Sine integral function The function Si(x) = ∫₀ˣ f(t) dt, where f(t) = {(sin t)/t if t ≠ 0, 1 if t = 0, is called the sine integral function.
b. Integrate the series to find a Taylor series for Si.

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Recall that the sine integral function is defined as \(\mathrm{Si}(x) = \int_0^x f(t) \, dt\), where \(f(t) = \frac{\sin t}{t}\) for \(t \neq 0\) and \(f(0) = 1\).

Start by expressing \(\sin t\) as its Taylor series expansion around \(t=0\): \(\sin t = \sum_{n=0}^\infty (-1)^n \frac{t^{2n+1}}{(2n+1)!}\).

Divide the series for \(\sin t\) by \(t\) to get the series for \(f(t) = \frac{\sin t}{t}\): \(f(t) = \sum_{n=0}^\infty (-1)^n \frac{t^{2n}}{(2n+1)!}\).

Integrate the series term-by-term from 0 to \(x\) to find the Taylor series for \(\mathrm{Si}(x)\): \(\mathrm{Si}(x) = \int_0^x f(t) \, dt = \int_0^x \sum_{n=0}^\infty (-1)^n \frac{t^{2n}}{(2n+1)!} \, dt\).

Interchange the integral and the summation (justified by uniform convergence on compact intervals) and integrate each term: \(\mathrm{Si}(x) = \sum_{n=0}^\infty (-1)^n \frac{1}{(2n+1)!} \int_0^x t^{2n} \, dt = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)(2n+1)!}\).

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Definition and Properties of the Sine Integral Function

The sine integral function Si(x) is defined as the integral from 0 to x of (sin t)/t dt, with a special value at t=0 to ensure continuity. Understanding this function involves recognizing it as an integral of a function with a removable discontinuity and its role in analysis and applications.

Taylor Series Expansion of Functions

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point, usually zero. To find the Taylor series of Si(x), one must express the integrand as a power series and then integrate term-by-term, ensuring convergence within the radius of expansion.

Term-by-Term Integration of Power Series

Integrating a power series term-by-term is valid within the interval of convergence and allows finding the integral of complex functions by integrating each term individually. This technique simplifies finding the Taylor series of integral-defined functions like Si(x) by integrating the series expansion of (sin t)/t.

Sine integral function The function Si(x) = ∫₀ˣ f(t) dt, where f(t) = {(sin t)/t if t ≠ 0, 1 if t = 0, is called the sine integral function.b. Integrate the series to find a Taylor series for Si.