or is a mathematician or something) may object to
this.] Of course, "reduced derivative" already means something, so
maybe it's not the best possible name.
Other options I came up with are "angular derivative" or "rotational
derivative", but I'm not so happy with these either. Angular
derivative in particular seems already to be a thing in complex
analysis, see for example here, here [PDF], or here. Suggestions are
welcome!
I chose Th\ThetaTh as the symbol for e2pe^{2\pi}e2p, because it doesn't
seem to be that widely used in math and physics (I checked here, here
, and here), it's kind of like a circle, which hints at rotations and
angles, and Thx\Theta^xThx also kind of resembles exe^xex, which is a
reminder of its definition. Apart from these extremely compelling
aesthetic arguments, I should note that Th\ThetaTh might be easy to
confuse with the lowercase th\thetath or eee in handwriting, so I'm
also open to suggestions for the symbol.
Product rule, chain rule
How do the basic rules of differentiation work with the reduced
derivative? In short: they work just fine. Just by using the
definition, the product rule is ddx(f(x)g(x))=12pddx(f(x)g(x))=12p
( [?] [?](ddxf(x))g(x)+f(x)(ddxg(x)) [?] [?])=(ddxf(x))g(x)+f(x)(ddxg(x)), \
begin{align*} \frac{\mathrm{d}}{\mathrm{d} x} \left(f(x) g(x)\right)
&= \frac{1}{2\pi} \frac{\mathrm{d}}{\mathrm{d} x} \left(f(x) g(x)\
right) \\ &= \frac{1}{2\pi} \left(\!\! \left(\frac{\mathrm{d}}{\
mathrm{d} x} f(x)\right) g(x) + f(x) \left(\frac{\mathrm{d}}{\mathrm
{d} x} g(x)\right) \!\!\right) \\ &= \left(\frac{\mathrm{d}}{\mathrm
{d} x} f(x)\right) g(x) + f(x) \left(\frac{\mathrm{d}}{\mathrm{d} x}
g(x)\right) , \end{align*} dxd (f(x)g(x)) =2p1 dxd (f(x)g(x))=2p1 ((d
xd f(x))g(x)+f(x)(dxd g(x)))=(dxd f(x))g(x)+f(x)(dxd g(x)), and the
chain rule is ddxf(g(x))=12pddxf(g(x))=12p ddyf(y)|y=g(x)[?]ddxg(x)=
ddyf(y)|y=g(x)[?]ddxg(x)=ddyf(y)|y=g(x)[?]ddxg(x), \begin{align*} \frac{\
mathrm{d}}{\mathrm{d} x} f(g(x)) &= \frac{1}{2\pi} \frac{\mathrm{d}}
{\mathrm{d} x} f(g(x)) \\ &= \frac{1}{2\pi} \, \frac{\mathrm{d}}{\
mathrm{d} y} f(y) \Big|_{y = g(x)} \cdot \frac{\mathrm{d}}{\mathrm{d}
x} g(x) \\ &= \frac{\mathrm{d}}{\mathrm{d} y} f(y) \Big|_{y = g(x)} \
cdot \frac{\mathrm{d}}{\mathrm{d} x} g(x) \\ &= \frac{\mathrm{d}}{\
mathrm{d} y} f(y) \Big|_{y = g(x)} \cdot \frac{\mathrm{d}}{\mathrm{d}
x} g(x) , \end{align*} dxd f(g(x)) =2p1 dxd f(g(x))=2p1 dyd f(y)|| y=
g(x) [?]dxd g(x)=dyd f(y)|| y=g(x) [?]dxd g(x)=dyd f(y)|| y=g(x) [?]dxd g(x
), where the last two rows are to show that you are free to choose
which derivative you put the bar on. I suppose the desirable choice
will depend on the specific functions f(x)f(x)f(x) and g(x)g(x)g(x).
Note that the product rule doesn't produce very pretty results if one
of the functions is not defined in terms of Thx\Theta^xThx, for example
the reduced derivative of x[?]usin[?](x)x \cdot \operatorname{usin}(x)x[?]
usin(x) (or regular sin[?](x)\sin(x)sin(x) for that matter) has an
extra factor of 1/2p1/2\pi1/2p in the ddxx\frac{\mathrm{d}}{\mathrm
{d} x} xdxd x term.
Why not also include the iii?
In most of the examples I showed, Thx\Theta^xThx actually appears in
the form of Thix\Theta^{ix}Thix. So why not define Th[?]e2pi\Theta \equiv
e^{2\pi i}Th[?]e2pi and get even cleaner formulae? The problem is of
course that e2pi=1e^{2\pi i} = 1e2pi=1, so (e2pi)x=1x=1(e^{2\pi i})^x
= 1^x = 1(e2pi)x=1x=1 for any xxx. But why does Thix\Theta^{i x}Thix
work then? This was a little bit surprising to me initially, but the
answer is that in general, (ez)w[?]ezw(e^z)^w \neq e^{z w}(ez)w=ezw
for complex zzz and www. The correct formula is instead (ez)w=ewln[?]
(ez)(e^{z})^w = e^{w \ln(e^z)}(ez)w=ewln(ez)^[V][ ] [See for example
here.], so that Thix=(e2p)ix=eixln[?](e2p)=e2pix,\Theta^{i x} = (e^{2\
pi})^{ix} = e^{ix \ln(e^{2\pi})} = e^{2\pi ix},Thix=(e2p)ix=eixln(e2p)
=e2pix, but (e2pi)x=exln[?](e2pi),(e^{2\pi i})^{x} = e^{x \ln\left(e^{2
\pi i}\right)},(e2pi)x=exln(e2pi), where ln[?](e2pi)=ln[?]1=0\ln\left(e^
{2\pi i}\right) = \ln 1 = 0ln(e2pi)=ln1=0, so we get ex[?]0=1e^{x \cdot
0} = 1ex[?]0=1. Note also that in general ln[?](z)\ln(z)ln(z) is
multi-valued for complex numbers.
Further reading
As I mentioned, there doesn't seem to be much discussion on treating
e2pe^{2\pi}e2p as a single symbol, so these are mostly about p\pip vs
t\taut.
* The Tau Manifesto by Michael Hartl of course.
* Al-Kashi's Constant [PDF] by Peter Harremoes lists lots of
formulae where 2p2\pi2p shows up.
* Trig Rerigged [PDF] by Thomas Colignatus discusses the functions
I called usin[?]\operatorname{usin}usin and ucos[?]\operatorname
{ucos}ucos.
* This post on the Wolfram blog by Georgia Fortuna does some
interesting quantitative analysis of whether using t\taut instead
of p\pip would make formulae simpler. The conclusion is that in
many cases it would not. While I'm not entirely convinced of the
methodology (they look at formulae found in math papers on arXiv
and determine their complexity by looking at the size of
expression trees produced by the Wolfram Language, which in my
opinion doesn't consider enough of the surrounding context), an
interesting point is bought up at the end: Treating 2pi2\pi i2pi
as a single constant instead of just 2p2\pi2p would simplify
formulae a lot more than changing from p\pip to t\taut. However,
I suspect what is actually happening is that formulae are
simplified because they involve e2pie^{2\pi i}e2pi!^[V][ ] [This
seems to be the case at least based on the word cloud of the
simplified formulae they show, although there are a couple of
examples where 2pi2\pi i2pi appears outside of an exponential.]
This would be another argument in favor of using Thix\Theta^{ix}Thi
x.
Thanks to Jami Kinnunen, Elsa Mannila, Viljami Pirttimaa, Aashish
Sah, Lauri Seppalainen, and Vasilii Vadimov for comments and
feedback.