Limit at Infinity of Sine Integral Function

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Theorem

Let $\Si: \R \to \R$ denote the sine integral function.


Then $\Si$ has a (finite) limit at infinity:

$\displaystyle \lim_{x \mathop \to +\infty} \map \Si x = \frac \pi 2$


Corollary

$\displaystyle \lim_{x \mathop \to -\infty} \Si \paren x = -\frac \pi 2$


Proof

The limit:

$\displaystyle \lim_{x \mathop \to +\infty} \map \Si x = \lim_{x \mathop \to +\infty} \int_{t \mathop \to 0}^{t \mathop = x} \frac {\sin t} t \rd t$

is the Dirichlet Integral.

$\blacksquare$


Sources