arctans¶
arctans is a library for manipulating arctans to generate Machin-like formulae and other formulae involving arctans.
Installing arctans¶
The latest release of arctans can be installed by running:
pip3 install arctans
Alternatively, you can install the latest version directly from GitHub with:
pip3 install git+https://github.com/mscroggs/arctans.git
Using arctans¶
arctans can be used to represent arctans and sums of arctans symbolically and generate arctan sums equivalent to a given set of sums.
For example, Machin’s formula for pi (\(\pi = 16\arctan(1/5) - 4\arctan(1/239)\)) can be created with:
from arctans import arctan, Rational
pi = 16 * arctan(Rational(1, 5)) - 4 * arctan(Rational(1, 239))
print(pi)
print(float(pi))
Or equivalently:
from arctans import arccotan
pi = 16 * arccotan(5) - 4 * arccotan(239)
print(pi)
print(float(pi))
As arccotangents of integers commonly appear in formulae for pi, when printing formulae represented using arctans, the shorthand notation [n] will be used to represent arccotan(n).
Once a formulae is expressed, new formulae that have the same value can be generated using the generate function, for example:
from arctans import arccotan, generate
pi = 16 * arccotan(5) - 4 * arccotan(239)
formulae = generate([pi])
for f in formulae:
print(f)
This will print a number of different arctan sum formulae, including 16*[4] + -16*[21] + -4*[239] (ie \(\pi = 16\arctan(1/4) - 16\arctan(1/21) - 4\arctan(1/239)\)).