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The normalization causes the definite integral of the function over the real numbers to equal 1 (whereas the same integral of the unnormalized sinc function has a value of π). As a further useful property, the zeros of the normalized sinc function are the nonzero integer values of x.
Generally, if the function is any trigonometric function, and is its derivative, ∫ a cos n x d x = a n sin n x + C {\displaystyle \int a\cos nx\,dx={\frac {a}{n}}\sin nx+C} In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration .
Since sinc is an even entire function (holomorphic over the entire complex plane), Si is entire, odd, and the integral in its definition can be taken along any path connecting the endpoints. By definition, Si( x ) is the antiderivative of sin x / x whose value is zero at x = 0 , and si( x ) is the antiderivative whose value is zero at x = ∞ .
In mathematics, a Borwein integral is an integral whose unusual properties were first presented by mathematicians David Borwein and Jonathan Borwein in 2001. [1] Borwein integrals involve products of (), where the sinc function is given by = / for not equal to 0, and =.
The sinc function is, however, integrable in the sense of the improper Riemann integral or the generalized Riemann or Henstock–Kurzweil integral. [ 1 ] [ 2 ] This can be seen by using Dirichlet's test for improper integrals .
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In numerical analysis and applied mathematics, sinc numerical methods are numerical techniques [1] for finding approximate solutions of partial differential equations and integral equations based on the translates of sinc function and Cardinal function C(f,h) which is an expansion of f defined by