Chapter 3
How Far Away Is Infinity?

§3.3.1 A Dead End: Calculating a Closed Form for (A^T) A

Session (p. 57).

g[j_, 
        l_] := (∑_(k = 1)^∞) (4/(((k^2) + (2 j - 1) k + (j - 1) (j - 2)) ((k^2) + (2 l - 1) k + (l - 1) (l - 2)))) // FullSimplify
g[3, 3]

§3.5 Summation Formulas: The Real Case

An exponential summation formula (p. 65).

ϕ[
          u_] := (E^((2/(((1 - u))^2)) - (1/(2 (u^2)))));
n = 60; 
    k = N[Range[n - 1, 1, -1]];
c = 
        Append[k + ϕ[(k/n)] + 1, 1];
w=((Append[1+(n^(-1))ϕ'[(k/n)],1])^(1/2));
A = 
        Table[((w[[j]] w[[k]])/((c[[j]] + c[[k]] - 1) (c[[j]] + c[[k]])/2 - (c[[k]] - 1))), {j, n}, {k, n}];
Norm[A]

§3.6 Summation Formulas: The Complex Case

Example (p. 67).

f[z_] := (z^(-4));
Z[t_] := (1 - I t)/2;
FullSimplify[
        Re[(1/(2  I)) f[Z[t]] Cot[π Z[t]] Z'[t]] // 
          ComplexExpand, {t ∈ Reals}] // TraditionalForm