Defining new operators with specific rules

edit

The following code seems to do the job. However, it looks very clumsy and I would be very happy to receive comments.

##################################################################
from sage.symbolic.operators import add_vararg, mul_vararg
##################################################################
def efdd(self, x):
    # x is a number:
    if x in RR or (I*x) in RR:
        return 0

    # Exponential exp(a)
    elif x.operator()==exp:
        return dd((log(x)).simplify())*x

    # Logarithm log(a)
    elif x.operator()==ln:
        return (1/exp(x))*dd(exp(x))

    # Single symbolic input
    elif x.is_symbol():
        if x.is_integer() or (I*x).is_integer() or x in RR or (I*x) in RR:
            return 0
        else:
            pass
    elif (-1*(x)).is_symbol():
        if x.is_integer() or (I*x).is_integer() or x in RR or (I*x) in RR:
            return 0
        else:
            return -dd(-x)

    # Power in input with plus (a1^c1)
    elif x.operator() == operator.pow:
        try:
            a, b = x.operands()
            if a.is_integer() or a in RR or (I*a) in RR:
                return a*(b^(a-1))*dd(b)
            else:
                return b*(a^(b-1))*dd(a)
        except:
            a, b, c = x.operands()
            if a.is_integer() or (I*a).is_integer() or a in RR or (I*a) in RR:
                return a*(b^(a-1))*dd(b)
            else:
                return b*(a^(b-1))*dd(a)

    # Power in input with minus (-a1^c1)
    elif (-1*x).operator() == operator.pow:
        try:
            a, b = (-1*x).operands()
            if a.is_integer() or (I*a).is_integer() or a in RR or (I*a) in RR:
                return -a*(b^(a-1))*dd(b)
            else:
                return -b*(a^(b-1))*dd(a)
        except:
            a, b, c = (-1*x).operands()
            if a.is_integer() or (I*a).is_integer() or a in RR or (I*a) in RR:
                return -a*(b^(a-1))*dd(b)
            else:
                return -b*(a^(b-1))*dd(a)

    # Summation in input
    elif x.operator() == add_vararg:
        oplist = x.operands()
        return sum(dd(op) for op in oplist)

    # Multiplication in input
    elif x.operator() == mul_vararg:
        ops=x.operands()
        result=0
        for i in range(len(ops)):
            prod=1
            for j in range(len(ops)):
                if i!=j:
                    prod=prod*ops[j]
            result=result+dd(ops[i])*prod
        return result

##################################################################
dd = function("dd",latex_name=r"\Delta",eval_func=efdd)
##################################################################

var('a1,a2,a3,a4')
var('c1,c2,c3')
assume(c1,'integer')
assume(c2,'integer')
assume(c3,'integer')

show("dd(a1+a2+a3)=",dd(a1+a2+a3))
show("dd(-a1+a2-a3)=",dd(-a1+a2-a3))
show("dd(a1*a2)=",dd(a1*a2))
show("dd(a1*a2-a3*a4)=",dd(a1*a2-a3*a4))
show("dd(c1*a1)=",dd(c1*a1))
show("dd(-c1*a2)=",dd(-c1*a2))
show("dd(a1^c1)=",dd(a1^c1))
show("dd(-a1^c1)=",dd(-a1^c1))
show("dd(1/a1)=",dd(1/a1))
show("dd(-1/a1)=",dd(-1/a1))
show("dd(1/(a1^2))=",dd(1/(a1^2)))
show("dd(c1*a1*a2)=",dd(c1*a1*a2))
show("dd(a1*a2*a3*a4)=",dd(a1*a2*a3*a4))
show("dd(c1*a2*a3*a4)=",dd(c1*a2*a3*a4))