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3

I believe the reason for the confusion is the way the Dot product operates. It is a Flat but non-commutative operation when the factors aren't just vectors. In particular, it contracts the last index of the first factor with the first index of the next factor, going from left to right. But in the form $a\cdot\varepsilon\cdot b$, this means that the ...

2

The first result can be obtained using Dirichlet regularization: Sum[n, {n, 1, Infinity}, Regularization -> "Dirichlet"] -(1/12) The second can not be obtained, though. I don't have enough smarts to know if this is because it would actually require different regularization, or that Mma just doesn't know how to handle this case.

1

If you look at the output of Solve[Z + Y #1 + X #1^2 - Conjugate[Y] #1^3 + Conjugate[Z] #1^4 &[x] == 0, x, Quartics -> True] you'll see that the solutions in radicals are far more complicated. How do I treat solutions in terms of Root? I leave them that way. Root is nice. It's handier to have solutions ordered by magnitude than by principal values ...

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