That one you found out already, it would make it much more consistent with how similar constants are used.
The gravitational constant looks like off by a factor of 4π
Not sure what you mean. Do you mean when comparing the equation for gravitational force to the electric force? Or do you mean when looking at the ‘intuitive’ way of writing the differential equation
nablag=rho?
In either case it seems that the choice of 4π is arbitrary on one equation or the other. For example choosing Gaussian units introduces a 4π in the electrical equation and makes it look more like the gravitational equation.
cosine seems more primitive than sine
They seem equally primitive by
sin2xcos2x=1
and
%20=%20cos(x%20-%20\pi/2))
The Riemann Zeta function ζ(s) negates s for reasons beyond me
I agree about gamma, cosine, and pi. I’m not troubled by the minus sign in the zeta function but suspect we should really be working with the related “xi function” whose symmetries are simpler. I’m not a very expert physicist but my guess is that the 4pi there is going to pop in one place or another and it doesn’t matter very much which you choose.
The only one of these that I actually get cross about is the gamma function. With all the others, there are tradeoffs—e.g., if you work with tau = 2pi instead of with pi, some things become simpler, some become more complicated, and on balance it’s probably a slight improvement. If you work with the factorial function instead of the gamma function, I think pretty much every formula I’ve ever seen that uses it becomes simpler (usually by the omission of an annoying “-1” term).
(But I’m not an analytic number theorist or a complex analyst—though I was kinda-sorta a bit of a complex analyst once—and it’s possible that the cognoscenti know of good reasons why gamma should stay the way it is.)
I would say cos is simpler than sin because its Taylor series has a factor of x knocked off.
In practice they tend to show up together, though. Often you can replace the pair with something like e^(i x), so maybe that should be considered the simplest.
That one you found out already, it would make it much more consistent with how similar constants are used.
Not sure what you mean. Do you mean when comparing the equation for gravitational force to the electric force? Or do you mean when looking at the ‘intuitive’ way of writing the differential equation
nablag=rho?
In either case it seems that the choice of 4π is arbitrary on one equation or the other. For example choosing Gaussian units introduces a 4π in the electrical equation and makes it look more like the gravitational equation.
They seem equally primitive by
sin2x cos2x=1
and
It doesn’t according to Wikipedia
I haven’t read up on that so I don’t really know. Seems arbitrary to me too.
I agree about gamma, cosine, and pi. I’m not troubled by the minus sign in the zeta function but suspect we should really be working with the related “xi function” whose symmetries are simpler. I’m not a very expert physicist but my guess is that the 4pi there is going to pop in one place or another and it doesn’t matter very much which you choose.
The only one of these that I actually get cross about is the gamma function. With all the others, there are tradeoffs—e.g., if you work with tau = 2pi instead of with pi, some things become simpler, some become more complicated, and on balance it’s probably a slight improvement. If you work with the factorial function instead of the gamma function, I think pretty much every formula I’ve ever seen that uses it becomes simpler (usually by the omission of an annoying “-1” term).
(But I’m not an analytic number theorist or a complex analyst—though I was kinda-sorta a bit of a complex analyst once—and it’s possible that the cognoscenti know of good reasons why gamma should stay the way it is.)
I would say
cosis simpler thansinbecause its Taylor series has a factor of x knocked off.In practice they tend to show up together, though. Often you can replace the pair with something like
e^(i x), so maybe that should be considered the simplest.