Module calculus

EXAMPLES:
    sage: is_SymbolicExpression(sin(x))
    True
    sage: is_SymbolicExpression(2/3)
    False
    sage: is_SymbolicExpression(sqrt(2))
    True

EXAMPLES:
    sage: is_SymbolicExpressionRing(QQ)
    False
    sage: is_SymbolicExpressionRing(SR)
    True

Return the symbolic expression ring.  There is one
globally defines symbolic expression ring in the
calculus module. 

EXAMPLES:
    sage: SymbolicExpressionRing()
    Symbolic Ring
    sage: SymbolicExpressionRing() is SR
    True

Return True if $x$ is a symbolic variable.

INPUT:
    x -- object
OUTPUT:
    bool -- True precisely if x is a symbolic variable.

EXAMPLES:
    sage: is_SymbolicVariable('x')
    False
    sage: is_SymbolicVariable(x)
    True

Create a symbolic variable with the name \emph{s}.

INPUTS:
    s -- a string, either a single variable name
         or a space or comma separated list of
         variable names

NOTE: sage.calculus.calculus.var is better suited for using var in library
code since it won't touch the global namespace. To create a new variable
in the global namespace, use sage.calculus.var.var. That is, use
sage.calculus.calculus.var when defining a symbolic variable for use in
sage.calculus.calculus functions or library code.

EXAMPLES:
Note that sage.calculus.calculus.var defines a variable which is locally
defined in the calculus namespace but is not globally defined:

    sage: alpha = 42; alpha
    42
    sage: sage.calculus.calculus.var('alpha')
    alpha
    sage: alpha
    42

The variable is still of type SymbolicVariable and belongs to the
symbolic expression ring:

    sage: type(alpha)
    <type 'sage.rings.integer.Integer'>
    sage: type(sage.calculus.calculus.var('alpha'))
    <class 'sage.calculus.calculus.SymbolicVariable'>
    sage: var('beta')
    beta
    sage: type(beta)
    <class 'sage.calculus.calculus.SymbolicVariable'>

TESTS:
    sage: var('xx')
    xx
    sage: var('.foo')
    Traceback (most recent call last):
    ...
    ValueError: variable name is not a valid Python identifier
    sage: var('.foo/x')
    Traceback (most recent call last):
    ...
    ValueError: variable name is not a valid Python identifier

Return True if x is a callable symbolic expression.

INPUT:
    x -- object

OUTPUT:
    bool 

EXAMPLES:
    sage: is_CallableSymbolicExpressionRing(QQ)
    False
    sage: var('x,y,z')
    (x, y, z)
    sage: is_CallableSymbolicExpressionRing(CallableSymbolicExpressionRing((x,y,z)))
    True

Returns true if \var{x} is a callable symbolic expression.

EXAMPLES:
    sage: var('a x y z')
    (a, x, y, z)
    sage: f(x,y) = a + 2*x + 3*y + z
    sage: is_CallableSymbolicExpression(f)
    True
    sage: is_CallableSymbolicExpression(a+2*x)
    False
    sage: def foo(n): return n^2
    ...
    sage: is_CallableSymbolicExpression(foo)
    False

Modified version of arctan function, since it is used by Maxima.

     \code{arctan2(y,x) = arctan(y/x)}

This is mainly for internal use.

TODO: entering 'atan2(1,2)' into Sage returns a  NameError that 'atan2'
is not defined despite the two lines following this function definition.
However, one can enter 'atan(1/2)' with no errors.

EXAMPLES:
    sage: arctan2 = sage.calculus.calculus.arctan2
    sage: arctan2(1,2)
    arctan(1/2)
    sage: float(arctan2(1,2))
    0.46364760900080609
    sage: arctan2(2,3)
    arctan(2/3)

Modified version of arctan function, since it is used by Maxima.

     \code{arctan2(y,x) = arctan(y/x)}

This is mainly for internal use.

TODO: entering 'atan2(1,2)' into Sage returns a  NameError that 'atan2'
is not defined despite the two lines following this function definition.
However, one can enter 'atan(1/2)' with no errors.

EXAMPLES:
    sage: arctan2 = sage.calculus.calculus.arctan2
    sage: arctan2(1,2)
    arctan(1/2)
    sage: float(arctan2(1,2))
    0.46364760900080609
    sage: arctan2(2,3)
    arctan(2/3)

The natural logarithm of x.

INPUT:
    x -- positive real number

OUTPUT:
    ln(x) -- real number

EXAMPLES:
    sage: ln(e^2)
    2 
    sage: ln(2)
    log(2)
    sage: ln(2.0)
    0.693147180559945

Return the logarithm of x to the given base.

Calls the \code{log} method of the object x when computing the logarithm,
thus allowing use of logarithm on any object containing a \code{log}
method. In other words, log works on more than just real numbers.

TODO: Add p-adic log example.

EXAMPLES:
    sage: log(e^2)
    2 
    sage: log(1024, 2); RDF(log(1024, 2))
    log(1024)/log(2)
    10.0
    sage: log(10, 4); RDF(log(10, 4))
    log(10)/log(4)
    1.66096404744

    sage: log(10, 2)
    log(10)/log(2)
    sage: n(log(10, 2))
    3.32192809488736
    sage: log(10, e)
    log(10)
    sage: n(log(10, e))
    2.30258509299405

The log function also works in finite fields as long as the base is
generator of the multiplicative group:
    sage: F = GF(13); g = F.multiplicative_generator(); g
    2
    sage: a = F(8)
    sage: log(a,g); g^log(a,g)
    3
    8
    sage: log(a,3)
    Traceback (most recent call last):
    ...
    ValueError: base (=3) for discrete log must generate multiplicative group

The polylogarithm function $\text{Li}_n(z) = \sum_{k=1}^{\infty} z^k / k^n$.

EXAMPLES:
    sage: polylog(2,1)
    pi^2/6
    sage: polylog(2,x^2+1)
    polylog(2, x^2 + 1)
    sage: polylog(4,0.5)
    polylog(4, 0.500000000000000)
    sage: float(polylog(4,0.5))
    0.51747906167389934

    sage: var('z')
    z
    sage: polylog(2,z).taylor(z, 1/2, 3)
    -(6*log(2)^2 - pi^2)/12 + 2*log(2)*(z - 1/2) + (-2*log(2) + 2)*(z - 1/2)^2 + (8*log(2) - 4)*(z - 1/2)^3/3

The dilogarithm function $\text{Li}_2(z) = \sum_{k=1}^{\infty} z^k / k^2$.

This is simply an alias for polylog(2, z).

EXAMPLES:
    sage: dilog(1)
    pi^2/6
    sage: dilog(1/2)
    pi^2/12 - log(2)^2/2
    sage: dilog(x^2+1)
    polylog(2, x^2 + 1)
    sage: float(dilog(1))
    1.6449340668482264
    sage: var('z')
    z
    sage: dilog(z).diff(z, 2)
    1/((1 - z)*z) + log(1 - z)/z^2
    sage: dilog(z).taylor(z, 1/2, 3)
    -(6*log(2)^2 - pi^2)/12 + 2*log(2)*(z - 1/2) + (-2*log(2) + 2)*(z - 1/2)^2 + (8*log(2) - 4)*(z - 1/2)^3/3

Create a formal symbolic function with the name \emph{s}.

EXAMPLES:
    sage: var('a, b')
    (a, b)
    sage: f = function('cr', a)
    sage: g = f.diff(a).integral(b)
    sage: g
    diff(cr(a), a, 1)*b
    sage: g(cr=cos)
    -sin(a)*b
    sage: g(cr=sin(x) + cos(x))
    (cos(a) - sin(a))*b

Basic arithmetic:
    sage: x = var('x')
    sage: h = function('f',x)
    sage: 2*f
    2*f
    sage: 2*h
    2*f(x)

This function is called to create formal wrappers of limits that
Maxima can't compute:

EXAMPLES:
    sage: a = lim(exp(x^2)*(1-erf(x)), x=infinity); a
    limit(e^x^2 - e^x^2*erf(x), x, +Infinity)
    sage: a = sage.calculus.calculus.dummy_limit(sin(x)/x, x, 0);a 
    limit(sin(x)/x, x, 0)

Given a string representation of a Maxima expression, parse it and
return the corresponding Sage symbolic expression.

INPUT:
    x -- a string
    equals_sub -- (default: False) if True, replace '=' by '==' in self
    maxima -- (default: the calculus package's Maxima) the Maxima
              interpreter to use.

EXAMPLES:
    sage: sage.calculus.calculus.symbolic_expression_from_maxima_string('x^%e + %e^%pi + %i + sin(0)')
    x^e + I + e^pi

Given an element of the calculus copy of the Maxima interface,
create the corresponding Sage symbolic expression.

EXAMPLES:
    sage: a = sage.calculus.calculus.maxima('x^(sqrt(y)+%pi) + sin(%e + %pi)')
    sage: sage.calculus.calculus.symbolic_expression_from_maxima_element(a)
    x^(sqrt(y) + pi) - sin(e)

Given a string expression that makes sense in Maxima, return the
corresponding Sage symbolic expression.  This is used mainly
internally by the Calculus module.

EXAMPLES:
    sage: sage.calculus.calculus.evaled_symbolic_expression_from_maxima_string('2*x + x^3 + y*z*sin(sqrt(x)*erf(theta))')
    sin(erf(theta)*sqrt(x))*y*z + x^3 + 2*x
    sage: sage.calculus.calculus.evaled_symbolic_expression_from_maxima_string('x^%e + %e^%pi + %i')
    x^e + I + e^pi

Return the first variable in expr or `x' if there are no variables
in expression.

EXAMPLES:
    sage: var('a,x,y')
    (a, x, y)
    sage: sage.calculus.calculus.first_var(a + y^x)
    a
    sage: sage.calculus.calculus.first_var(y^x - x^3)
    x

Used internally when creating a string of options to pass to Maxima.

INPUT:
    v -- an object
OUTPUT:
    a string.

The main use of this is to turn Python bools into lower case strings.

EXAMPLES:
    sage: sage.calculus.calculus.mapped_opts(True)
    'true'
    sage: sage.calculus.calculus.mapped_opts(False)
    'false'
    sage: sage.calculus.calculus.mapped_opts('bar')
    'bar'

Used internally to create a string of options to pass to Maxima.

EXAMPLES:
    sage: sage.calculus.calculus.maxima_options(an_option=True, another=False, foo='bar')
    'an_option=true,foo=bar,another=false'

infixops

Value:

{<built-in function add>: '+', <built-in function div>: '/', <built-in function mul>: '*', <built-in function pow>: '^', <built-in function sub>: '-'}

arc_functions

Value:

['asin', 'acos', 'atan', 'asinh', 'acosh', 'atanh', 'acoth', 'asech', ...

cache

Value:

{<class 'sage.calculus.calculus.SymbolicExpressionRing_class'>: Symbol ic Ring}

symbols

Value:

{<built-in function add>: ' + ', <built-in function div>: '/', <built-in function mul>: '*', <built-in function neg>: '-', <built-in function pow>: '^', <built-in function sub>: ' - '}

symtable

Value:

{'%e': 'e', '%gamma': 'euler_gamma', '%i': 'I', '%pi': 'pi', 'li[2]': 'polylog2', 'li[3]': 'polylog3'}

syms_cur

Value:

{'A': A, 'B': B, 'C': C, 'D': D, 'E': E, 'F': F, 'G': G, 'H': H, ...

syms_default

Value:

{'abs': abs, 'acos': arccos, 'acosh': arccosh, 'acot': arccot, 'acoth': arccoth, 'acsc': arccsc, 'acsch': arccsch, 'asec': arcsec, ...

SR_parser

Value:

Parser(make_int= lambda x: SymbolicConstant(Integer(x)), make_float= l ambda x: SymbolicConstant(create_RealNumber(x)), make_var= _find_var, make_function= _find_func)