transcendental-range 0.9.0

Generate ranges of transcendental numbers from algebraic generators and transcendental functions.

transcendental-range

Generate ranges of transcendental numbers.

A Python port of the Wolfram Language resource function TranscendentalRange (version 1.1.0), returning exact sympy expressions.

pip install transcendental-range

from transcendental_range import transcendental_range

transcendental_range(10)
# [E, 2*E, exp(2), 3*E]

Overview

transcendental_range creates ranges of transcendental numbers — numbers that provably cannot be the root of any polynomial equation with integer coefficients. Whole families of such numbers follow from classical theorems (Lindemann–Weierstrass, Gelfond–Schneider, Baker), and this function generates them systematically over 27 methods covering exponential, logarithmic, trigonometric, hyperbolic and inverse forms, plus algebraic powers:

>>> from transcendental_range import transcendental_range
>>> transcendental_range(10)
[E, 2*E, exp(2), 3*E]

>>> transcendental_range(-2, 2)
[-2*exp(-1), -exp(-1), -2*exp(-2), -exp(-2), exp(-2), 2*exp(-2), exp(-1), 2*exp(-1)]

>>> from sympy import Rational
>>> transcendental_range(0, 4, Rational(1, 2))          # with step
[exp(1/2)/2, E/2, exp(1/2), exp(3/2)/2, 3*exp(1/2)/2, E, 2*exp(1/2), exp(2)/2]

>>> transcendental_range(100, 1, -1)                    # descending
[3*exp(3), ..., 2*exp(2), ..., E]

Form	Description
transcendental_range(x)	transcendentals t = b f(a) with 1 ≤ t ≤ x, a and b in range(1, x+1)
transcendental_range(x, y)	with x ≤ t ≤ y, a and b in range(x, y+1)
transcendental_range(x, y, s)	a and b in range(x, y+1, s) (negative s for descending)
transcendental_range(x, y, s, d)	minimum difference d between successive elements

Every element is an exact sympy expression, sorted by numerical value; numerically coincident alternatives are reduced to the representative with the smallest function argument. The implementation uses a monotonicity-aware scan that is far more efficient than the naive outer product (see Performance).

Options

Option	Default	Description
method	'exp'	transcendental function generating the numbers
multiplicity	1	number of terms to combine linearly
generators_domain	'rationals'	domain of the algebraic generators
farey_range	False	step denominators as in the Farey sequence
formula_complexity_threshold	inf	limit the complexity of the expressions
working_precision	15	precision of all internal numerical evaluations

method

Method	Generated forms
'exp'	exponential forms b ea
'log'	logarithmic forms b log(a)
'power'	power forms ab, for irrational b only
'sin', 'cos', 'tan', …	trigonometric forms b sin(a), b cos(a), …
'asin', 'acos', 'atan', …	inverse trigonometric forms
'sinh', 'cosh', 'tanh', …	hyperbolic forms
'asinh', 'acosh', 'atanh', …	inverse hyperbolic forms
a list of the above	combined range
'all'	all of the above types

transcendental_range(-10, 10, method='sinh')
# [-8*sinh(1), -7*sinh(1), -2*sinh(2), -6*sinh(1), ..., 2*sinh(2), 7*sinh(1), 8*sinh(1)]

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

Different types can be freely combined:

transcendental_range(-4, 4, method=['acot', 'exp'])
# [-pi, -E, -3*pi/4, -4*acot(2), -pi/2, -4*exp(-1), ..., 4*acot(2), 3*pi/4, E, pi]

transcendental_range(3, method='all')
# [coth(3), 3*acsc(3), coth(2), 3*acoth(3), pi/3, 2*cos(1), log(3), csc(2), ...]

The method 'power' produces nontrivial results together with generators_domain='algebraics' (only irrational exponents generate transcendental powers, by Gelfond–Schneider):

transcendental_range(-3, 3, method='power', generators_domain='algebraics')
# [3**(-2*sqrt(2)), 2**(-3*sqrt(2)), 3**(-sqrt(7)), 7**(-sqrt(2)), ...]

multiplicity

Linear combinations of the generated numbers are still transcendental:

from sympy import Rational

transcendental_range(0, 6, Rational(1, 2), multiplicity=2)
# [exp(1/2), exp(1/2)/2 + E/2, 3*exp(1/2)/2, E, E/2 + exp(1/2), ...]

For the method 'power' the combinations are products; different methods of a list combine only within themselves.

generators_domain

With 'algebraics', the generator arguments and coefficients extend from the rationals of range(x, y, s) to the real algebraic numbers of algebraic_range:

transcendental_range(8, generators_domain='algebraics')
# [E, sqrt(2)*E, exp(sqrt(2)), sqrt(3)*E, 2*E, exp(sqrt(3)), ...]

The rational range is always a subset of the algebraic one.

farey_range

Step denominators can follow the Farey sequence, as generated by the farey package; the resulting range is a strict superset of the union of the plain ranges with the corresponding steps:

transcendental_range(1, 10, Rational(1, 3), farey_range=True)
# [E, 4*E/3, exp(4/3), 3*E/2, exp(3/2), 5*E/3, 4*exp(4/3)/3, ...]

A Farey step must be an integer ≥ 1 or of the form 1/n (or −1/n for a descending range): anything else raises FareyStepError.

formula_complexity_threshold

The output can be restricted to expressions below a heuristic complexity score (from algebraic_range.formula_complexity):

transcendental_range(0, 8, Rational(1, 2), generators_domain='algebraics',
                     formula_complexity_threshold=5)
# 14 simple elements, against 382 with the default infinite threshold

working_precision

Symbolically distinct elements may numerically collide at machine precision, in which case only one representative is kept:

transcendental_range(20, 25, method='tanh')
# [21*tanh(20), 22*tanh(20), 23*tanh(20), 24*tanh(20), 25*tanh(20)]

Raising the precision resolves the collisions:

transcendental_range(20, 25, method='tanh', working_precision=30)
# [21*tanh(20), 21*tanh(21), ..., 25*tanh(24), 25*tanh(25)]  (30 elements)

Properties and relations

transcendental_range(x, y, s) equals the outer product of b e^a over the generator range, restricted to the bounds, cleared of algebraic values, deduplicated and sorted — but the actual monotonicity-aware implementation is far more efficient than the outer product over large exponentially-growing ranges, and every method is verified against that naive baseline in the test suite.

Performance

The Python port reproduces the WL output exactly (80 604 elements match one for one) within 2–3× of the native WL timings. See benchmark/BENCHMARK.md for per-method timings and methodology.

Possible issues

By definition, the range arguments must be algebraic numbers:

from sympy import E
transcendental_range(0, E, Rational(1, 3))
# NotAlgebraicError: the range arguments provided are not all algebraic numbers

transcendental_range(0, 3, Rational(1, 3))   # ceiling(E) = 3
# [exp(1/3)/3, exp(2/3)/3, E/3, 2*exp(1/3)/3, ...]

Applications

transcendental_range was designed especially as a search space for find-closed-form, for exhaustively searching closed forms of raw numeric values in terms of arbitrary mathematical functions with transcendental arguments. Since find-closed-form 0.5.0 (pip install "find-closed-form[ranges]") this works out of the box through the search_range option, with search_range_options forwarded to the generator.

Formulae like ζ(√e) — a higher mathematical function wrapped around a transcendental number — are out of reach of the default rational search, but are recovered from their bare machine digits over this range:

from find_closed_form import find_closed_form

find_closed_form(2.1638308208408383, search_range="Transcendental")
# zeta(exp(1/2))

The range elements themselves are candidate closed forms, matched up to algebraic factors and addends — here recognizing the Gelfond–Schneider constant among the 'power' elements over algebraic generators:

find_closed_form(2.665144142690225, search_range="Transcendental",
                 search_range_options={"method": "power",
                                       "generators_domain": "algebraics"})
# 2**sqrt(2)

An explicit transcendental_range(...) output can also be passed directly as fixed search_arguments.

References

1. Alan Baker, Transcendental Number Theory, Cambridge University Press, 1975.

See also

For the full documentation see the Wolfram Language resource function this package ports, and CHANGELOG.md for version history.

Dependencies

• sympy ≥ 1.12 (and mpmath ≥ 1.3)
• algebraic-range ≥ 0.9.0
• farey ≥ 0.7.0

Author

Daniele Gregori

License

MIT