Si
Sine integral function
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Si(x
)
Si(x)
represents the sine integral $$\underset{0}{\overset{x}{\int}}\frac{\mathrm{sin}\left(t\right)}{t}dt$$.
If x
is a floatingpoint number, then Si(x)
returns
floatingpoint results. The special values Si(0) = 0
, Si(∞)
= π/2
, and Si(∞) = π/2
are
implemented. For all other arguments, Si
returns
symbolic function calls.
If x
is a negative integer or a negative
rational number, then Si(x) = Si(x)
. The Si
function
also uses this reflection rule when the argument is a symbolic product
involving such a factor. See Example 2.
The float attribute of Si
is a kernel function,
thus, floatingpoint evaluation is fast.
When called with a floatingpoint argument, the functions are
sensitive to the environment variable DIGITS
which determines
the numerical working precision.
Most calls with exact arguments return themselves unevaluated:
Si(0), Si(1), Si(sqrt(2)), Si(x + 1), Si(infinity)
To approximate exact results with floatingpoint numbers, use float
:
float(Si(1)), float(Si(sqrt(2)))
Alternatively, use a floatingpoint value as an argument:
Si(5.0), Si(1.0), Si(2.0 + 10.0*I)
For negative real numbers and products involving such numbers, Si
applies
the reflection rule Si(x) = Si(x)
:
Si(3), Si(3/7), Si(sqrt(2)), Si(x/7), Si(0.3*x)
No such “normalization” occurs for complex numbers or arguments that are not products:
Si( 3  I), Si(3 + I), Si(x  1), Si(1  x)
diff
, float
, limit
, series
, and other functions
handle expressions involving Si
:
diff(Si(x), x, x, x), float(ln(3 + Si(sqrt(PI))))
limit(Si(2*x^2/(1+x)), x = infinity)
series(Si(x), x = 0)
series(Si(x), x = infinity, 3)

Arithmetical expression.
x
Si
, Ssi
, and Shi
are
entire functions.
Ssi(x) = Si(x)  π
for all x
in
the complex plane.
i*Si(x) = Shi(i*x)
for all x
in
the complex plane.
Reference: M. Abramowitz and I. Stegun, “Handbook of Mathematical Functions”, Dover Publications Inc., New York (1965).