# nt.quantity concept – Continuing an analytic continuation of the Dirichlet \$eta\$-duty? retort

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## nt.quantity concept – Continuing an analytic continuation of the Dirichlet \$eta\$-duty?

The Dirichlet $$eta$$-duty is outlined as:
$$eta(s) = sum_{n=1}^infty frac{(-1)^{n+1}}{n^s} qquad Re(s) > 0$$

and has the whole analytical continuation:

$$eta(s) = sum_{n=1}^N frac{(-1)^{n+1}}{n^s} + frac{(-1)^N}{2} int_{-infty}^{infty} frac{(N+1/2 +ix)^{-s}}{cosh(pi x)},dx qquad s in mathbb{C} tag{1}$$

precise for all integers $$N ge 0$$.

Wondered what would befall for traverse $$N$$ and create numerically that:

$$eta(s) = sum_{n=1}^{-N-1} frac{(-1)^{n+1}}{n^s} + (-1)^{s+1},frac{(-1)^N}{2} int_{-infty}^{infty} frac{(N+1/2 +ix)^{-s}}{cosh(pi x)},dxqquad s in mathbb{Z} tag{2}$$

precise for all integers $$N < 0$$.

level to: occupy the sums to breathe zero when their stop values are $$< 1$$.

Question:

Is there a path to too broaden equation (2) to $$s in mathbb{C}$$? If workable, I endure this is able to require some mental continuation of the $$(-1)^{s+1}$$ issue. Experimented with features affection $$cosleft(pi(s+1)capable)$$, however no success but.

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