# sagemath/sagelib

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 from sage.symbolic.function_factory import function as new_function from sage.symbolic.ring import SR def var(*args, **kwds): r""" Create a symbolic variable with the name *s*. INPUT: - args - A single string var('x y'), a list of strings var(['x','y']), or multiple strings var('x', 'y'). A single string can be either a single variable name, or a space or comma separated list of variable names. In a list or tuple of strings, each entry is one variable. If multiple arguments are specified, each argument is taken to be one variable. Spaces before or after variable names are ignored. - kwds - keyword arguments can be given to specify domain and custom latex_name for variables. See EXAMPLES for usage. .. note:: The new variable is both returned and automatically injected into the global namespace. If you need symbolic variable in library code, it is better to use either SR.var() or SR.symbol(). OUTPUT: If a single symbolic variable was created, the variable itself. Otherwise, a tuple of symbolic variables. The variable names are checked to be valid Python identifiers and a ValueError is raised otherwise. EXAMPLES: Here are the different ways to define three variables x, y, and z in a single line:: sage: var('x y z') (x, y, z) sage: var('x, y, z') (x, y, z) sage: var(['x', 'y', 'z']) (x, y, z) sage: var('x', 'y', 'z') (x, y, z) sage: var('x'), var('y'), var(z) (x, y, z) We define some symbolic variables:: sage: var('n xx yy zz') (n, xx, yy, zz) Then we make an algebraic expression out of them:: sage: f = xx^n + yy^n + zz^n; f xx^n + yy^n + zz^n By default, var returns a complex variable. To define real or positive variables we can specify the domain as:: sage: x = var('x', domain=RR); x; x.conjugate() x x sage: y = var('y', domain='real'); y.conjugate() y sage: y = var('y', domain='positive'); y.abs() y Custom latex expression can be assigned to variable:: sage: x = var('sui', latex_name="s_{u,i}"); x._latex_() 's_{u,i}' In notebook, we can also colorize latex expression:: sage: x = var('sui', latex_name="\\color{red}{s_{u,i}}"); x._latex_() '\\color{red}{s_{u,i}}' We can substitute a new variable name for n:: sage: f(n = var('sigma')) xx^sigma + yy^sigma + zz^sigma If you make an important built-in variable into a symbolic variable, you can get back the original value using restore:: sage: var('QQ RR') (QQ, RR) sage: QQ QQ sage: restore('QQ') sage: QQ Rational Field We make two new variables separated by commas:: sage: var('theta, gamma') (theta, gamma) sage: theta^2 + gamma^3 gamma^3 + theta^2 The new variables are of type Expression, and belong to the symbolic expression ring:: sage: type(theta) sage: parent(theta) Symbolic Ring TESTS:: sage: var('q',ns=False) Traceback (most recent call last): ... NotImplementedError: The new (Pynac) symbolics are now the only symbolics; please do not use keyword ns any longer. sage: q Traceback (most recent call last): ... NameError: name 'q' is not defined sage: var('q',ns=1) doctest:...: DeprecationWarning: The new (Pynac) symbolics are now the only symbolics; please do not use keyword 'ns' any longer. q """ if len(args)==1: name = args[0] else: name = args G = globals() # this is the reason the code must be in Cython. if kwds.has_key('ns'): if kwds['ns']: from sage.misc.misc import deprecation deprecation("The new (Pynac) symbolics are now the only symbolics; please do not use keyword 'ns' any longer.") else: raise NotImplementedError, "The new (Pynac) symbolics are now the only symbolics; please do not use keyword ns any longer." kwds.pop('ns') v = SR.var(name, **kwds) if isinstance(v, tuple): for x in v: G[repr(x)] = x else: G[repr(v)] = v return v def function(s, *args, **kwds): r""" Create a formal symbolic function with the name *s*. INPUT: - s - a string, either a single variable name, or a space or comma separated list of variable names. - **kwds - keyword arguments. Either one of the following two keywords can be used to customize latex representation of symbolic functions: (1) latex_name=LaTeX where LaTeX is any valid latex expression. Ex: f = function('f', x, latex_name="\\mathcal{F}") See EXAMPLES for more. (2) print_latex_func=my_latex_print where my_latex_print is any callable function that returns a valid latex expression. Ex: f = function('f', x, print_latex_func=my_latex_print) See EXAMPLES for an explicit usage. .. note:: The new function is both returned and automatically injected into the global namespace. If you use this function in library code, it is better to use sage.symbolic.function_factory.function, since it won't touch the global namespace. EXAMPLES:: We create a formal function called supersin:: sage: f = function('supersin', x) sage: f supersin(x) We can immediately use supersin in symbolic expressions:: sage: y, z, A = var('y z A') sage: supersin(y+z) + A^3 A^3 + supersin(y + z) We can define other functions in terms of supersin:: sage: g(x,y) = supersin(x)^2 + sin(y/2) sage: g (x, y) |--> supersin(x)^2 + sin(1/2*y) sage: g.diff(y) (x, y) |--> 1/2*cos(1/2*y) sage: k = g.diff(x); k (x, y) |--> 2*supersin(x)*D[0](supersin)(x) Custom typesetting of symbolic functions in LaTeX:: (1) Either using latex_name keyword:: sage: riemann(x) = function('riemann', x, latex_name="\\mathcal{R}") sage: latex(riemann(x)) \mathcal{R}\left(x\right) (2) Or passing a custom callable function that returns a latex expression:: sage: mu,nu = var('mu,nu') sage: def my_latex_print(self, *args): return "\\psi_{%s}"%(', '.join(map(latex, args))) sage: psi(mu,nu) = function('psi', mu, nu, print_latex_func=my_latex_print) sage: latex(psi(mu,nu)) \psi_{\mu, \nu} In Sage 4.0, you must now use the :meth:substitute_function method to replace functions:: sage: k.substitute_function(supersin, sin) 2*sin(x)*cos(x) """ if len(args) > 0: return function(s, **kwds)(*args) G = globals() # this is the reason the code must be in Cython. v = new_function(s, **kwds) if isinstance(v, tuple): for x in v: G[repr(x)] = x else: G[repr(v)] = v return v def clear_vars(): """ Delete all 1-letter symbolic variables that are predefined at startup of Sage. Any one-letter global variables that are not symbolic variables are not cleared. EXAMPLES:: sage: var('x y z') (x, y, z) sage: (x+y)^z (x + y)^z sage: k = 15 sage: clear_vars() sage: (x+y)^z Traceback (most recent call last): ... NameError: name 'x' is not defined sage: expand((e + i)^2) 2*I*e + e^2 - 1 sage: k 15 """ G = globals() from sage.symbolic.ring import is_SymbolicVariable for i in range(65,65+26) + range(97,97+26): if G.has_key(chr(i)) and is_SymbolicVariable(G[chr(i)]): # We check to see if there is a corresponding pyobject # associated with the expression. This will work for # constants which we want to keep, but will fail for # variables that we want to delete. try: G[chr(i)].pyobject() except TypeError: del G[chr(i)]