arctans

Arctans.

Submodules

Attributes

j

one

zero

__version__

Classes

Integer

An integer.

Rational

A rational number.

GaussianInteger

A Gaussian integer.

GaussianRational

A Gaussian rational.

Functions

arccotan(→ AbstractTerm)

Symbolic arccotangent.

arctan(→ AbstractTerm)

Symbolic arctangent.

is_irreducible(→ bool)

Check if arctan(n) is irreducible.

reduce(→ arctans.arctans.AbstractTerm)

Express an arctan as a sum of irreducible integral arccotangents.

convert_rational(→ arctans.arctans.AbstractTerm)

Convert a rational arccotangent into a sum of integral arccotangents.

generate(→ list[arctans.arctans.AbstractTerm])

Generate new formulae.

Package Contents

arctans.arccotan(a: arctans.numbers.AbstractNumber | int) → AbstractTerm

Symbolic arccotangent.

Parameters:

a – The argument of the arccotan

Returns:

arccotan(a)

arctans.arctan(a: arctans.numbers.AbstractNumber | int) → AbstractTerm

Symbolic arctangent.

Parameters:

a – The argument of the arctan

Returns:

arctan(a)

class arctans.Integer(i: int)

Bases: RealNumber

An integer.

_i
__str__()
__repr__()
as_latex() → str

Represent in LaTeX.

property numerator: AbstractNumber

Numerator.

property denominator: Integer

Denominator.

__int__() → int
__float__() → float
__complex__() → complex
_to_same_type(other: Any) → Integer

Convert other to the same type as self.

_add(other: Self) → AbstractNumber

Add something of the same type to this.

_sub(other: Self) → AbstractNumber

Subtract something of the same type from this.

_mul(other: Self) → AbstractNumber

Multiply something of the same type by this.

_truediv(other: Self) → AbstractNumber

Divide this by something of the same type.

_pow(other: int) → AbstractNumber

Raise to an integer power.

_mod(other: Self) → AbstractNumber

Find the remainder when dividing this by something of the same type.

_floordiv(other: Self) → AbstractNumber

Find the remainder when dividing this by something of the same type.

_eq(other: Self) → bool

Check if something of the same type is equal to this.

__abs__()
__hash__()
class arctans.Rational(numerator: int, denominator: int)

Bases: RealNumber

A rational number.

_num
_den
as_latex() → str

Represent in LaTeX.

__str__()
__repr__()
property numerator: AbstractNumber

Numerator.

property denominator: Integer

Denominator.

__int__() → int
__float__() → float
__complex__() → complex
_to_same_type(other: Any) → Rational

Convert other to the same type as self.

_add(other: Self) → AbstractNumber

Add something of the same type to this.

_sub(other: Self) → AbstractNumber

Subtract something of the same type from this.

_mul(other: Self) → AbstractNumber

Multiply something of the same type by this.

_truediv(other: Self) → AbstractNumber

Divide this by something of the same type.

_pow(other: int) → AbstractNumber

Raise to an integer power.

_eq(other: Self) → bool

Check if something of the same type is equal to this.

__abs__()
class arctans.GaussianInteger(re: int, im: int)

Bases: AbstractNumber

A Gaussian integer.

_re
_im
__str__()
__repr__()
property real: RealNumber

Real part.

property imag: RealNumber

Imaginary part.

conjugate() → AbstractNumber

Compute the complex conjugate.

as_latex() → str

Represent in LaTeX.

property numerator: AbstractNumber

Numerator.

property denominator: Integer

Denominator.

__int__() → int
__float__() → float
__complex__() → complex
_to_same_type(other: Any) → GaussianInteger

Convert other to the same type as self.

_add(other: Self) → AbstractNumber

Add something of the same type to this.

_sub(other: Self) → AbstractNumber

Subtract something of the same type from this.

_mul(other: Self) → AbstractNumber

Multiply something of the same type by this.

_truediv(other: Self) → AbstractNumber

Divide this by something of the same type.

_mod(other: Self) → AbstractNumber

Find the remainder when dividing this by something of the same type.

_floordiv(other: Self) → AbstractNumber

Find the remainder when dividing this by something of the same type.

_eq(other: Self) → bool

Check if something of the same type is equal to this.

class arctans.GaussianRational(re_numerator: int, re_denominator, im_numerator: int, im_denominator: int)

Bases: AbstractNumber

A Gaussian rational.

_re_num
_re_den
_im_num
_im_den
__str__()
__repr__()
property real: RealNumber

Real part.

property imag: RealNumber

Imaginary part.

as_latex() → str

Represent in LaTeX.

conjugate() → AbstractNumber

Compute the complex conjugate.

property numerator: AbstractNumber

Numerator.

property denominator: Integer

Denominator.

__int__() → int
__float__() → float
__complex__() → complex
_to_same_type(other: Any) → GaussianRational

Convert other to the same type as self.

_add(other: Self) → AbstractNumber

Add something of the same type to this.

_sub(other: Self) → AbstractNumber

Subtract something of the same type from this.

_mul(other: Self) → AbstractNumber

Multiply something of the same type by this.

_truediv(other: Self) → AbstractNumber

Divide this by something of the same type.

_eq(other: Self) → bool

Check if something of the same type is equal to this.

arctans.j
arctans.one
arctans.zero
arctans.is_irreducible(n: arctans.numbers.Integer | int) → bool

Check if arctan(n) is irreducible.

An arctan is irreducible iff it cannot be written as a weighted sum of integer arccotangents, or equivalently arctan(n) is irreducible iff the largest prime factor of 1 + n**2 is greater than or equal to 2*n.

Parameters:

n – An integer

Returns:

True if n is irreducible

arctans.reduce(a: arctans.arctans.AbstractTerm) → arctans.arctans.AbstractTerm

Express an arctan as a sum of irreducible integral arccotangents.

Parameters:

a – An arctan or sum of arctans

Returns:

A sum of irreducible integral arccotangents

arctans.convert_rational(a: arctans.arctans.AbstractTerm) → arctans.arctans.AbstractTerm

Convert a rational arccotangent into a sum of integral arccotangents.

Parameters:

a – An arctan or sum of arctans

Returns:

A sum of integral arccotangents

arctans.generate(known_formula: arctans.arctans.AbstractTerm | Sequence[arctans.arctans.AbstractTerm], *, min_denominator: int = 1, max_denominator: int = 100, min_numerator: int = 1, max_numerator: int = 1, max_terms: int | None = None, max_coefficient_denominator: int | None = None, printing: bool = False) → list[arctans.arctans.AbstractTerm]

Generate new formulae.

Parameters:

  • known_formula – Known formula or formulae that all have the same value

  • min_numerator – The minimum numerator to use for arctan arguments

  • max_numerator – The maximum numerator to use for arctan arguments

  • min_denominator – The minimum denominator to use for arctan arguments

  • max_denominator – The maximum denominator to use for arctan arguments

  • max_terms – The maximum number of arctan terms to include in the new formulae

  • max_coefficient_denominator – The maximum allowbale denominator to use in the coefficients in the new formulae

  • printing – Print information about progress

Returns:

A list of new formulae that have the same value as the known formula(e)

arctans.__version__ = '1.0.0'