Problem 3) Expansions of the sine integral [8 points]
Lecture notes, Chapter 3, Problem 14 (page 83):
The sine integral is defined as Si$(x) := \int_0^x \frac{\sin(t)}{t} dt$ (consider $x$ as real and positive). (1)
a) Obtain an expansion of Si$(x)$ useful for small $x$.
Hint: Expand the integrand in (1) in a power series in $t$ and integrate term-by-term, as in
Example 11.1, equations (11.2a), (11.2b), page 70.
b) Obtain an expansion of Si$(x)$ useful for large $x$.
Hint: Using that Si$(x) \to \pi/2$ for $x \to \infty$ (see Example 9.3, page 62),
define the complementary sine integral Sic$(x) :=$Si$(x) - \frac{\pi}{2} = -\int_x^\infty \frac{\sin(t)}{t} dt$. (2)
(Sic$(x)$ is also denoted si$(x)$, using a small letter \"s\", but we use the notation Sic$(x)$
to avoid confusion with Si$(x)$). Integrate (2) by parts multiple times to obtain a power
series in 1/x, as in Example 11.1, text on page 71. For example,
Sic$(x) = -\int_x^\infty \frac{\sin(t)}{t} dt = -\frac{\cos(x)}{x} + \int_x^\infty \frac{\cos(t)}{t^2} dt$, and so on. Proceed as in Example 11.1
to obtain an asymptotic series in 1/x.
