Slightly more meat edit : I want to solve this integral
$$ J_{k=1} = \int dt \sqrt(2 t^6 - 2 t^5 - t^4 + t^2 + 1) - 1 $$ I know about some approximation schemes, but I want to grow bigger so I am venturing to learn what seasoned mathematicians would do. The below polynomial is cute
$$2 t^6 - 2 t^5 - t^4 + t^2 + 1$$ So I have decided to think about it for a while. I feel like may be if I can factor this somehow things will be great because I can then be able to may be split rewrite the integral in some fashion.
So I have gone the route of thinking of checking what sort of roots are possible and so forth. I had some conversation and did deep self reflection about the possibility of looking at splitting fields, and Galois groups and so forth. A few non-trivial things came up. I am considering this route because It will be cute to write a paper with some Hasse lattice. However the only lattice I have now is a baby version associated with complex roots.
I am hoping that I will then be able to factor out a "square" then reduce the rest to may be a cubic, then apply standard elliptic integration tricks which I have already practiced. May be I can also then have yet another lattice. A lattice of poles.
Dirty Numerics :: Silly complex games (via well known algebra theorem hehe) ::: roots $$ 0.64830 + 0.59502i$$ $$ 0.64830 - 0.59502i$$ $$-0.82428 + 0.53732i$$ $$-0.82428 - 0.53732i$$ $$-0.32402 + 0.74964i$$ $$-0.32402 - 0.74964i$$
Can I even write the above in a symbolic fashion, I CAS(ed) these out after some thinking.
Edit :
Consider a lattice with basis $w$ and $w'$ and some transformation between them governed by a $2 \times 2$ matrix. One may use standard techniques to "accidentally stumble"(or just follow well-known algorithms for this) on the properties that can be easily found on wikipedia. At any rate. This is a somewhat cool story.
Even cooler is the fact that there is some lattice out there $K$ for a meromorphic function $f(z)$
notwithstanding
$$z_i \sim z_j;$$
$$z_i-z_j \in K$$
The new story I am touching on has a lot of parts. I want to know how one passes from an $f(z)$ to a $K$ You may call the lattice Kevin for simplicity :).
How do Weierstrass functions come into the picture? Would be nice to know
Version 0.1
I have an integral in Jacobi form. It is something that can be reduced to an Jacobi elliptic integral of the second kind, or some sum of such beasts. This is somewhat cool. What I want to do is turn this into Weierstrass form. I am considering doing this for a few reasons, one of them is that I want to be able to draw and talk about the associated lattice. Can someone
a) sketch for me the general algorithm for doing the transformation between the forms. I noticed there is a distinction between Weierstrass form that specifies "periods" and that which specifies "invariants" see for example this link : http://mathworld.wolfram.com/WeierstrassEllipticFunction.html
b) How does the lattice come in?
Think of me as a struggling engineer (really a baby brain) attempting to learn basic math. Is there an extremely simple example of this done somewhere and what would be a good reference for this?