pg_utils.pg_model.expansion.InnerProduct1D

class pg_utils.pg_model.expansion.InnerProduct1D(*args)[source]

Bases: Expr

1-D (integration) inner product \(\langle a, b \rangle\)

Variables:

_opd_A (sympy.Expr) – left operand
_opd_B (sympy.Expr) – right operand
_wt (sympy.Expr) – weight
_int_var (sympy.Symbol) – integration variable
_bound (List[sympy.Expr]) – integration bounds

__init__(opd_A: Expr, opd_B: Expr, wt: Expr, int_var: Symbol, lower: Expr, upper: Expr) → None[source]

Initialization

Parameters:

opd_A (sympy.Expr) – left operand
opd_B (sympy.Expr) – right operand
wt (sympy.Expr) – weight
int_var (sympy.Symbol) – integration variable
lower (sympy.Expr) – lower bound of integration
upper (sympy.Expr) – upper bound of integration

Methods

`__init__`(opd_A, opd_B, wt, int_var, lower, upper)	Initialization
`adjoint`()
`apart`([x])	See the apart function in sympy.polys
`args_cnc`([cset, warn, split_1])	Return [commutative factors, non-commutative factors] of self.
`as_base_exp`()
`as_coeff_Add`([rational])	Efficiently extract the coefficient of a summation.
`as_coeff_Mul`([rational])	Efficiently extract the coefficient of a product.
`as_coeff_add`(*deps)	Return the tuple (c, args) where self is written as an Add, `a`.
`as_coeff_exponent`(x)	`cx*e -> c,e` where x can be any symbolic expression.
`as_coeff_mul`(deps, *kwargs)	Return the tuple (c, args) where self is written as a Mul, `m`.
`as_coefficient`(expr)	Extracts symbolic coefficient at the given expression.
`as_coefficients_dict`(*syms)	Return a dictionary mapping terms to their Rational coefficient.
`as_content_primitive`([radical, clear])	This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive).
`as_dummy`()	Return the expression with any objects having structurally bound symbols replaced with unique, canonical symbols within the object in which they appear and having only the default assumption for commutativity being True.
`as_expr`(*gens)	Convert a polynomial to a SymPy expression.
`as_independent`(deps, *hint)	A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps.
`as_leading_term`(*symbols[, logx, cdir])	Returns the leading (nonzero) term of the series expansion of self.
`as_numer_denom`()	expression -> a/b -> a, b
`as_ordered_factors`([order])	Return list of ordered factors (if Mul) else [self].
`as_ordered_terms`([order, data])	Transform an expression to an ordered list of terms.
`as_poly`(gens, *args)	Converts `self` to a polynomial or returns `None`.
`as_powers_dict`()	Return self as a dictionary of factors with each factor being treated as a power.
`as_real_imag`([deep])	Performs complex expansion on 'self' and returns a tuple containing collected both real and imaginary parts.
`as_terms`()	Transform an expression to a list of terms.
`aseries`([x, n, bound, hir])	Asymptotic Series expansion of self.
`atoms`(*types)	Returns the atoms that form the current object.
`cancel`(gens, *args)	See the cancel function in sympy.polys
`change_variable`(new_var, int_var_expr, inv_expr)	Change the integration variable
`class_key`()	Nice order of classes.
`coeff`(x[, n, right, _first])	Returns the coefficient from the term(s) containing `x**n`.
`collect`(syms[, func, evaluate, exact, ...])	See the collect function in sympy.simplify
`combsimp`()	See the combsimp function in sympy.simplify
`commute_factor_in`(term[, opd])	Move a factor into the inner product
`commute_factor_out`(term[, opd])	Move a factor out of the inner product
`compare`(other)	Return -1, 0, 1 if the object is smaller, equal, or greater than other.
`compute_leading_term`(x[, logx])	as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first.
`conjugate`()	Returns the complex conjugate of 'self'.
`copy`()
`could_extract_minus_sign`()	Return True if self has -1 as a leading factor or has more literal negative signs than positive signs in a sum, otherwise False.
`count`(query)	Count the number of matching subexpressions.
`count_ops`([visual])	wrapper for count_ops that returns the operation count.
`diff`(symbols, *assumptions)
`dir`(x, cdir)
`doit`(**hints)	Return un-evaluated integral form
`dummy_eq`(other[, symbol])	Compare two expressions and handle dummy symbols.
`equals`(other[, failing_expression])	Return True if self == other, False if it does not, or None.
`evalf`([n, subs, maxn, chop, strict, quad, ...])	Evaluate the given formula to an accuracy of n digits.
`expand`([deep, modulus, power_base, ...])	Expand an expression using hints.
`extract_additively`(c)	Return self - c if it's possible to subtract c from self and make all matching coefficients move towards zero, else return None.
`extract_branch_factor`([allow_half])	Try to write self as `exp_polar(2piIn)z` in a nice way.
`extract_multiplicatively`(c)	Return None if it's not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self.
`factor`(gens, *args)	See the factor() function in sympy.polys.polytools
`find`(query[, group])	Find all subexpressions matching a query.
`fourier_series`([limits])	Compute fourier sine/cosine series of self.
`fps`([x, x0, dir, hyper, order, rational, full])	Compute formal power power series of self.
`fromiter`(args, **assumptions)	Create a new object from an iterable.
`gammasimp`()	See the gammasimp function in sympy.simplify
`getO`()	Returns the additive O(..) symbol if there is one, else None.
`getn`()	Returns the order of the expression.
`has`(*patterns)	Test whether any subexpression matches any of the patterns.
`has_free`(*patterns)	return True if self has object(s) `x` as a free expression else False.
`integral_form`()	Get the explicit integral form
`integrand`()	Get the explicit form of the integrand
`integrate`(args, *kwargs)	See the integrate function in sympy.integrals
`invert`(g, gens, *args)	Return the multiplicative inverse of `self` mod `g` where `self` (and `g`) may be symbolic expressions).
`is_algebraic_expr`(*syms)	This tests whether a given expression is algebraic or not, in the given symbols, syms.
`is_constant`(wrt, *flags)	Return True if self is constant, False if not, or None if the constancy could not be determined conclusively.
`is_hypergeometric`(k)
`is_meromorphic`(x, a)	This tests whether an expression is meromorphic as a function of the given symbol `x` at the point `a`.
`is_polynomial`(*syms)	Return True if self is a polynomial in syms and False otherwise.
`is_rational_function`(*syms)	Test whether function is a ratio of two polynomials in the given symbols, syms.
`leadterm`(x[, logx, cdir])	Returns the leading term ax*b as a tuple (a, b).
`limit`(x, xlim[, dir])	Compute limit x->xlim.
`lseries`([x, x0, dir, logx, cdir])	Wrapper for series yielding an iterator of the terms of the series.
`match`(pattern[, old])	Pattern matching.
`matches`(expr[, repl_dict, old])	Helper method for match() that looks for a match between Wild symbols in self and expressions in expr.
`n`([n, subs, maxn, chop, strict, quad, verbose])	Evaluate the given formula to an accuracy of n digits.
`normal`()	expression -> a/b
`nseries`([x, x0, n, dir, logx, cdir])	Wrapper to _eval_nseries if assumptions allow, else to series.
`nsimplify`([constants, tolerance, full])	See the nsimplify function in sympy.simplify
`powsimp`(args, *kwargs)	See the powsimp function in sympy.simplify
`primitive`()	Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add).
`radsimp`(**kwargs)	See the radsimp function in sympy.simplify
`ratsimp`()	See the ratsimp function in sympy.simplify
`rcall`(*args)	Apply on the argument recursively through the expression tree.
`refine`([assumption])	See the refine function in sympy.assumptions
`removeO`()	Removes the additive O(..) symbol if there is one
`replace`(query, value[, map, simultaneous, exact])	Replace matching subexpressions of `self` with `value`.
`rewrite`(*args[, deep])	Rewrite self using a defined rule.
`round`([n])	Return x rounded to the given decimal place.
`separate`([deep, force])	See the separate function in sympy.simplify
`serialize`()	Serialize inner product to a dictionary
`series`([x, x0, n, dir, logx, cdir])	Series expansion of "self" around `x = x0` yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None.
`simplify`(**kwargs)	See the simplify function in sympy.simplify
`sort_key`([order])
`split`([opd])	Split the inner product into sum of inner products.
`subs`(args, *kwargs)	Substitutes old for new in an expression after sympifying args.
`taylor_term`(n, x, *previous_terms)	General method for the taylor term.
`together`(args, *kwargs)	See the together function in sympy.polys
`transpose`()
`trigsimp`(**args)	See the trigsimp function in sympy.simplify
`xreplace`(rule)	Replace occurrences of objects within the expression.

Attributes

`args`	Returns a tuple of arguments of 'self'.
`assumptions0`	Return object type assumptions.
`canonical_variables`	Return a dictionary mapping any variable defined in `self.bound_symbols` to Symbols that do not clash with any free symbols in the expression.
`default_assumptions`
`expr_free_symbols`	Like `free_symbols`, but returns the free symbols only if they are contained in an expression node.
`free_symbols`	Return from the atoms of self those which are free symbols.
`func`	The top-level function in an expression.
`is_Add`
`is_AlgebraicNumber`
`is_Atom`
`is_Boolean`
`is_Derivative`
`is_Dummy`
`is_Equality`
`is_Float`
`is_Function`
`is_Indexed`
`is_Integer`
`is_MatAdd`
`is_MatMul`
`is_Matrix`
`is_Mul`
`is_Not`
`is_Number`
`is_NumberSymbol`
`is_Order`
`is_Piecewise`
`is_Point`
`is_Poly`
`is_Pow`
`is_Rational`
`is_Relational`
`is_Symbol`
`is_Vector`
`is_Wild`
`is_algebraic`
`is_antihermitian`
`is_commutative`
`is_comparable`	Return True if self can be computed to a real number (or already is a real number) with precision, else False.
`is_complex`
`is_composite`
`is_even`
`is_extended_negative`
`is_extended_nonnegative`
`is_extended_nonpositive`
`is_extended_nonzero`
`is_extended_positive`
`is_extended_real`
`is_finite`
`is_hermitian`
`is_imaginary`
`is_infinite`
`is_integer`
`is_irrational`
`is_negative`
`is_noninteger`
`is_nonnegative`
`is_nonpositive`
`is_nonzero`
`is_number`	Returns True if `self` has no free symbols and no undefined functions (AppliedUndef, to be precise).
`is_odd`
`is_polar`
`is_positive`
`is_prime`
`is_rational`
`is_real`
`is_scalar`
`is_symbol`
`is_transcendental`
`is_zero`
`kind`

apart(x=None, **args)[source]: See the apart function in sympy.polys

property args: tuple[Basic, ...][source]

Returns a tuple of arguments of ‘self’.

>>> from sympy import cot
>>> from sympy.abc import x, y

>>> cot(x).args
(x,)

>>> cot(x).args[0]
x

>>> (x*y).args
(x, y)

>>> (x*y).args[1]
y

Never use self._args, always use self.args. Only use _args in __new__ when creating a new function. Do not override .args() from Basic (so that it is easy to change the interface in the future if needed).

args_cnc(cset=False, warn=True, split_1=True)[source]

Return [commutative factors, non-commutative factors] of self.

self is treated as a Mul and the ordering of the factors is maintained. If cset is True the commutative factors will be returned in a set. If there were repeated factors (as may happen with an unevaluated Mul) then an error will be raised unless it is explicitly suppressed by setting warn to False.

Note: -1 is always separated from a Number unless split_1 is False.

>>> from sympy import symbols, oo
>>> A, B = symbols('A B', commutative=0)
>>> x, y = symbols('x y')
>>> (-2*x*y).args_cnc()
[[-1, 2, x, y], []]
>>> (-2.5*x).args_cnc()
[[-1, 2.5, x], []]
>>> (-2*x*A*B*y).args_cnc()
[[-1, 2, x, y], [A, B]]
>>> (-2*x*A*B*y).args_cnc(split_1=False)
[[-2, x, y], [A, B]]
>>> (-2*x*y).args_cnc(cset=True)
[{-1, 2, x, y}, []]

The arg is always treated as a Mul:

>>> (-2 + x + A).args_cnc()
[[], [x - 2 + A]]
>>> (-oo).args_cnc() # -oo is a singleton
[[-1, oo], []]

as_coeff_Add(rational=False) → tuple['Number', Expr][source]: Efficiently extract the coefficient of a summation.

as_coeff_Mul(rational: bool = False) → tuple['Number', Expr][source]: Efficiently extract the coefficient of a product.

as_coeff_add(*deps) → tuple[Expr, tuple[Expr, ...]][source]

Return the tuple (c, args) where self is written as an Add, a.

c should be a Rational added to any terms of the Add that are independent of deps.

args should be a tuple of all other terms of a; args is empty if self is a Number or if self is independent of deps (when given).

This should be used when you do not know if self is an Add or not but you want to treat self as an Add or if you want to process the individual arguments of the tail of self as an Add.

if you know self is an Add and want only the head, use self.args[0];
if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail.
if you want to split self into an independent and dependent parts use self.as_independent(*deps)

>>> from sympy import S
>>> from sympy.abc import x, y
>>> (S(3)).as_coeff_add()
(3, ())
>>> (3 + x).as_coeff_add()
(3, (x,))
>>> (3 + x + y).as_coeff_add(x)
(y + 3, (x,))
>>> (3 + y).as_coeff_add(x)
(y + 3, ())

as_coeff_exponent(x) → tuple[Expr, Expr][source]: c*x**e -> c,e where x can be any symbolic expression.

as_coeff_mul(*deps, **kwargs) → tuple[Expr, tuple[Expr, ...]][source]

Return the tuple (c, args) where self is written as a Mul, m.

c should be a Rational multiplied by any factors of the Mul that are independent of deps.

args should be a tuple of all other factors of m; args is empty if self is a Number or if self is independent of deps (when given).

This should be used when you do not know if self is a Mul or not but you want to treat self as a Mul or if you want to process the individual arguments of the tail of self as a Mul.

if you know self is a Mul and want only the head, use self.args[0];
if you do not want to process the arguments of the tail but need the tail then use self.as_two_terms() which gives the head and tail;
if you want to split self into an independent and dependent parts use self.as_independent(*deps)

>>> from sympy import S
>>> from sympy.abc import x, y
>>> (S(3)).as_coeff_mul()
(3, ())
>>> (3*x*y).as_coeff_mul()
(3, (x, y))
>>> (3*x*y).as_coeff_mul(x)
(3*y, (x,))
>>> (3*y).as_coeff_mul(x)
(3*y, ())

as_coefficient(expr)[source]

Extracts symbolic coefficient at the given expression. In other words, this functions separates ‘self’ into the product of ‘expr’ and ‘expr’-free coefficient. If such separation is not possible it will return None.

>>> from sympy import E, pi, sin, I, Poly
>>> from sympy.abc import x

>>> E.as_coefficient(E)
1
>>> (2*E).as_coefficient(E)
2
>>> (2*sin(E)*E).as_coefficient(E)

Two terms have E in them so a sum is returned. (If one were desiring the coefficient of the term exactly matching E then the constant from the returned expression could be selected. Or, for greater precision, a method of Poly can be used to indicate the desired term from which the coefficient is desired.)

>>> (2*E + x*E).as_coefficient(E)
x + 2
>>> _.args[0]  # just want the exact match
2
>>> p = Poly(2*E + x*E); p
Poly(x*E + 2*E, x, E, domain='ZZ')
>>> p.coeff_monomial(E)
2
>>> p.nth(0, 1)
2

Since the following cannot be written as a product containing E as a factor, None is returned. (If the coefficient 2*x is desired then the coeff method should be used.)

>>> (2*E*x + x).as_coefficient(E)
>>> (2*E*x + x).coeff(E)
2*x

>>> (E*(x + 1) + x).as_coefficient(E)

>>> (2*pi*I).as_coefficient(pi*I)
2
>>> (2*I).as_coefficient(pi*I)

coeff: return sum of terms have a given factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used

as_coefficients_dict(*syms)[source]

Return a dictionary mapping terms to their Rational coefficient. Since the dictionary is a defaultdict, inquiries about terms which were not present will return a coefficient of 0.

If symbols syms are provided, any multiplicative terms independent of them will be considered a coefficient and a regular dictionary of syms-dependent generators as keys and their corresponding coefficients as values will be returned.

>>> from sympy.abc import a, x, y
>>> (3*x + a*x + 4).as_coefficients_dict()
{1: 4, x: 3, a*x: 1}
>>> _[a]
0
>>> (3*a*x).as_coefficients_dict()
{a*x: 3}
>>> (3*a*x).as_coefficients_dict(x)
{x: 3*a}
>>> (3*a*x).as_coefficients_dict(y)
{1: 3*a*x}

as_content_primitive(radical=False, clear=True)[source]

This method should recursively remove a Rational from all arguments and return that (content) and the new self (primitive). The content should always be positive and Mul(*foo.as_content_primitive()) == foo. The primitive need not be in canonical form and should try to preserve the underlying structure if possible (i.e. expand_mul should not be applied to self).

>>> from sympy import sqrt
>>> from sympy.abc import x, y, z

>>> eq = 2 + 2*x + 2*y*(3 + 3*y)

The as_content_primitive function is recursive and retains structure:

>>> eq.as_content_primitive()
(2, x + 3*y*(y + 1) + 1)

Integer powers will have Rationals extracted from the base:

>>> ((2 + 6*x)**2).as_content_primitive()
(4, (3*x + 1)**2)
>>> ((2 + 6*x)**(2*y)).as_content_primitive()
(1, (2*(3*x + 1))**(2*y))

Terms may end up joining once their as_content_primitives are added:

>>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive()
(11, x*(y + 1))
>>> ((3*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive()
(9, x*(y + 1))
>>> ((3*(z*(1 + y)) + 2.0*x*(3 + 3*y))).as_content_primitive()
(1, 6.0*x*(y + 1) + 3*z*(y + 1))
>>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))**2).as_content_primitive()
(121, x**2*(y + 1)**2)
>>> ((x*(1 + y) + 0.4*x*(3 + 3*y))**2).as_content_primitive()
(1, 4.84*x**2*(y + 1)**2)

Radical content can also be factored out of the primitive:

>>> (2*sqrt(2) + 4*sqrt(10)).as_content_primitive(radical=True)
(2, sqrt(2)*(1 + 2*sqrt(5)))

If clear=False (default is True) then content will not be removed from an Add if it can be distributed to leave one or more terms with integer coefficients.

>>> (x/2 + y).as_content_primitive()
(1/2, x + 2*y)
>>> (x/2 + y).as_content_primitive(clear=False)
(1, x/2 + y)

as_dummy()[source]

Return the expression with any objects having structurally bound symbols replaced with unique, canonical symbols within the object in which they appear and having only the default assumption for commutativity being True. When applied to a symbol a new symbol having only the same commutativity will be returned.

>>> from sympy import Integral, Symbol
>>> from sympy.abc import x
>>> r = Symbol('r', real=True)
>>> Integral(r, (r, x)).as_dummy()
Integral(_0, (_0, x))
>>> _.variables[0].is_real is None
True
>>> r.as_dummy()
_r

Any object that has structurally bound variables should have a property, bound_symbols that returns those symbols appearing in the object.

as_expr(*gens)[source]

Convert a polynomial to a SymPy expression.

>>> from sympy import sin
>>> from sympy.abc import x, y

>>> f = (x**2 + x*y).as_poly(x, y)
>>> f.as_expr()
x**2 + x*y

>>> sin(x).as_expr()
sin(x)

as_independent(*deps, **hint) → tuple[Expr, Expr][source]

A mostly naive separation of a Mul or Add into arguments that are not are dependent on deps. To obtain as complete a separation of variables as possible, use a separation method first, e.g.:

separatevars() to change Mul, Add and Pow (including exp) into Mul
.expand(mul=True) to change Add or Mul into Add
.expand(log=True) to change log expr into an Add

The only non-naive thing that is done here is to respect noncommutative ordering of variables and to always return (0, 0) for self of zero regardless of hints.

For nonzero self, the returned tuple (i, d) has the following interpretation:

i will has no variable that appears in deps
d will either have terms that contain variables that are in deps, or be equal to 0 (when self is an Add) or 1 (when self is a Mul)
if self is an Add then self = i + d
if self is a Mul then self = i*d
otherwise (self, S.One) or (S.One, self) is returned.

To force the expression to be treated as an Add, use the hint as_Add=True

– self is an Add

>>> from sympy import sin, cos, exp
>>> from sympy.abc import x, y, z

>>> (x + x*y).as_independent(x)
(0, x*y + x)
>>> (x + x*y).as_independent(y)
(x, x*y)
>>> (2*x*sin(x) + y + x + z).as_independent(x)
(y + z, 2*x*sin(x) + x)
>>> (2*x*sin(x) + y + x + z).as_independent(x, y)
(z, 2*x*sin(x) + x + y)

– self is a Mul

>>> (x*sin(x)*cos(y)).as_independent(x)
(cos(y), x*sin(x))

non-commutative terms cannot always be separated out when self is a Mul

>>> from sympy import symbols
>>> n1, n2, n3 = symbols('n1 n2 n3', commutative=False)
>>> (n1 + n1*n2).as_independent(n2)
(n1, n1*n2)
>>> (n2*n1 + n1*n2).as_independent(n2)
(0, n1*n2 + n2*n1)
>>> (n1*n2*n3).as_independent(n1)
(1, n1*n2*n3)
>>> (n1*n2*n3).as_independent(n2)
(n1, n2*n3)
>>> ((x-n1)*(x-y)).as_independent(x)
(1, (x - y)*(x - n1))

– self is anything else:

>>> (sin(x)).as_independent(x)
(1, sin(x))
>>> (sin(x)).as_independent(y)
(sin(x), 1)
>>> exp(x+y).as_independent(x)
(1, exp(x + y))

– force self to be treated as an Add:

>>> (3*x).as_independent(x, as_Add=True)
(0, 3*x)

– force self to be treated as a Mul:

>>> (3+x).as_independent(x, as_Add=False)
(1, x + 3)
>>> (-3+x).as_independent(x, as_Add=False)
(1, x - 3)

Note how the below differs from the above in making the constant on the dep term positive.

>>> (y*(-3+x)).as_independent(x)
(y, x - 3)

– use .as_independent() for true independence testing instead: of .has(). The former considers only symbols in the free symbols while the latter considers all symbols

>>> from sympy import Integral
>>> I = Integral(x, (x, 1, 2))
>>> I.has(x)
True
>>> x in I.free_symbols
False
>>> I.as_independent(x) == (I, 1)
True
>>> (I + x).as_independent(x) == (I, x)
True

Note: when trying to get independent terms, a separation method might need to be used first. In this case, it is important to keep track of what you send to this routine so you know how to interpret the returned values

>>> from sympy import separatevars, log
>>> separatevars(exp(x+y)).as_independent(x)
(exp(y), exp(x))
>>> (x + x*y).as_independent(y)
(x, x*y)
>>> separatevars(x + x*y).as_independent(y)
(x, y + 1)
>>> (x*(1 + y)).as_independent(y)
(x, y + 1)
>>> (x*(1 + y)).expand(mul=True).as_independent(y)
(x, x*y)
>>> a, b=symbols('a b', positive=True)
>>> (log(a*b).expand(log=True)).as_independent(b)
(log(a), log(b))

.separatevars(), .expand(log=True), sympy.core.add.Add.as_two_terms(), sympy.core.mul.Mul.as_two_terms(), .as_coeff_add(), .as_coeff_mul()

as_leading_term(*symbols, logx=None, cdir=0)[source]

Returns the leading (nonzero) term of the series expansion of self.

The _eval_as_leading_term routines are used to do this, and they must always return a non-zero value.

>>> from sympy.abc import x
>>> (1 + x + x**2).as_leading_term(x)
1
>>> (1/x**2 + x + x**2).as_leading_term(x)
x**(-2)

as_numer_denom()[source]

expression -> a/b -> a, b

This is just a stub that should be defined by an object’s class methods to get anything else.

normal: return a/b instead of (a, b)

as_ordered_factors(order=None)[source]: Return list of ordered factors (if Mul) else [self].

as_ordered_terms(order=None, data=False)[source]

Transform an expression to an ordered list of terms.

>>> from sympy import sin, cos
>>> from sympy.abc import x

>>> (sin(x)**2*cos(x) + sin(x)**2 + 1).as_ordered_terms()
[sin(x)**2*cos(x), sin(x)**2, 1]

as_poly(*gens, **args)[source]

Converts self to a polynomial or returns None.

>>> from sympy import sin
>>> from sympy.abc import x, y

>>> print((x**2 + x*y).as_poly())
Poly(x**2 + x*y, x, y, domain='ZZ')

>>> print((x**2 + x*y).as_poly(x, y))
Poly(x**2 + x*y, x, y, domain='ZZ')

>>> print((x**2 + sin(y)).as_poly(x, y))
None

as_powers_dict()[source]

Return self as a dictionary of factors with each factor being treated as a power. The keys are the bases of the factors and the values, the corresponding exponents. The resulting dictionary should be used with caution if the expression is a Mul and contains non- commutative factors since the order that they appeared will be lost in the dictionary.

as_ordered_factors: An alternative for noncommutative applications,: returning an ordered list of factors.
args_cnc: Similar to as_ordered_factors, but guarantees separation: of commutative and noncommutative factors.

as_real_imag(deep=True, **hints)[source]

Performs complex expansion on ‘self’ and returns a tuple containing collected both real and imaginary parts. This method cannot be confused with re() and im() functions, which does not perform complex expansion at evaluation.

However it is possible to expand both re() and im() functions and get exactly the same results as with a single call to this function.

>>> from sympy import symbols, I

>>> x, y = symbols('x,y', real=True)

>>> (x + y*I).as_real_imag()
(x, y)

>>> from sympy.abc import z, w

>>> (z + w*I).as_real_imag()
(re(z) - im(w), re(w) + im(z))

as_terms()[source]: Transform an expression to a list of terms.

aseries(x=None, n=6, bound=0, hir=False)[source]

Asymptotic Series expansion of self. This is equivalent to self.series(x, oo, n).

selfExpression: The expression whose series is to be expanded.
xSymbol: It is the variable of the expression to be calculated.
nValue: The value used to represent the order in terms of x**n, up to which the series is to be expanded.
hirBoolean: Set this parameter to be True to produce hierarchical series. It stops the recursion at an early level and may provide nicer and more useful results.
boundValue, Integer: Use the bound parameter to give limit on rewriting coefficients in its normalised form.

>>> from sympy import sin, exp
>>> from sympy.abc import x

>>> e = sin(1/x + exp(-x)) - sin(1/x)

>>> e.aseries(x)
(1/(24*x**4) - 1/(2*x**2) + 1 + O(x**(-6), (x, oo)))*exp(-x)

>>> e.aseries(x, n=3, hir=True)
-exp(-2*x)*sin(1/x)/2 + exp(-x)*cos(1/x) + O(exp(-3*x), (x, oo))

>>> e = exp(exp(x)/(1 - 1/x))

>>> e.aseries(x)
exp(exp(x)/(1 - 1/x))

>>> e.aseries(x, bound=3) 
exp(exp(x)/x**2)*exp(exp(x)/x)*exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x**2)*exp(exp(x))

For rational expressions this method may return original expression without the Order term. >>> (1/x).aseries(x, n=8) 1/x

Expr: Asymptotic series expansion of the expression.

This algorithm is directly induced from the limit computational algorithm provided by Gruntz. It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first to look for the most rapidly varying subexpression w of a given expression f and then expands f in a series in w. Then same thing is recursively done on the leading coefficient till we get constant coefficients.

If the most rapidly varying subexpression of a given expression f is f itself, the algorithm tries to find a normalised representation of the mrv set and rewrites f using this normalised representation.

If the expansion contains an order term, it will be either O(x ** (-n)) or O(w ** (-n)) where w belongs to the most rapidly varying expression of self.

Expr.aseries: See the docstring of this function for complete details of this wrapper.

property assumptions0[source]

Return object type assumptions.

For example:

Symbol(‘x’, real=True) Symbol(‘x’, integer=True)

are different objects. In other words, besides Python type (Symbol in this case), the initial assumptions are also forming their typeinfo.

>>> from sympy import Symbol
>>> from sympy.abc import x
>>> x.assumptions0
{'commutative': True}
>>> x = Symbol("x", positive=True)
>>> x.assumptions0
{'commutative': True, 'complex': True, 'extended_negative': False,
 'extended_nonnegative': True, 'extended_nonpositive': False,
 'extended_nonzero': True, 'extended_positive': True, 'extended_real':
 True, 'finite': True, 'hermitian': True, 'imaginary': False,
 'infinite': False, 'negative': False, 'nonnegative': True,
 'nonpositive': False, 'nonzero': True, 'positive': True, 'real':
 True, 'zero': False}

atoms(*types)[source]

Returns the atoms that form the current object.

By default, only objects that are truly atomic and cannot be divided into smaller pieces are returned: symbols, numbers, and number symbols like I and pi. It is possible to request atoms of any type, however, as demonstrated below.

>>> from sympy import I, pi, sin
>>> from sympy.abc import x, y
>>> (1 + x + 2*sin(y + I*pi)).atoms()
{1, 2, I, pi, x, y}

If one or more types are given, the results will contain only those types of atoms.

>>> from sympy import Number, NumberSymbol, Symbol
>>> (1 + x + 2*sin(y + I*pi)).atoms(Symbol)
{x, y}

>>> (1 + x + 2*sin(y + I*pi)).atoms(Number)
{1, 2}

>>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol)
{1, 2, pi}

>>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol, I)
{1, 2, I, pi}

Note that I (imaginary unit) and zoo (complex infinity) are special types of number symbols and are not part of the NumberSymbol class.

The type can be given implicitly, too:

>>> (1 + x + 2*sin(y + I*pi)).atoms(x) # x is a Symbol
{x, y}

Be careful to check your assumptions when using the implicit option since S(1).is_Integer = True but type(S(1)) is One, a special type of SymPy atom, while type(S(2)) is type Integer and will find all integers in an expression:

>>> from sympy import S
>>> (1 + x + 2*sin(y + I*pi)).atoms(S(1))
{1}

>>> (1 + x + 2*sin(y + I*pi)).atoms(S(2))
{1, 2}

Finally, arguments to atoms() can select more than atomic atoms: any SymPy type (loaded in core/__init__.py) can be listed as an argument and those types of “atoms” as found in scanning the arguments of the expression recursively:

>>> from sympy import Function, Mul
>>> from sympy.core.function import AppliedUndef
>>> f = Function('f')
>>> (1 + f(x) + 2*sin(y + I*pi)).atoms(Function)
{f(x), sin(y + I*pi)}
>>> (1 + f(x) + 2*sin(y + I*pi)).atoms(AppliedUndef)
{f(x)}

>>> (1 + x + 2*sin(y + I*pi)).atoms(Mul)
{I*pi, 2*sin(y + I*pi)}

cancel(*gens, **args)[source]: See the cancel function in sympy.polys

property canonical_variables[source]

Return a dictionary mapping any variable defined in self.bound_symbols to Symbols that do not clash with any free symbols in the expression.

>>> from sympy import Lambda
>>> from sympy.abc import x
>>> Lambda(x, 2*x).canonical_variables
{x: _0}

change_variable(new_var: Symbol, int_var_expr: Expr, inv_expr: Expr, jac_positive: bool = True, merge: bool = False, simplify: bool = False) → InnerProduct1D[source]

Change the integration variable

Parameters:

new_var (sympy.Symbol) – the new variable to be integrated over
int_var_expr (sympy.Expr) – the current variable expressed in new variable
inv_expr (sympy.Expr) – inverse expression of int_var_expr one needs to explicitly state the inverse expression
jac_positive (bool) – whether to assume the Jacobian is positive, default is True. If False, the absolute value of the Jacobian will be taken before taking the integral
merge (bool) – whether to merge the weight with the second operand. Default is False. Although not reasonable in the definition of inner products, it can be useful when the inner product is converted to integral form in the end.
simplify (bool) – whether to simplify the operands

Returns:

InnerProduct1D object with changed integration variable

classmethod class_key()[source]: Nice order of classes.

coeff(x, n=1, right=False, _first=True)[source]

Returns the coefficient from the term(s) containing x**n. If n is zero then all terms independent of x will be returned.

When x is noncommutative, the coefficient to the left (default) or right of x can be returned. The keyword ‘right’ is ignored when x is commutative.

>>> from sympy import symbols
>>> from sympy.abc import x, y, z

You can select terms that have an explicit negative in front of them:

>>> (-x + 2*y).coeff(-1)
x
>>> (x - 2*y).coeff(-1)
2*y

You can select terms with no Rational coefficient:

>>> (x + 2*y).coeff(1)
x
>>> (3 + 2*x + 4*x**2).coeff(1)
0

You can select terms independent of x by making n=0; in this case expr.as_independent(x)[0] is returned (and 0 will be returned instead of None):

>>> (3 + 2*x + 4*x**2).coeff(x, 0)
3
>>> eq = ((x + 1)**3).expand() + 1
>>> eq
x**3 + 3*x**2 + 3*x + 2
>>> [eq.coeff(x, i) for i in reversed(range(4))]
[1, 3, 3, 2]
>>> eq -= 2
>>> [eq.coeff(x, i) for i in reversed(range(4))]
[1, 3, 3, 0]

You can select terms that have a numerical term in front of them:

>>> (-x - 2*y).coeff(2)
-y
>>> from sympy import sqrt
>>> (x + sqrt(2)*x).coeff(sqrt(2))
x

The matching is exact:

>>> (3 + 2*x + 4*x**2).coeff(x)
2
>>> (3 + 2*x + 4*x**2).coeff(x**2)
4
>>> (3 + 2*x + 4*x**2).coeff(x**3)
0
>>> (z*(x + y)**2).coeff((x + y)**2)
z
>>> (z*(x + y)**2).coeff(x + y)
0

In addition, no factoring is done, so 1 + z*(1 + y) is not obtained from the following:

>>> (x + z*(x + x*y)).coeff(x)
1

If such factoring is desired, factor_terms can be used first:

>>> from sympy import factor_terms
>>> factor_terms(x + z*(x + x*y)).coeff(x)
z*(y + 1) + 1

>>> n, m, o = symbols('n m o', commutative=False)
>>> n.coeff(n)
1
>>> (3*n).coeff(n)
3
>>> (n*m + m*n*m).coeff(n) # = (1 + m)*n*m
1 + m
>>> (n*m + m*n*m).coeff(n, right=True) # = (1 + m)*n*m
m

If there is more than one possible coefficient 0 is returned:

>>> (n*m + m*n).coeff(n)
0

If there is only one possible coefficient, it is returned:

>>> (n*m + x*m*n).coeff(m*n)
x
>>> (n*m + x*m*n).coeff(m*n, right=1)
1

as_coefficient: separate the expression into a coefficient and factor as_coeff_Add: separate the additive constant from an expression as_coeff_Mul: separate the multiplicative constant from an expression as_independent: separate x-dependent terms/factors from others sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used

collect(syms, func=None, evaluate=True, exact=False, distribute_order_term=True)[source]: See the collect function in sympy.simplify

combsimp()[source]: See the combsimp function in sympy.simplify

commute_factor_in(term: Expr, opd: int = 1) → InnerProduct1D[source]

Move a factor into the inner product

A manual operation to move a factor from outside the inner product to its operand. Realization of “linearity”

Parameters:

term (sympy.Expr) – the factor to be moved in
opd (int) – index for the operand. Either 0 or 1, default = 1. If opd = 0, the factor will be moved to the first operand; if opd = 1, the factor will be moved to the second operand

commute_factor_out(term: Expr, opd: int = 1) → Expr[source]

Move a factor out of the inner product

A manual operation to move a factor from an operand to outside the inner product. Realization of “linearity”

Parameters:

term (sympy.Expr) – the factor to be moved out
opd (int) – index for the operand. Either 0 or 1, default = 1. If opd = 0, the factor will be moved out of the first operand; if opd = 1, the factor will be moved out of the second operand

compare(other)[source]

Return -1, 0, 1 if the object is smaller, equal, or greater than other.

Not in the mathematical sense. If the object is of a different type from the “other” then their classes are ordered according to the sorted_classes list.

>>> from sympy.abc import x, y
>>> x.compare(y)
-1
>>> x.compare(x)
0
>>> y.compare(x)
1

compute_leading_term(x, logx=None)[source]: as_leading_term is only allowed for results of .series() This is a wrapper to compute a series first.

conjugate()[source]: Returns the complex conjugate of ‘self’.

could_extract_minus_sign()[source]

Return True if self has -1 as a leading factor or has more literal negative signs than positive signs in a sum, otherwise False.

>>> from sympy.abc import x, y
>>> e = x - y
>>> {i.could_extract_minus_sign() for i in (e, -e)}
{False, True}

Though the y - x is considered like -(x - y), since it is in a product without a leading factor of -1, the result is false below:

>>> (x*(y - x)).could_extract_minus_sign()
False

To put something in canonical form wrt to sign, use signsimp:

>>> from sympy import signsimp
>>> signsimp(x*(y - x))
-x*(x - y)
>>> _.could_extract_minus_sign()
True

count(query)[source]: Count the number of matching subexpressions.

count_ops(visual=None)[source]: wrapper for count_ops that returns the operation count.

doit(**hints)[source]: Return un-evaluated integral form

dummy_eq(other, symbol=None)[source]

Compare two expressions and handle dummy symbols.

>>> from sympy import Dummy
>>> from sympy.abc import x, y

>>> u = Dummy('u')

>>> (u**2 + 1).dummy_eq(x**2 + 1)
True
>>> (u**2 + 1) == (x**2 + 1)
False

>>> (u**2 + y).dummy_eq(x**2 + y, x)
True
>>> (u**2 + y).dummy_eq(x**2 + y, y)
False

equals(other, failing_expression=False)[source]

Return True if self == other, False if it does not, or None. If failing_expression is True then the expression which did not simplify to a 0 will be returned instead of None.

If self is a Number (or complex number) that is not zero, then the result is False.

If self is a number and has not evaluated to zero, evalf will be used to test whether the expression evaluates to zero. If it does so and the result has significance (i.e. the precision is either -1, for a Rational result, or is greater than 1) then the evalf value will be used to return True or False.

evalf(n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False)[source]

Evaluate the given formula to an accuracy of n digits.

subsdict, optional

Substitute numerical values for symbols, e.g. subs={x:3, y:1+pi}. The substitutions must be given as a dictionary.

maxnint, optional

Allow a maximum temporary working precision of maxn digits.

chopbool or number, optional

Specifies how to replace tiny real or imaginary parts in subresults by exact zeros.

When True the chop value defaults to standard precision.

Otherwise the chop value is used to determine the magnitude of “small” for purposes of chopping.

>>> from sympy import N
>>> x = 1e-4
>>> N(x, chop=True)
0.000100000000000000
>>> N(x, chop=1e-5)
0.000100000000000000
>>> N(x, chop=1e-4)
0

strictbool, optional

Raise PrecisionExhausted if any subresult fails to evaluate to full accuracy, given the available maxprec.

quadstr, optional

Choose algorithm for numerical quadrature. By default, tanh-sinh quadrature is used. For oscillatory integrals on an infinite interval, try quad='osc'.

verbosebool, optional

Print debug information.

When Floats are naively substituted into an expression, precision errors may adversely affect the result. For example, adding 1e16 (a Float) to 1 will truncate to 1e16; if 1e16 is then subtracted, the result will be 0. That is exactly what happens in the following:

>>> from sympy.abc import x, y, z
>>> values = {x: 1e16, y: 1, z: 1e16}
>>> (x + y - z).subs(values)
0

Using the subs argument for evalf is the accurate way to evaluate such an expression:

>>> (x + y - z).evalf(subs=values)
1.00000000000000

expand(deep=True, modulus=None, power_base=True, power_exp=True, mul=True, log=True, multinomial=True, basic=True, **hints)[source]

Expand an expression using hints.

See the docstring of the expand() function in sympy.core.function for more information.

property expr_free_symbols[source]

Like free_symbols, but returns the free symbols only if they are contained in an expression node.

>>> from sympy.abc import x, y
>>> (x + y).expr_free_symbols 
{x, y}

If the expression is contained in a non-expression object, do not return the free symbols. Compare:

>>> from sympy import Tuple
>>> t = Tuple(x + y)
>>> t.expr_free_symbols 
set()
>>> t.free_symbols
{x, y}

extract_additively(c)[source]

Return self - c if it’s possible to subtract c from self and make all matching coefficients move towards zero, else return None.

>>> from sympy.abc import x, y
>>> e = 2*x + 3
>>> e.extract_additively(x + 1)
x + 2
>>> e.extract_additively(3*x)
>>> e.extract_additively(4)
>>> (y*(x + 1)).extract_additively(x + 1)
>>> ((x + 1)*(x + 2*y + 1) + 3).extract_additively(x + 1)
(x + 1)*(x + 2*y) + 3

extract_multiplicatively coeff as_coefficient

extract_branch_factor(allow_half=False)[source]

Try to write self as exp_polar(2*pi*I*n)*z in a nice way. Return (z, n).

>>> from sympy import exp_polar, I, pi
>>> from sympy.abc import x, y
>>> exp_polar(I*pi).extract_branch_factor()
(exp_polar(I*pi), 0)
>>> exp_polar(2*I*pi).extract_branch_factor()
(1, 1)
>>> exp_polar(-pi*I).extract_branch_factor()
(exp_polar(I*pi), -1)
>>> exp_polar(3*pi*I + x).extract_branch_factor()
(exp_polar(x + I*pi), 1)
>>> (y*exp_polar(-5*pi*I)*exp_polar(3*pi*I + 2*pi*x)).extract_branch_factor()
(y*exp_polar(2*pi*x), -1)
>>> exp_polar(-I*pi/2).extract_branch_factor()
(exp_polar(-I*pi/2), 0)

If allow_half is True, also extract exp_polar(I*pi):

>>> exp_polar(I*pi).extract_branch_factor(allow_half=True)
(1, 1/2)
>>> exp_polar(2*I*pi).extract_branch_factor(allow_half=True)
(1, 1)
>>> exp_polar(3*I*pi).extract_branch_factor(allow_half=True)
(1, 3/2)
>>> exp_polar(-I*pi).extract_branch_factor(allow_half=True)
(1, -1/2)

extract_multiplicatively(c)[source]

Return None if it’s not possible to make self in the form c * something in a nice way, i.e. preserving the properties of arguments of self.

>>> from sympy import symbols, Rational

>>> x, y = symbols('x,y', real=True)

>>> ((x*y)**3).extract_multiplicatively(x**2 * y)
x*y**2

>>> ((x*y)**3).extract_multiplicatively(x**4 * y)

>>> (2*x).extract_multiplicatively(2)
x

>>> (2*x).extract_multiplicatively(3)

>>> (Rational(1, 2)*x).extract_multiplicatively(3)
x/6

factor(*gens, **args)[source]: See the factor() function in sympy.polys.polytools

find(query, group=False)[source]: Find all subexpressions matching a query.

fourier_series(limits=None)[source]

Compute fourier sine/cosine series of self.

See the docstring of the fourier_series in sympy.series.fourier for more information.

fps(x=None, x0=0, dir=1, hyper=True, order=4, rational=True, full=False)[source]

Compute formal power power series of self.

See the docstring of the fps function in sympy.series.formal for more information.

property free_symbols: set[Basic][source]

Return from the atoms of self those which are free symbols.

Not all free symbols are Symbol. Eg: IndexedBase(‘I’)[0].free_symbols

For most expressions, all symbols are free symbols. For some classes this is not true. e.g. Integrals use Symbols for the dummy variables which are bound variables, so Integral has a method to return all symbols except those. Derivative keeps track of symbols with respect to which it will perform a derivative; those are bound variables, too, so it has its own free_symbols method.

Any other method that uses bound variables should implement a free_symbols method.

classmethod fromiter(args, **assumptions)[source]

Create a new object from an iterable.

This is a convenience function that allows one to create objects from any iterable, without having to convert to a list or tuple first.

>>> from sympy import Tuple
>>> Tuple.fromiter(i for i in range(5))
(0, 1, 2, 3, 4)

property func[source]

The top-level function in an expression.

The following should hold for all objects:

>> x == x.func(*x.args)

>>> from sympy.abc import x
>>> a = 2*x
>>> a.func
<class 'sympy.core.mul.Mul'>
>>> a.args
(2, x)
>>> a.func(*a.args)
2*x
>>> a == a.func(*a.args)
True

gammasimp()[source]: See the gammasimp function in sympy.simplify

getO()[source]: Returns the additive O(..) symbol if there is one, else None.

getn()[source]

Returns the order of the expression.

The order is determined either from the O(…) term. If there is no O(…) term, it returns None.

>>> from sympy import O
>>> from sympy.abc import x
>>> (1 + x + O(x**2)).getn()
2
>>> (1 + x).getn()

has(*patterns)[source]

Test whether any subexpression matches any of the patterns.

>>> from sympy import sin
>>> from sympy.abc import x, y, z
>>> (x**2 + sin(x*y)).has(z)
False
>>> (x**2 + sin(x*y)).has(x, y, z)
True
>>> x.has(x)
True

Note has is a structural algorithm with no knowledge of mathematics. Consider the following half-open interval:

>>> from sympy import Interval
>>> i = Interval.Lopen(0, 5); i
Interval.Lopen(0, 5)
>>> i.args
(0, 5, True, False)
>>> i.has(4)  # there is no "4" in the arguments
False
>>> i.has(0)  # there *is* a "0" in the arguments
True

Instead, use contains to determine whether a number is in the interval or not:

>>> i.contains(4)
True
>>> i.contains(0)
False

Note that expr.has(*patterns) is exactly equivalent to any(expr.has(p) for p in patterns). In particular, False is returned when the list of patterns is empty.

>>> x.has()
False

has_free(*patterns)[source]

return True if self has object(s) x as a free expression else False.

>>> from sympy import Integral, Function
>>> from sympy.abc import x, y
>>> f = Function('f')
>>> g = Function('g')
>>> expr = Integral(f(x), (f(x), 1, g(y)))
>>> expr.free_symbols
{y}
>>> expr.has_free(g(y))
True
>>> expr.has_free(*(x, f(x)))
False

This works for subexpressions and types, too:

>>> expr.has_free(g)
True
>>> (x + y + 1).has_free(y + 1)
True

integral_form()[source]: Get the explicit integral form

integrand()[source]: Get the explicit form of the integrand

integrate(*args, **kwargs)[source]: See the integrate function in sympy.integrals

invert(g, *gens, **args)[source]

Return the multiplicative inverse of self mod g where self (and g) may be symbolic expressions).

sympy.core.numbers.mod_inverse, sympy.polys.polytools.invert

is_algebraic_expr(*syms)[source]

This tests whether a given expression is algebraic or not, in the given symbols, syms. When syms is not given, all free symbols will be used. The rational function does not have to be in expanded or in any kind of canonical form.

This function returns False for expressions that are “algebraic expressions” with symbolic exponents. This is a simple extension to the is_rational_function, including rational exponentiation.

>>> from sympy import Symbol, sqrt
>>> x = Symbol('x', real=True)
>>> sqrt(1 + x).is_rational_function()
False
>>> sqrt(1 + x).is_algebraic_expr()
True

This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be an algebraic expression to become one.

>>> from sympy import exp, factor
>>> a = sqrt(exp(x)**2 + 2*exp(x) + 1)/(exp(x) + 1)
>>> a.is_algebraic_expr(x)
False
>>> factor(a).is_algebraic_expr()
True

is_rational_function()

property is_comparable[source]

Return True if self can be computed to a real number (or already is a real number) with precision, else False.

>>> from sympy import exp_polar, pi, I
>>> (I*exp_polar(I*pi/2)).is_comparable
True
>>> (I*exp_polar(I*pi*2)).is_comparable
False

A False result does not mean that self cannot be rewritten into a form that would be comparable. For example, the difference computed below is zero but without simplification it does not evaluate to a zero with precision:

>>> e = 2**pi*(1 + 2**pi)
>>> dif = e - e.expand()
>>> dif.is_comparable
False
>>> dif.n(2)._prec
1

is_constant(*wrt, **flags)[source]

Return True if self is constant, False if not, or None if the constancy could not be determined conclusively.

If an expression has no free symbols then it is a constant. If there are free symbols it is possible that the expression is a constant, perhaps (but not necessarily) zero. To test such expressions, a few strategies are tried:

1) numerical evaluation at two random points. If two such evaluations give two different values and the values have a precision greater than 1 then self is not constant. If the evaluations agree or could not be obtained with any precision, no decision is made. The numerical testing is done only if wrt is different than the free symbols.

2) differentiation with respect to variables in ‘wrt’ (or all free symbols if omitted) to see if the expression is constant or not. This will not always lead to an expression that is zero even though an expression is constant (see added test in test_expr.py). If all derivatives are zero then self is constant with respect to the given symbols.

3) finding out zeros of denominator expression with free_symbols. It will not be constant if there are zeros. It gives more negative answers for expression that are not constant.

If neither evaluation nor differentiation can prove the expression is constant, None is returned unless two numerical values happened to be the same and the flag failing_number is True – in that case the numerical value will be returned.

If flag simplify=False is passed, self will not be simplified; the default is True since self should be simplified before testing.

>>> from sympy import cos, sin, Sum, S, pi
>>> from sympy.abc import a, n, x, y
>>> x.is_constant()
False
>>> S(2).is_constant()
True
>>> Sum(x, (x, 1, 10)).is_constant()
True
>>> Sum(x, (x, 1, n)).is_constant()
False
>>> Sum(x, (x, 1, n)).is_constant(y)
True
>>> Sum(x, (x, 1, n)).is_constant(n)
False
>>> Sum(x, (x, 1, n)).is_constant(x)
True
>>> eq = a*cos(x)**2 + a*sin(x)**2 - a
>>> eq.is_constant()
True
>>> eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0
True

>>> (0**x).is_constant()
False
>>> x.is_constant()
False
>>> (x**x).is_constant()
False
>>> one = cos(x)**2 + sin(x)**2
>>> one.is_constant()
True
>>> ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1
True

is_meromorphic(x, a)[source]

This tests whether an expression is meromorphic as a function of the given symbol x at the point a.

This method is intended as a quick test that will return None if no decision can be made without simplification or more detailed analysis.

>>> from sympy import zoo, log, sin, sqrt
>>> from sympy.abc import x

>>> f = 1/x**2 + 1 - 2*x**3
>>> f.is_meromorphic(x, 0)
True
>>> f.is_meromorphic(x, 1)
True
>>> f.is_meromorphic(x, zoo)
True

>>> g = x**log(3)
>>> g.is_meromorphic(x, 0)
False
>>> g.is_meromorphic(x, 1)
True
>>> g.is_meromorphic(x, zoo)
False

>>> h = sin(1/x)*x**2
>>> h.is_meromorphic(x, 0)
False
>>> h.is_meromorphic(x, 1)
True
>>> h.is_meromorphic(x, zoo)
True

Multivalued functions are considered meromorphic when their branches are meromorphic. Thus most functions are meromorphic everywhere except at essential singularities and branch points. In particular, they will be meromorphic also on branch cuts except at their endpoints.

>>> log(x).is_meromorphic(x, -1)
True
>>> log(x).is_meromorphic(x, 0)
False
>>> sqrt(x).is_meromorphic(x, -1)
True
>>> sqrt(x).is_meromorphic(x, 0)
False

property is_number[source]

Returns True if self has no free symbols and no undefined functions (AppliedUndef, to be precise). It will be faster than if not self.free_symbols, however, since is_number will fail as soon as it hits a free symbol or undefined function.

>>> from sympy import Function, Integral, cos, sin, pi
>>> from sympy.abc import x
>>> f = Function('f')

>>> x.is_number
False
>>> f(1).is_number
False
>>> (2*x).is_number
False
>>> (2 + Integral(2, x)).is_number
False
>>> (2 + Integral(2, (x, 1, 2))).is_number
True

Not all numbers are Numbers in the SymPy sense:

>>> pi.is_number, pi.is_Number
(True, False)

If something is a number it should evaluate to a number with real and imaginary parts that are Numbers; the result may not be comparable, however, since the real and/or imaginary part of the result may not have precision.

>>> cos(1).is_number and cos(1).is_comparable
True

>>> z = cos(1)**2 + sin(1)**2 - 1
>>> z.is_number
True
>>> z.is_comparable
False

sympy.core.basic.Basic.is_comparable

is_polynomial(*syms)[source]

Return True if self is a polynomial in syms and False otherwise.

This checks if self is an exact polynomial in syms. This function returns False for expressions that are “polynomials” with symbolic exponents. Thus, you should be able to apply polynomial algorithms to expressions for which this returns True, and Poly(expr, *syms) should work if and only if expr.is_polynomial(*syms) returns True. The polynomial does not have to be in expanded form. If no symbols are given, all free symbols in the expression will be used.

This is not part of the assumptions system. You cannot do Symbol(‘z’, polynomial=True).

>>> from sympy import Symbol, Function
>>> x = Symbol('x')
>>> ((x**2 + 1)**4).is_polynomial(x)
True
>>> ((x**2 + 1)**4).is_polynomial()
True
>>> (2**x + 1).is_polynomial(x)
False
>>> (2**x + 1).is_polynomial(2**x)
True
>>> f = Function('f')
>>> (f(x) + 1).is_polynomial(x)
False
>>> (f(x) + 1).is_polynomial(f(x))
True
>>> (1/f(x) + 1).is_polynomial(f(x))
False

>>> n = Symbol('n', nonnegative=True, integer=True)
>>> (x**n + 1).is_polynomial(x)
False

This function does not attempt any nontrivial simplifications that may result in an expression that does not appear to be a polynomial to become one.

>>> from sympy import sqrt, factor, cancel
>>> y = Symbol('y', positive=True)
>>> a = sqrt(y**2 + 2*y + 1)
>>> a.is_polynomial(y)
False
>>> factor(a)
y + 1
>>> factor(a).is_polynomial(y)
True

>>> b = (y**2 + 2*y + 1)/(y + 1)
>>> b.is_polynomial(y)
False
>>> cancel(b)
y + 1
>>> cancel(b).is_polynomial(y)
True

See also lseries().

>>> from sympy import sin, log, Symbol
>>> from sympy.abc import x, y
>>> sin(x).nseries(x, 0, 6)
x - x**3/6 + x**5/120 + O(x**6)
>>> log(x+1).nseries(x, 0, 5)
x - x**2/2 + x**3/3 - x**4/4 + O(x**5)

Handling of the logx parameter — in the following example the expansion fails since sin does not have an asymptotic expansion at -oo (the limit of log(x) as x approaches 0):

>>> e = sin(log(x))
>>> e.nseries(x, 0, 6)
Traceback (most recent call last):
...
PoleError: ...
...
>>> logx = Symbol('logx')
>>> e.nseries(x, 0, 6, logx=logx)
sin(logx)

In the following example, the expansion works but only returns self unless the logx parameter is used:

>>> e = x**y
>>> e.nseries(x, 0, 2)
x**y
>>> e.nseries(x, 0, 2, logx=logx)
exp(logx*y)

nsimplify(constants=(), tolerance=None, full=False)[source]: See the nsimplify function in sympy.simplify

powsimp(*args, **kwargs)[source]: See the powsimp function in sympy.simplify

primitive()[source]

Return the positive Rational that can be extracted non-recursively from every term of self (i.e., self is treated like an Add). This is like the as_coeff_Mul() method but primitive always extracts a positive Rational (never a negative or a Float).

>>> from sympy.abc import x
>>> (3*(x + 1)**2).primitive()
(3, (x + 1)**2)
>>> a = (6*x + 2); a.primitive()
(2, 3*x + 1)
>>> b = (x/2 + 3); b.primitive()
(1/2, x + 6)
>>> (a*b).primitive() == (1, a*b)
True

radsimp(**kwargs)[source]: See the radsimp function in sympy.simplify

ratsimp()[source]: See the ratsimp function in sympy.simplify

rcall(*args)[source]

Apply on the argument recursively through the expression tree.

This method is used to simulate a common abuse of notation for operators. For instance, in SymPy the following will not work:

(x+Lambda(y, 2*y))(z) == x+2*z,

however, you can use:

>>> from sympy import Lambda
>>> from sympy.abc import x, y, z
>>> (x + Lambda(y, 2*y)).rcall(z)
x + 2*z

refine(assumption=True)[source]: See the refine function in sympy.assumptions

removeO()[source]: Removes the additive O(..) symbol if there is one

replace(query, value, map=False, simultaneous=True, exact=None)[source]

Replace matching subexpressions of self with value.

If map = True then also return the mapping {old: new} where old was a sub-expression found with query and new is the replacement value for it. If the expression itself does not match the query, then the returned value will be self.xreplace(map) otherwise it should be self.subs(ordered(map.items())).

Traverses an expression tree and performs replacement of matching subexpressions from the bottom to the top of the tree. The default approach is to do the replacement in a simultaneous fashion so changes made are targeted only once. If this is not desired or causes problems, simultaneous can be set to False.

In addition, if an expression containing more than one Wild symbol is being used to match subexpressions and the exact flag is None it will be set to True so the match will only succeed if all non-zero values are received for each Wild that appears in the match pattern. Setting this to False accepts a match of 0; while setting it True accepts all matches that have a 0 in them. See example below for cautions.

The list of possible combinations of queries and replacement values is listed below:

Initial setup

>>> from sympy import log, sin, cos, tan, Wild, Mul, Add
>>> from sympy.abc import x, y
>>> f = log(sin(x)) + tan(sin(x**2))

1.1. type -> type

obj.replace(type, newtype)

When object of type type is found, replace it with the result of passing its argument(s) to newtype.

>>> f.replace(sin, cos)
log(cos(x)) + tan(cos(x**2))
>>> sin(x).replace(sin, cos, map=True)
(cos(x), {sin(x): cos(x)})
>>> (x*y).replace(Mul, Add)
x + y

1.2. type -> func

obj.replace(type, func)

When object of type type is found, apply func to its argument(s). func must be written to handle the number of arguments of type.

>>> f.replace(sin, lambda arg: sin(2*arg))
log(sin(2*x)) + tan(sin(2*x**2))
>>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args)))
sin(2*x*y)

2.1. pattern -> expr

obj.replace(pattern(wild), expr(wild))

Replace subexpressions matching pattern with the expression written in terms of the Wild symbols in pattern.

>>> a, b = map(Wild, 'ab')
>>> f.replace(sin(a), tan(a))
log(tan(x)) + tan(tan(x**2))
>>> f.replace(sin(a), tan(a/2))
log(tan(x/2)) + tan(tan(x**2/2))
>>> f.replace(sin(a), a)
log(x) + tan(x**2)
>>> (x*y).replace(a*x, a)
y

Matching is exact by default when more than one Wild symbol is used: matching fails unless the match gives non-zero values for all Wild symbols:

>>> (2*x + y).replace(a*x + b, b - a)
y - 2
>>> (2*x).replace(a*x + b, b - a)
2*x

When set to False, the results may be non-intuitive:

>>> (2*x).replace(a*x + b, b - a, exact=False)
2/x

2.2. pattern -> func

obj.replace(pattern(wild), lambda wild: expr(wild))

All behavior is the same as in 2.1 but now a function in terms of pattern variables is used rather than an expression:

>>> f.replace(sin(a), lambda a: sin(2*a))
log(sin(2*x)) + tan(sin(2*x**2))

3.1. func -> func

obj.replace(filter, func)

Replace subexpression e with func(e) if filter(e) is True.

>>> g = 2*sin(x**3)
>>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2)
4*sin(x**9)

The expression itself is also targeted by the query but is done in such a fashion that changes are not made twice.

>>> e = x*(x*y + 1)
>>> e.replace(lambda x: x.is_Mul, lambda x: 2*x)
2*x*(2*x*y + 1)

When matching a single symbol, exact will default to True, but this may or may not be the behavior that is desired:

Here, we want exact=False:

>>> from sympy import Function
>>> f = Function('f')
>>> e = f(1) + f(0)
>>> q = f(a), lambda a: f(a + 1)
>>> e.replace(*q, exact=False)
f(1) + f(2)
>>> e.replace(*q, exact=True)
f(0) + f(2)

But here, the nature of matching makes selecting the right setting tricky:

>>> e = x**(1 + y)
>>> (x**(1 + y)).replace(x**(1 + a), lambda a: x**-a, exact=False)
x
>>> (x**(1 + y)).replace(x**(1 + a), lambda a: x**-a, exact=True)
x**(-x - y + 1)
>>> (x**y).replace(x**(1 + a), lambda a: x**-a, exact=False)
x
>>> (x**y).replace(x**(1 + a), lambda a: x**-a, exact=True)
x**(1 - y)

It is probably better to use a different form of the query that describes the target expression more precisely:

>>> (1 + x**(1 + y)).replace(
... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1,
... lambda x: x.base**(1 - (x.exp - 1)))
...
x**(1 - y) + 1

subs: substitution of subexpressions as defined by the objects: themselves.
xreplace: exact node replacement in expr tree; also capable of: using matching rules

rewrite(*args, deep=True, **hints)[source]

Rewrite self using a defined rule.

Rewriting transforms an expression to another, which is mathematically equivalent but structurally different. For example you can rewrite trigonometric functions as complex exponentials or combinatorial functions as gamma function.

This method takes a pattern and a rule as positional arguments. pattern is optional parameter which defines the types of expressions that will be transformed. If it is not passed, all possible expressions will be rewritten. rule defines how the expression will be rewritten.

argsrule, or pattern and rule.

pattern is a type or an iterable of types.
rule can be any object.

deepbool, optional.

If True, subexpressions are recursively transformed. Default is True.

If pattern is unspecified, all possible expressions are transformed.

>>> from sympy import cos, sin, exp, I
>>> from sympy.abc import x
>>> expr = cos(x) + I*sin(x)
>>> expr.rewrite(exp)
exp(I*x)

Pattern can be a type or an iterable of types.

>>> expr.rewrite(sin, exp)
exp(I*x)/2 + cos(x) - exp(-I*x)/2
>>> expr.rewrite([cos,], exp)
exp(I*x)/2 + I*sin(x) + exp(-I*x)/2
>>> expr.rewrite([cos, sin], exp)
exp(I*x)

Rewriting behavior can be implemented by defining _eval_rewrite() method.

>>> from sympy import Expr, sqrt, pi
>>> class MySin(Expr):
...     def _eval_rewrite(self, rule, args, **hints):
...         x, = args
...         if rule == cos:
...             return cos(pi/2 - x, evaluate=False)
...         if rule == sqrt:
...             return sqrt(1 - cos(x)**2)
>>> MySin(MySin(x)).rewrite(cos)
cos(-cos(-x + pi/2) + pi/2)
>>> MySin(x).rewrite(sqrt)
sqrt(1 - cos(x)**2)

Defining _eval_rewrite_as_[...]() method is supported for backwards compatibility reason. This may be removed in the future and using it is discouraged.

>>> class MySin(Expr):
...     def _eval_rewrite_as_cos(self, *args, **hints):
...         x, = args
...         return cos(pi/2 - x, evaluate=False)
>>> MySin(x).rewrite(cos)
cos(-x + pi/2)

round(n=None)[source]

Return x rounded to the given decimal place.

If a complex number would results, apply round to the real and imaginary components of the number.

>>> from sympy import pi, E, I, S, Number
>>> pi.round()
3
>>> pi.round(2)
3.14
>>> (2*pi + E*I).round()
6 + 3*I

The round method has a chopping effect:

>>> (2*pi + I/10).round()
6
>>> (pi/10 + 2*I).round()
2*I
>>> (pi/10 + E*I).round(2)
0.31 + 2.72*I

The Python round function uses the SymPy round method so it will always return a SymPy number (not a Python float or int):

>>> isinstance(round(S(123), -2), Number)
True

separate(deep=False, force=False)[source]: See the separate function in sympy.simplify

serialize() → dict[source]: Serialize inner product to a dictionary

series(x=None, x0=0, n=6, dir='+', logx=None, cdir=0)[source]

Series expansion of “self” around x = x0 yielding either terms of the series one by one (the lazy series given when n=None), else all the terms at once when n != None.

Returns the series expansion of “self” around the point x = x0 with respect to x up to O((x - x0)**n, x, x0) (default n is 6).

If x=None and self is univariate, the univariate symbol will be supplied, otherwise an error will be raised.

exprExpression: The expression whose series is to be expanded.
xSymbol: It is the variable of the expression to be calculated.
x0Value: The value around which x is calculated. Can be any value from -oo to oo.
nValue: The value used to represent the order in terms of x**n, up to which the series is to be expanded.
dirString, optional: The series-expansion can be bi-directional. If dir="+", then (x->x0+). If dir="-", then (x->x0-). For infinite ``x0 (oo or -oo), the dir argument is determined from the direction of the infinity (i.e., dir="-" for oo).
logxoptional: It is used to replace any log(x) in the returned series with a symbolic value rather than evaluating the actual value.
cdiroptional: It stands for complex direction, and indicates the direction from which the expansion needs to be evaluated.

>>> from sympy import cos, exp, tan
>>> from sympy.abc import x, y
>>> cos(x).series()
1 - x**2/2 + x**4/24 + O(x**6)
>>> cos(x).series(n=4)
1 - x**2/2 + O(x**4)
>>> cos(x).series(x, x0=1, n=2)
cos(1) - (x - 1)*sin(1) + O((x - 1)**2, (x, 1))
>>> e = cos(x + exp(y))
>>> e.series(y, n=2)
cos(x + 1) - y*sin(x + 1) + O(y**2)
>>> e.series(x, n=2)
cos(exp(y)) - x*sin(exp(y)) + O(x**2)

If n=None then a generator of the series terms will be returned.

>>> term=cos(x).series(n=None)
>>> [next(term) for i in range(2)]
[1, -x**2/2]

For dir=+ (default) the series is calculated from the right and for dir=- the series from the left. For smooth functions this flag will not alter the results.

>>> abs(x).series(dir="+")
x
>>> abs(x).series(dir="-")
-x
>>> f = tan(x)
>>> f.series(x, 2, 6, "+")
tan(2) + (1 + tan(2)**2)*(x - 2) + (x - 2)**2*(tan(2)**3 + tan(2)) +
(x - 2)**3*(1/3 + 4*tan(2)**2/3 + tan(2)**4) + (x - 2)**4*(tan(2)**5 +
5*tan(2)**3/3 + 2*tan(2)/3) + (x - 2)**5*(2/15 + 17*tan(2)**2/15 +
2*tan(2)**4 + tan(2)**6) + O((x - 2)**6, (x, 2))

>>> f.series(x, 2, 3, "-")
tan(2) + (2 - x)*(-tan(2)**2 - 1) + (2 - x)**2*(tan(2)**3 + tan(2))
+ O((x - 2)**3, (x, 2))

For rational expressions this method may return original expression without the Order term. >>> (1/x).series(x, n=8) 1/x

ExprExpression: Series expansion of the expression about x0

TypeError: If “n” and “x0” are infinity objects
PoleError: If “x0” is an infinity object

simplify(**kwargs)[source]: See the simplify function in sympy.simplify

split(opd: int = 1) → Expr[source]

Split the inner product into sum of inner products.

A manual operation of the distributive property of inner products to split the inner product whose argument is a sum of several terms into the sum of several inner products.

subs(*args, **kwargs)[source]

Substitutes old for new in an expression after sympifying args.

args is either:

two arguments, e.g. foo.subs(old, new)
one iterable argument, e.g. foo.subs(iterable). The iterable may be

o an iterable container with (old, new) pairs. In this case the
replacements are processed in the order given with successive patterns possibly affecting replacements already made.

o a dict or set whose key/value items correspond to old/new pairs.
In this case the old/new pairs will be sorted by op count and in case of a tie, by number of args and the default_sort_key. The resulting sorted list is then processed as an iterable container (see previous).

If the keyword simultaneous is True, the subexpressions will not be evaluated until all the substitutions have been made.

>>> from sympy import pi, exp, limit, oo
>>> from sympy.abc import x, y
>>> (1 + x*y).subs(x, pi)
pi*y + 1
>>> (1 + x*y).subs({x:pi, y:2})
1 + 2*pi
>>> (1 + x*y).subs([(x, pi), (y, 2)])
1 + 2*pi
>>> reps = [(y, x**2), (x, 2)]
>>> (x + y).subs(reps)
6
>>> (x + y).subs(reversed(reps))
x**2 + 2

>>> (x**2 + x**4).subs(x**2, y)
y**2 + y

To replace only the x**2 but not the x**4, use xreplace:

>>> (x**2 + x**4).xreplace({x**2: y})
x**4 + y

To delay evaluation until all substitutions have been made, set the keyword simultaneous to True:

>>> (x/y).subs([(x, 0), (y, 0)])
0
>>> (x/y).subs([(x, 0), (y, 0)], simultaneous=True)
nan

This has the added feature of not allowing subsequent substitutions to affect those already made:

>>> ((x + y)/y).subs({x + y: y, y: x + y})
1
>>> ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True)
y/(x + y)

In order to obtain a canonical result, unordered iterables are sorted by count_op length, number of arguments and by the default_sort_key to break any ties. All other iterables are left unsorted.

>>> from sympy import sqrt, sin, cos
>>> from sympy.abc import a, b, c, d, e

>>> A = (sqrt(sin(2*x)), a)
>>> B = (sin(2*x), b)
>>> C = (cos(2*x), c)
>>> D = (x, d)
>>> E = (exp(x), e)

>>> expr = sqrt(sin(2*x))*sin(exp(x)*x)*cos(2*x) + sin(2*x)

>>> expr.subs(dict([A, B, C, D, E]))
a*c*sin(d*e) + b

The resulting expression represents a literal replacement of the old arguments with the new arguments. This may not reflect the limiting behavior of the expression:

>>> (x**3 - 3*x).subs({x: oo})
nan

>>> limit(x**3 - 3*x, x, oo)
oo

If the substitution will be followed by numerical evaluation, it is better to pass the substitution to evalf as

>>> (1/x).evalf(subs={x: 3.0}, n=21)
0.333333333333333333333

rather than

>>> (1/x).subs({x: 3.0}).evalf(21)
0.333333333333333314830

as the former will ensure that the desired level of precision is obtained.

replace: replacement capable of doing wildcard-like matching,: parsing of match, and conditional replacements
xreplace: exact node replacement in expr tree; also capable of: using matching rules

sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision

taylor_term(n, x, *previous_terms)[source]

General method for the taylor term.

This method is slow, because it differentiates n-times. Subclasses can redefine it to make it faster by using the “previous_terms”.

together(*args, **kwargs)[source]: See the together function in sympy.polys

trigsimp(**args)[source]: See the trigsimp function in sympy.simplify

xreplace(rule)[source]

Replace occurrences of objects within the expression.

ruledict-like: Expresses a replacement rule

xreplace : the result of the replacement

>>> from sympy import symbols, pi, exp
>>> x, y, z = symbols('x y z')
>>> (1 + x*y).xreplace({x: pi})
pi*y + 1
>>> (1 + x*y).xreplace({x: pi, y: 2})
1 + 2*pi

Replacements occur only if an entire node in the expression tree is matched:

>>> (x*y + z).xreplace({x*y: pi})
z + pi
>>> (x*y*z).xreplace({x*y: pi})
x*y*z
>>> (2*x).xreplace({2*x: y, x: z})
y
>>> (2*2*x).xreplace({2*x: y, x: z})
4*z
>>> (x + y + 2).xreplace({x + y: 2})
x + y + 2
>>> (x + 2 + exp(x + 2)).xreplace({x + 2: y})
x + exp(y) + 2

xreplace does not differentiate between free and bound symbols. In the following, subs(x, y) would not change x since it is a bound symbol, but xreplace does:

>>> from sympy import Integral
>>> Integral(x, (x, 1, 2*x)).xreplace({x: y})
Integral(y, (y, 1, 2*y))

Trying to replace x with an expression raises an error:

>>> Integral(x, (x, 1, 2*x)).xreplace({x: 2*y}) 
ValueError: Invalid limits given: ((2*y, 1, 4*y),)

replace: replacement capable of doing wildcard-like matching,: parsing of match, and conditional replacements
subs: substitution of subexpressions as defined by the objects: themselves.