could we just invent new sets of functions to help ease symbolic integration?

[Al-Layth]

currently we have
polynomials,
trigonometric functions
hyperbolic functions
exponential functions

and their reciprocals and inverses as the standard library of "elementary functions" and there are many integrals quite difficult to evaluate

would it make it any easier if we defined new function sets, based on operations of these functions as tools, developed their algebraic identities and derivatives/integrals for the purpose of making the evaluation of various integrands (that would be expressed in just the elementary functions) easier?

i envision a world where I trivially evaluate a difficult looking integral with my invented function sets, then when its evaluated I use the identities between my invented functions and the standard elementary functions to express them in terms of elementary functions and voila ?

as ai type this i realise this sounds a lot like u substitution with extra steps :/

thoughts?

 

[topsquark]

currently we have
polynomials,
trigonometric functions
hyperbolic functions
exponential functions

and their reciprocals and inverses as the standard library of "elementary functions" and there are many integrals quite difficult to evaluate

would it make it any easier if we defined new function sets, based on operations of these functions as tools, developed their algebraic identities and derivatives/integrals for the purpose of making the evaluation of various integrands (that would be expressed in just the elementary functions) easier?

i envision a world where I trivially evaluate a difficult looking integral with my invented function sets, then when its evaluated I use the identities between my invented functions and the standard elementary functions to express them in terms of elementary functions and voila ?

as ai type this i realise this sounds a lot like u substitution with extra steps :/

thoughts?

The problem is that last step, when you use your identities. What identities? If you defining a new function to solve a problem that can't be solved using a given set, it is very likely that there are no simple (if any) such identities.

For example
[imath]\int_0^5 e^{-x^2} \, dx = D(5) - D(0)[/imath]
where D(x) is the "Dan" function.

How do we transform D(x) into more elementary functions?

-Dan

 

[Al-Layth]

The problem is that last step, when you use your identities. What identities? If you defining a new function to solve a problem that can't be solved using a given set, it is very likely that there are no simple (if any) such identities.

For example
[imath]\int_0^5 e^{-x^2} \, dx = D(5) - D(0)[/imath]
where D(x) is the "Dan" function.

How do we transform D(x) into more elementary functions?

-Dan

Those identities would be the DEFINITIONS of the new functions. They would be defined based on operations between the standard library of elementary function, for example ( lets suppose hyperboloc functions are not part of the standard elementaries:

then we may see exponentials and write them in terms of Hyperbolics…if it helped in simplifying thr integral. We would then use the
(1) algebraic identities
(2) derived derivatives and integrals for the various hyperbolic functions

to help us evaluate the integral. Then we would use the definitions to convert them
Back into exponential form.

thats what Im thinking of, but with a newly invented set of functions of course defined with the objective of easing symbolic
Integration.

 

[topsquark]

Those identities would be the DEFINITIONS of the new functions. They would be defined based on operations between the standard library of elementary function, for example ( lets suppose hyperboloc functions are not part of the standard elementaries:

then we may see exponentials and write them in terms of Hyperbolics…if it helped in simplifying thr integral. We would then use the
(1) algebraic identities
(2) derived derivatives and integrals for the various hyperbolic functions

to help us evaluate the integral. Then we would use the definitions to convert them
Back into exponential form.

thats what Im thinking of, but with a newly invented set of functions of course defined with the objective of easing symbolic
Integration.

My point is, how do you actually do all of that. Most of these functions that you would be creating would be infinite series. There's no problem with that, and this has already been done with Legendre polynomials, Laguerre polynomials, confluent hypergeometric functions, etc. But you can't use these to solve for a more basic form. The best you can hope for is to use them to generate a numerical solution.

For example, one of these is actually considered elementary: sin(x). It is essentially a numerically calculated function. But how do you use it to find a, say, polynomial solution to an equation? Unless your problem happens to be just the right type, this is impossible.

-Dan