Exploring Sine Integral function

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Sine Integral function is a special function given by formula Si(x) = Integral(0; x; sin(x)/x ) on R. Let's explore Sine Integral using Math Center Level 2 . Open Graphing Calculator 2D Numeric, type sin(x)/x into Edit Formula window, and set a = 0 . Click button Take. Check check-box area and uncheck derivatives. Click button Draw.

Sine Integral with first derivative


Sine Integral with first derivative zoomed


As we see, the Sine Integral (pink graph) is a smooth function. Its first derivative (red graph) is sinc function (integrated function). Since sinc function has derivatives of all orders, Sine Integral also has derivatives of all orders. Thus we can use Taylor series for calculation of Sine Integral. Of course, calculation of Taylor series at arbitrary x is impractical. We have to use Taylor series centered at zero, that is Maclaurin series. Let's calculate Maclaurin series for Sine Integral of orders 3 and 4:

Sine Integral with Maclaurin series


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Sine Integral with Maclaurin series zoomed


As we see, the Maclaurin polynomials of third (green graph) and fourth (blue graph) degree give a good approximation for x in range [-3, 3]. Obviously the Sine Integral is not a periodic function. Thus for approximation of Sine Integral of large |x|  we have to use Taylor polynomials of  greater degrees. But calculation of Taylor polynomials of high degrees requires high precision. Otherwise rounding errors become too big. We can calculate Sine Integral with help of Scientific Calculator Precision 90 .



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