I have a simple recursive function:
$f(i+1)=\frac{1}{2}f(i)+\frac{1}{2}a$
both f(i) and a are numbers between 0 and 1, while a can change its value while iterating over the function f.
So lets say, this function is calculated once in a time interval $dt_0$ (thats the baseline), how would I need to change the factors $\frac{1}{2}$, so that for a given time t, the value of f is always the same, even if $dt$ is changing during the iterations.
So basically I want a smooth asymptotic transition from $f$ to $a$ during a time interval $dt_0$, but it has to be independet on $dt$, for $dt$ smaller and larger $dt_0$
Many thanks in advance and best regards