# Just define any, real function

I know a way to define any function (say, a (x)) and then apply an assumption to it (say, a (x) is real). My code will be inserted below.

``````
A(x_) = Function(x, a(x))
Assumptions(A(x) (Element) Reals

In(216):= Integrate(x A(x), {x, -1, 1}) // FullSimplify
Out(216)= 0
``````

However, it becomes very "chaotic". When I try to define operators for arbitrary functions, the output is as follows:

``````    Function(x, a(x))
``````

``````    a(x)
``````

Is there a better way?

If what I just wrote is not detailed enough, here is the exact code:

``````    In(197):= Clear("Global`*")

In(198):= (*-v- Arbitrary Real functions*)

In(199):= (Alpha)(x_) = Function(x, a(x))

Out(199)= Function(x, a(x))

In(200):= (Beta)(x_) = Function(x, b(x));

In(201):= (Gamma)(x_) = Function(x, cc(x));

In(202):= (Rho)(x_) = Function(x, r(x));

In(203):= f(n_, x_) = Function({x, n}, ff(x, n));

In(204):= F(n_, x_, y_) = E^(I n y) f(n, x);

In(205):= (*-^- Arbitrary Real functions*)

In(206):= (*-v- Differential Operators*)

In(207):= (Lambda)n(n_,
x_) = ((Alpha)(x)*D(#, {x, 1}) + (Beta)(x) + n*(Rho)(x)) &;

In(208):= (Mu)n(n_,
x_) = ((Alpha)(x)*D(#, {x, 1}) + (Gamma)(x) - n*(Rho)(x)) &;

In(220):=
Tp(x_, y_) = (E^(
I y) ((Alpha)(x)*D(#, {x, 1}) + (Beta)(x) -
I (Rho)(x)*D(#, {y, 1}))) &;

In(210):=
Tm(x_, y_) = (-E^(-I y) ((Alpha)(x)*D(#, {x, 1}) + (Gamma)(x) +
I (Rho)(x)*D(#, {y, 1}))) &;

In(211):= T0(y_) = (-I D(#, {y, 1})) &;

In(212):= (*-^- Differential Operators*)

In(213):= (*-v- Assumptions*)

In(214):=
Assumptions((Alpha)(x) (Element) Reals, (Beta)(x) (Element)
Reals, (Gamma)(x) (Element) Reals, (Rho)(x) (Element) Reals,
F(n, x) (Element) Complexes);

In(215):= (*-^- Assumptions*)

In(217):= Tm(x, y)(Tp(x, y)(F(n, x, y))) // FullSimplify

Out(217)= -E^(-I y) Function(x, cc(x)) +
Function(x, b(x)) Function(x, r(x)) +
E^(I n y) (-(Function(x, a(x))^2 -
Function(x, a(x)) Function(x,
r(x)) + (1 + I) (1 + (1 + I) n) Function(x,
r(x))^2) Function({x, n}, 0) +
n (1 + n) Function(x, r(x))^2 Function({x, n}, ff(x, n))) -
Function(x,
0) (Function(x, a(x)) + I Function(x, r(x))) (1 +
E^(I n y) ((1 - I) Function({x, n}, 0) +
n Function({x, n}, ff(x, n))))
$$```$$
``````