In Calculus, one of the most basic things that can be solved for, is that a principal function receives a parameter, multiplies it by a multiplier, and then passes the product to a nested function, of which either the derivative or the integral can subsequently be found. But what needs to be done over the multiplier, is opposite for integration, from what it was for differentiation. The following two work-sheets illustrate:
Please pardon the poor typesetting of the EPUB File. It’s the result of some compatibility issues (with EPUB readers which do not support EPUB3 that uses MathML.)
This realization also explains how, When the sinc function has been discretized in a certain way and applied as a low-pass filter, I can know that its Nominal Gain, or, its D.C. gain will be close to (2). The assumption which I was making about the low-pass filter was, that the sinc function will make its first zero-crossings near the centre-point, two input samples away, and that it will have an additional zero-crossing every two input samples after that.
This is not how every filter based on the sinc function will be designed; it was only how one specific filter would have been designed.
This means that, a phenomenon which would normally happen over an interval of (π), happens over an interval of (2). Additionally I read, that when the sinc function is Mathematically pure, and has not been translated into Engineering equivalents, its integral approaches (π). Just to be obtuse, the interval of the (Engineering) function’s parameter has been multiplied by (π/2), to arrive at the value which must be fed to the true trig function.
Thus, that half-band filter, that employs 2x over-sampling, will have an integral that approaches (2), not (π).