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# Symbolic functions

`sage.symbolic.function_factory.``deprecated_custom_evalf_wrapper`(func)

  This is used while pickling old symbolic functions that define a custom evalf method.

  The protocol for numeric evaluation functions was changed to include a `parent` argument instead of `prec`. This function creates a wrapper around the old custom method, which extracts the precision information from the given `parent`, and passes it on to the old function.

  EXAMPLES:

      sage: from sage.symbolic.function_factory import deprecated_custom_evalf_wrapper as dcew
      sage: def old_func(x, prec=0): print "x: %s, prec: %s"%(x,prec)
      sage: new_func = dcew(old_func)
      sage: new_func(5, parent=RR)
      x: 5, prec: 53
      sage: new_func(0r, parent=ComplexField(100))
      x: 0, prec: 100

`sage.symbolic.function_factory.``eval_on_operands`(f)

  Given a method `f` return a new method which takes a single symbolic expression argument and appends operands of the given expression to the arguments of `f`.

  EXAMPLES:

      sage: def f(ex, x, y):
      ....:     '''
      ....:     Some documentation.
      ....:     '''
      ....:     return x + 2*y
      ....:
      sage: f(None, x, 1)
      x + 2
      sage: from sage.symbolic.function_factory import eval_on_operands
      sage: g = eval_on_operands(f)
      sage: g(x + 1)
      x + 2
      sage: g.__doc__.strip()
      'Some documentation.'

`sage.symbolic.function_factory.``function`(s, *args, **kwds)

  Create a formal symbolic function with the name s.

  INPUT:

  - `args` - arguments to the function, if specified returns the new function evaluated at the given arguments
  - `nargs=0` - number of arguments the function accepts, defaults to variable number of arguments, or 0
  - `latex_name` - name used when printing in latex mode
  - `conversions` - a dictionary specifying names of this function in other systems, this is used by the interfaces internally during conversion
  - `eval_func` - method used for automatic evaluation
  - `evalf_func` - method used for numeric evaluation
  - `evalf_params_first` - bool to indicate if parameters should be evaluated numerically before calling the custom evalf function
  - `conjugate_func` - method used for complex conjugation
  - `real_part_func` - method used when taking real parts
  - `imag_part_func` - method used when taking imaginary parts
  - `derivative_func` - method to be used for (partial) derivation This method should take a keyword argument deriv_param specifying the index of the argument to differentiate w.r.t
  - `tderivative_func` - method to be used for derivatives
  - `power_func` - method used when taking powers This method should take a keyword argument power_param specifying the exponent
  - `series_func` - method used for series expansion This method should expect keyword arguments - `order` - order for the expansion to be computed - `var` - variable to expand w.r.t. - `at` - expand at this value
  - `print_func` - method for custom printing
  - `print_latex_func` - method for custom printing in latex mode

  Note that custom methods must be instance methods, i.e., expect the instance of the symbolic function as the first argument.

  EXAMPLES:

      sage: var('a, b')
      (a, b)
      sage: f = function('cr', a)
      sage: g = f.diff(a).integral(b)
      sage: g
      b*D[0](cr)(a)
      sage: foo = function("foo", nargs=2)
      sage: x,y,z = var("x y z")
      sage: foo(x, y) + foo(y, z)^2
      foo(y, z)^2 + foo(x, y)

  In Sage 4.0, you need to use `substitute_function()` to replace all occurrences of a function with another:

      sage: g.substitute_function(cr, cos)
      -b*sin(a)

      sage: g.substitute_function(cr, (sin(x) + cos(x)).function(x))
      b*(cos(a) - sin(a))

  In Sage 4.0, basic arithmetic with unevaluated functions is no longer supported:

      sage: x = var('x')
      sage: f = function('f')
      sage: 2*f
      Traceback (most recent call last):
      ...
      TypeError: unsupported operand parent(s) for '*': 'Integer Ring' and '<class 'sage.symbolic.function_factory.NewSymbolicFunction'>'

  You now need to evaluate the function in order to do the arithmetic:

      sage: 2*f(x)
      2*f(x)

  We create a formal function of one variable, write down an expression that involves first and second derivatives, and extract off coefficients.

      sage: r, kappa = var('r,kappa')
      sage: psi = function('psi', nargs=1)(r); psi
      psi(r)
      sage: g = 1/r^2*(2*r*psi.derivative(r,1) + r^2*psi.derivative(r,2)); g
      (r^2*D[0, 0](psi)(r) + 2*r*D[0](psi)(r))/r^2
      sage: g.expand()
      2*D[0](psi)(r)/r + D[0, 0](psi)(r)
      sage: g.coeff(psi.derivative(r,2))
      1
      sage: g.coeff(psi.derivative(r,1))
      2/r

  Defining custom methods for automatic or numeric evaluation, derivation, conjugation, etc. is supported:

      sage: def ev(self, x): return 2*x
      sage: foo = function("foo", nargs=1, eval_func=ev)
      sage: foo(x)
      2*x
      sage: foo = function("foo", nargs=1, eval_func=lambda self, x: 5)
      sage: foo(x)
      5
      sage: def ef(self, x): pass
      sage: bar = function("bar", nargs=1, eval_func=ef)
      sage: bar(x)
      bar(x)

      sage: def evalf_f(self, x, parent=None, algorithm=None): return 6
      sage: foo = function("foo", nargs=1, evalf_func=evalf_f)
      sage: foo(x)
      foo(x)
      sage: foo(x).n()
      6

      sage: foo = function("foo", nargs=1, conjugate_func=ev)
      sage: foo(x).conjugate()
      2*x

      sage: def deriv(self, *args,**kwds): print args, kwds; return args[kwds['diff_param']]^2
      sage: foo = function("foo", nargs=2, derivative_func=deriv)
      sage: foo(x,y).derivative(y)
      (x, y) {'diff_param': 1}
      y^2

      sage: def pow(self, x, power_param=None): print x, power_param; return x*power_param
      sage: foo = function("foo", nargs=1, power_func=pow)
      sage: foo(y)^(x+y)
      y x + y
      (x + y)*y

      sage: def expand(self, *args, **kwds): print args, kwds; return sum(args[0]^i for i in range(kwds['order']))
      sage: foo = function("foo", nargs=1, series_func=expand)
      sage: foo(y).series(y, 5)
      (y,) {'var': y, 'options': 0, 'at': 0, 'order': 5}
      y^4 + y^3 + y^2 + y + 1

      sage: def my_print(self, *args): return "my args are: " + ', '.join(map(repr, args))
      sage: foo = function('t', nargs=2, print_func=my_print)
      sage: foo(x,y^z)
      my args are: x, y^z

      sage: latex(foo(x,y^z))
      t\left(x, y^{z}\right)
      sage: foo = function('t', nargs=2, print_latex_func=my_print)
      sage: foo(x,y^z)
      t(x, y^z)
      sage: latex(foo(x,y^z))
      my args are: x, y^z
      sage: foo = function('t', nargs=2, latex_name='foo')
      sage: latex(foo(x,y^z))
      foo\left(x, y^{z}\right)

  Chain rule:

      sage: def print_args(self, *args, **kwds): print "args:",args; print "kwds:",kwds; return args[0]
      sage: foo = function('t', nargs=2, tderivative_func=print_args)
      sage: foo(x,x).derivative(x)
      args: (x, x)
      kwds: {'diff_param': x}
      x
      sage: foo = function('t', nargs=2, derivative_func=print_args)
      sage: foo(x,x).derivative(x)
      args: (x, x)
      kwds: {'diff_param': 0}
      args: (x, x)
      kwds: {'diff_param': 1}
      2*x

  TESTS:

  Make sure that trac ticket #15860 is fixed and whitespaces are removed:

      sage: function('A, B')
      (A, B)
      sage: B
      B
      sage: C, D, E = function(' C  D E')
      sage: C(D(x))
      C(D(x))
      sage: E
      E

`sage.symbolic.function_factory.``function_factory`(name, nargs=0, latex_name=None, conversions=None, evalf_params_first=True, eval_func=None, evalf_func=None, conjugate_func=None, real_part_func=None, imag_part_func=None, derivative_func=None, tderivative_func=None, power_func=None, series_func=None, print_func=None, print_latex_func=None)

  Create a formal symbolic function. For an explanation of the arguments see the documentation for the method `function()`.

  EXAMPLES:

      sage: from sage.symbolic.function_factory import function_factory
      sage: f = function_factory('f', 2, '\\foo', {'mathematica':'Foo'})
      sage: f(2,4)
      f(2, 4)
      sage: latex(f(1,2))
      \foo\left(1, 2\right)
      sage: f._mathematica_init_()
      'Foo'

      sage: def evalf_f(self, x, parent=None, algorithm=None): return x*.5r
      sage: g = function_factory('g',1,evalf_func=evalf_f)
      sage: g(2)
      g(2)
      sage: g(2).n()
      1.00000000000000

`sage.symbolic.function_factory.``unpickle_function`(name, nargs, latex_name, conversions, evalf_params_first, pickled_funcs)

  This is returned by the `__reduce__` method of symbolic functions to be called during unpickling to recreate the given function.

  It calls `function_factory()` with the supplied arguments.

  EXAMPLES:

      sage: from sage.symbolic.function_factory import unpickle_function
      sage: nf = unpickle_function('f', 2, '\\foo', {'mathematica':'Foo'}, True, [])
      sage: nf
      f
      sage: nf(1,2)
      f(1, 2)
      sage: latex(nf(x,x))
      \foo\left(x, x\right)
      sage: nf._mathematica_init_()
      'Foo'

      sage: from sage.symbolic.function import pickle_wrapper
      sage: def evalf_f(self, x, parent=None, algorithm=None): return 2r*x + 5r
      sage: def conjugate_f(self, x): return x/2r
      sage: nf = unpickle_function('g', 1, None, None, True, [None, pickle_wrapper(evalf_f), pickle_wrapper(conjugate_f)] + [None]*8)
      sage: nf
      g
      sage: nf(2)
      g(2)
      sage: nf(2).n()
      9.00000000000000
      sage: nf(2).conjugate()
      1

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