knitr::opts_chunk$set(echo = TRUE)
library(Ryacas)
yacas
comes with a number of rules all defined in the yacas
directory of the installed package:
system.file(package = "Ryacas", "yacas")
For example in the sums.rep
folder, a number of rules for sums are defined in the code.ys
file.
As an example, the fact that
[
\sum_{k = 1}^n (2k-1) = n^2
]
is defined in yacas
as
SumFunc(_k,1,_n,2*_k-1, n^2 );
and the geometric sum is defined as
SumFunc(_k,0,_n,(r_IsFreeOf(k))^(_k), (1-r^(n+1))/(1-r) );
These can be verified:
yac_str("Sum(i, 1, m, 2*i-1)") yac_str("Sum(i, 0, m, 2^i)")
There are also rules in yacas
that are able to let the user change some limits of some sums, e.g. for the geometric sum:
yac_str("Sum(i, 1, m, 2^i)")
But what about changing the limit of the first sum? I.e. instead of
[
\sum_{k = 1}^n (2k-1) = n^2
]
then know that
[
\sum_{k = 0}^n (2k-1) = -1 + \sum_{k = 1}^n (2k-1) = n^2 - 1 .
]
But what does yacas
say?
yac_str("Sum(i, 0, m, 2*i-1)")
We can then add our own rule by:
yac_silent("SumFunc(_k,0,_n,2*_k-1, n^2 - 1)")
And then try again:
yac_str("Sum(i, 0, m, 2*i-1)")
A good source of inspiration for writing custom rules is reading the included rules,
but there is a lot to programming in yacas
and we refer to yacas
's documentation,
specifically the chapter Programming in Yacas.
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