transcendental-range 0.8.1

Generate transcendental numbers within a range using algebraic generators and transcendental functions.

transcendental-range

Generate ranges of transcendental numbers.

Python port of the Wolfram Language resource function TranscendentalRange.

Usage

from transcendental_range import transcendental_range

transcendental_range(x) gives all transcendental numbers of the form t = b e^a with 1 <= t <= x and a, b algebraics in range(1, x).

transcendental_range(x, y) gives all transcendental numbers of the form t = b e^a with x <= t <= y and a, b algebraics in range(x, y).

transcendental_range(x, y, s) uses a step parameter s for the generator range.

transcendental_range(x, y, s, d) requires a minimum step d between successive elements.

Details

• transcendental_range can systematically generate various types of transcendental numbers.

• Transcendental numbers are irrational numbers that cannot be expressed as solutions of any polynomial equation with integer coefficients (that is, they are not algebraic numbers).

• By default, the range elements are generated according to the Lindemann-Weierstrass theorem, that is linear combinations of exponentials t = b e^a, for all algebraic arguments and coefficients a, b that are members of range(x, y), or possibly range(x, y, s) using a step parameter s.

• For all the available transcendental types, the generated exact numbers have been mathematically proven to be transcendental through the theorems of Lindemann-Weierstrass, Gelfond-Schneider and Baker (cf. Baker, 1975 - Transcendental Number Theory).

• The elements are returned as exact sympy expressions, sorted by numerical value.

Options

transcendental_range accepts the following keyword arguments:

Option	Default	Description
method	'exp'	Function for generating transcendental numbers
generators_domain	'rationals'	Type of algebraic generators for the function arguments
farey_range	False	Set the step denominators as in the Farey sequence
formula_complexity_threshold	math.inf	Limit the complexity of the expressions
working_precision	15	Precision for all internal numerical evaluations

method

The method option specifies the transcendental function used to generate numbers. Available values:

Method	Form
'exp'	b exp(a)
'log'	b log(a)
'power'	a^b
'sin', 'cos', 'tan', 'cot', 'sec', 'csc'	b f(a)
'sinh', 'cosh', 'tanh', 'coth', 'sech', 'csch'	b f(a)
'asin', 'acos', 'atan', 'acot', 'asec', 'acsc'	b f(a)
'asinh', 'acosh', 'atanh', 'acoth', 'asech', 'acsch'	b f(a)
list of the above	combined range
'all'	all of the above types

For the method 'power', only algebraic irrational generators in the exponent b can produce transcendental numbers (Gelfond-Schneider theorem).

generators_domain

The generators_domain option specifies whether the arguments a, b should belong to the rationals ('rationals') or to the algebraics ('algebraics'), as generated by range and algebraic_range respectively. These are restricted to be real numbers.

formula_complexity_threshold

The output can be restricted by setting a threshold for the complexity of the numeric expressions involved. The corresponding numerical values are assigned through a heuristic recipe provided by formula_complexity from the algebraic-range package.

Examples

from transcendental_range import transcendental_range

transcendental_range(5)
# [exp(1), 2*exp(1)/3, exp(2)/3, 3*exp(1)/4, exp(1)/2, ...]

Generate transcendental numbers using the logarithm:

transcendental_range(-2, 2, method='log')
# [-2*log(2), -log(2), log(2), 2*log(2)]

Use the 'power' method with algebraic generators to produce numbers of the form a^b with irrational exponents (Gelfond-Schneider theorem):

from sympy import Rational

transcendental_range(-1, Rational(3, 2), Rational(1, 2),
                     method='power', generators_domain='algebraics')
# [(1/2)**sqrt(2), (1/2)**(sqrt(2)/2), (sqrt(2)/2)**(sqrt(2)/2),
#  2**(sqrt(2)/4), (3/2)**(sqrt(2)/2)]

Generate transcendental numbers using the inverse tangent (multiples of pi appear naturally):

transcendental_range(-3, 3, method='atan')
# [-2*atan(3), -3*pi/4, -2*atan(2), -pi/2, -atan(3), -atan(2),
#  -pi/4, pi/4, atan(2), atan(3), pi/2, 2*atan(2), 3*pi/4, 2*atan(3)]

Use step, minimum separation and formula complexity threshold together:

transcendental_range(0, 20, Rational(1, 4), 2, formula_complexity_threshold=6)
# [exp(1/4)/4, 3*exp(1/2)/2, exp(3/2), 4*exp(1/2), 13*E/4,
#  4*E, 19*E/4, 11*E/2, 25*E/4, 7*E]

For more examples and details, see the Wolfram Language documentation.

Dependencies

• sympy >= 1.12
• algebraic-range >= 0.8.0

Author

Daniele Gregori

License

MIT