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2021-10-28 02:27:46 +0200 received badge  Popular Question (source)
2020-05-10 21:20:08 +0200 commented answer Self composition of a function with symbolic variable

Ahh, my mistake was that I was also declaring z as a variable. Why don't I need to do this? Does sage "know" that this is over the complex field?

2020-05-10 21:16:30 +0200 received badge  Student (source)
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2020-05-10 21:14:19 +0200 asked a question Self composition of a function with symbolic variable

I have a function in one variable with a placeholder variable, and would like to find the nth iterate. For example, using:

$f(z) = z^3 + a z^2$, a function in variable $z$ with placeholder coefficient $a$.

then I would like to see

$f(f(z)) = (z^3 + az^2)^3 + a(z^3 + az^2)^2$

However I am struggling to perform this in Sage.

f(f) returns (a*z^2 + z^3)*z^2 + z^3, so it substitutes for $a$ instead of $z$, and declaring f(f(z)) returns the function only in terms of $z$.

How can we self compose functions with a placeholder variable for a coefficient?