SIafgeleide

SIafgeleide refers to the derivative of the sine integral function. The sine integral Si(x) is defined for real x by Si(x) = ∫_0^x (sin t)/t dt, with Si(0) = 0 and Si(x) an odd function. The SIafgeleide is the derivative d/dx Si(x), which equals sin x / x for x ≠ 0; at x = 0 the derivative equals 1 by the limit, so the derivative can be extended continuously to Si'(0) = 1. The function sin x / x is commonly called the sinc function in its unnormalized form.

This derivative property is a direct consequence of the Fundamental Theorem of Calculus, since Si(x) is defined as an integral with an upper limit x. The SIafgeleide is an even function, since sin x / x is even.

The derivative has a standard power series expansion: Si'(x) = ∑_{n=0}^∞ (-1)^n x^{2n}/(2n+1)!, which converges for all real x. Conversely, Si(x) itself has the series Si(x) = ∑_{n=0}^∞ (-1)^n x^{2n+1}/[(2n+1)(2n+1)!].

As x tends to infinity, Si(x) approaches π/2, while as x tends to negative infinity it approaches -π/2. The derivative Si'(x) tends to 0 as |x| grows large.

Applications of the SIafgeleide appear in areas such as Fourier analysis, wave propagation, and signal processing, where the sinc function and the sine integral play roles in shaping oscillatory integrals and filtering behavior.