# Is there any way to define an as-yet-unknown function?

I'd like to know if there's a way of declaring functions in sage that are as-yet unknown. For example, let's say I have a function

p = R*T/v - a(T)/v/(v+b)


And I would like to be able to take a derivative like this

deriv(p,T)


and be given something back a partial derivative something like

R/v - diff(a(T),T)/v/(v+b)


However at present I can't seem to put an abstract function a(T) into my expression or find anything in the documentation that says how this is done.

As I recall there was a way to do this with wxMaxima, so maybe I just haven't found the trick in Sage yet.

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Sort by » oldest newest most voted Answer from 2012. See further down for 2020 update.

You can define a function using the keyword function. The script below gives the answer you expect.

T = var('T')
a = function('a', T)
R = var('R')
v = var('v')
b = var('b')
p = R*T/v - a(T)/v/(v+b)
diff(p, T)


[2020 update] Nowadays, function no longer takes the variable as an argument.

Use a = function('a') instead of a = function('a', T).

Example:

sage: version()
'SageMath version 9.0, Release Date: 2020-01-01'
sage: b, v, R, T = SR.var('b v R T')
sage: a = function('a')
sage: p = R*T/v - a(T)/v/(v+b)
sage: diff(p, T)
R/v - diff(a(T), T)/((b + v)*v)

more

Great. seems to work fine. The output is a bit weird though, in typeset sage, the diff(a(T),T) is written as D [ 0 ] ( a ) ( T ), not particular intuitive. But it works! Thank you.