Since $\sigma_c = \inf s$, $\eta \leq \sigma_c$. In older texts, the abscissa is sometimes used in a more general sense as a number determining the position of a point along a line.
Physicists and astronomers sometimes use the term to refer to the axis itself instead of the distance along it.
Mathematical definition of abscissa series#
Hardy and Riesz, for instance, give as an example on page 9 the series $\Sigma (-1)^n (\log n)^+s=s,$$ The - (horizontal) coordinate of a point in a two dimensional coordinate system. The situation is much more subtle in the context of Dirichlet series, however-a power series converges absolutely or diverges except possibly on a subset of a single circle, where it may converge conditionally (I think though that that subset can be quite complicated) a Dirichlet series, on the other hand, may converge conditionally (and not absolutely) on a strip or even on the entire plane. Basically, this formula is an analogue of the root test for the convergence of a power series (in fact, in the case $\lambda_n = n$ the two tests coincide). Here are a few remarks (mostly paraphrasing that book) to clarify the import of these formulas for other people who stumble across this question and wonder about its motivation. Riesz's The General Theory of Dirichlet's Series. As I mentioned in a comment, this proof is a simple application of ideas found on page 7 of Hardy & M. This x-coordinate is called the abscissa. For point P, it is +5 on the positive x-axis. It is measured along the x-axis which is positive along the positive direction and negative along the negative direction.
Here is a proof of the remaining inequality (the one not proved in your answer, ireallydonknow). In the cartesian system, the x-coordinate of a point is its perpendicular distance from the y-axis. I apologize for the delay in writing this up. (geometry) The first of the two terms by which a point is referred to, in a system of fixed rectilinear coordinate (Cartesian coordinate) axes.
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