# Computing with the residue in Sagemath

I have done the following computation to calculate $W12(z1,z2)$ I am expecting the result

$$ \begin{aligned} W_{1,2}(z_1 , z_2) &= Res_{z =0}\, K(z)\frac{1}{z-z_1} \left( W_{0,3}(z,-z,z_2) + W_{0,2}(z,z_2)W_{1,1}(-z)+ W_{0,2}(-z,z_2)W_{1,1}(z)\right) \\ &= 1/8\, {\frac { \left( 2\,a{{\it z1}}^{2}+5\,{a}^{2} \right) {{\it z2}}^ {4}+ \left( 2\,a{{\it z1}}^{4}+3\,{a}^{2}{{\it z1}}^{2} \right) {{\it z2}}^{2}+5\,{a}^{2}{{\it z1}}^{4}}{{{\it z1}}^{6}{{\it z2}}^{6}}} \end{aligned} $$ I got the correct result for W11, W03 with the code. For computing W12 I have to use the subsitution function, but at the end I not getting the correct result. I got the result I have computed in Maple. So please let me know what mistake I am making in the following code.

```
var('z,b,z1,z2,z3');
assume(b > 0);
y = lambda z: 2*arcsinh( z / (sqrt(2)*b) ) / sqrt(z^2 + 2*b^2)
K = lambda z: 1 / (z*(y(z) - y(-z)))
W02 = lambda z1, z2: 1/(z1 - z2)^2
W11 = lambda z1: ( K(z) / (z - z1) * (W02(z1,z2).substitute(z1==z, z2==-z))*derivative(-z,z) ).residue(z == 0).canonicalize_radical()
W03 = lambda z1, z2, z3 : ( K(z) / (z - z1) *( W02(z, z2)*W02(-z,z3)+W02(z, z3)*W02(-z,z2))*derivative(-z,z)
).residue(z == 0).canonicalize_radical()
W12 = lambda z1, z2: ( K(z) / (z - z1) *(W03(z1,z2,z2).substitute(z1==z , z2==-z, z3 ==z2)
+W02(z1, z2).substitute(z1==z)*W11(z1).substitute(z1==-z)
+W02(z1, z2).substitute(z1==-z)*W11(z1).substitute(z1==z))
* derivative(-z,z)).residue(z == 0).canonicalize_radical()
```

Would you mind

defining$W_{i, j}(z_1, z_2)$ ?It's a recursive definition $W_{g,n}(z_1 , \ldots ,z_n) = Res_{z=0} \left ( W_{g-1 , n+1}(z,-z,z2,\ldots z_n) +\sum_{g_1 + g_2 = g , n_1 + n_2 = n-1 }W_{g1 , n1}()W_{g_2, n_2}() \right)$$. I writing the recursive code in general. So it might be more with that progarmme, but I am stuck here.

Your definition isn't complete :

It doesn't give the initial definitions for starting values of $g$ and $n$.

it doesn't specify the use of the arguments $z_1,\dots z_p$.

The term $W_{g-1 , n+1}(z,-z,z2,\ldots z_n)$ isn't specially clear, introducing a new argument $z$ and missing (?) $z_1$.

By $\sum_{g_1 + g_2 = g , n_1 + n_2 = n-1 }W_{g1 , n1}()W_{g_2, n_2}() $, do you mean

$ \sum_{g_1 + g_2 = g} \sum_{n_1 + n_2 = n-1 } W_{g1 , n1} (z_1,\dots z_p)W_{g_2, n_2}(z_1,\dots z_p) $

or something else ?

In the latter, what about $z$ ?

This might be due to LaTeX mistyping or cunt n' paste mishaps. Could you clarify by writing

completedefinitions, avoiding any abbreviation ? I'm not especially good at divination...In my thesis https://bridges.monash.edu/articles/t... Page 30 eq 3.6 have the expression So in this case $x(z) = z^2$ $y(z)$ as I have defined in the code. $k(z)$ I have defined in the code Page 31 I gave a picture to make the understanding clear.

So the initial data given is $w_{0,1}$ and $w_{0,2}$ and the rest we compute by the recursion. I am writing a code for this recursion. I have computed it in the maple files, I don't know how to share it here.