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Summation, which includes both spatial summation and temporal summation, is the process that determines whether or not an action potential will be generated by the combined effects of excitatory and inhibitory signals, both from multiple simultaneous inputs (spatial summation), and from repeated inputs (temporal summation).
The sinc function as audio, at 2000 Hz (±1.5 seconds around zero) In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by = .. Alternatively, the unnormalized sinc function is often called the sampling function, indicated as Sa(x).
When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β. Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas ...
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, is taken to have the value
Illustration of the sum formula. Draw a horizontal line (the x -axis); mark an origin O. Draw a line from O at an angle α {\displaystyle \alpha } above the horizontal line and a second line at an angle β {\displaystyle \beta } above that; the angle between the second line and the x -axis is α + β . {\displaystyle \alpha +\beta .}
This identity stays true by analytic continuation everywhere the integral converges, letting we obtain the Abel–Plana formula = = (() + ()). The case ƒ (0) ≠ 0 is obtained similarly, replacing ∫ a − 1 ∞ a ∞ f ( z ) e − 2 i π z − 1 d z {\textstyle \int _{a^{-1}\infty }^{a\infty }{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz} by two ...
The constants a, b, c, p, q and r (only five of them are independent) can be determined by assuming that the formula must be exactly valid when x = 0, π/6, π/2, π, and further assuming that it has to satisfy the property that sin(x) = sin(π − x). [2] [3] This procedure produces the formula expressed using radian measure of angles.
This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and is also called Euler's formula in this more general case. [1] Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering.