What exactly does RootSum mean?

I have difficulties understanding what RootSum object is mathematically, as it is not really well explained anywhere. It came up in a solution to a quite complicated system of differential equations in my Code. The normal "Root" is just some representation of the solution f==0 right? If I use Normal, you can turn RootSum into a function of Roots. for example in my case:

Left of the comma is f and the "1" on the right signifies it is the first root?

Now RootSum[f, form] depends on 2 functions, how do I have to understand this notation to get to a Number as an output? Especially what "form[x]" means.

• Have you read the document of RootSum, especially the examples under Scope section?: reference.wolfram.com/language/ref/RootSum.html Commented May 16 at 9:22
• To be more specific, I mean the 6 examples following the line "Sums over roots of polynomials with inexact number coefficients". Commented May 16 at 12:23

RootSum[#^2 - 1 &, f] // FullSimplify
(*f[-1] + f[1]*)

RootSum[#^3 - 1 &, Sin] // ToRadicals
(*Sin[1] - Sin[(-1)^(1/3)] + Sin[(-1)^(2/3)]*)


You can see that RootSum[f, form] returns form[x1]+form[x2]+... where x1,x2,... are the root of f[x]==0

• Better use Normal[] to convert RootSum to Root objects.
– Acus
Commented May 16 at 10:40
• Thank you, this clarified it. I tried FullSimplify but f is too complicated, so nothing happens. So this did not become clear to me so far. Commented May 16 at 12:15