Ssi
Shifted sine integral function
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Ssi(x
)
Ssi(x)
represents the shifted sine integral $$\mathrm{Si}\left(x\right)\frac{\pi}{2}$$.
The special values Ssi(0) = π/2
, Ssi(∞)
= 0
, Ssi( ∞) = π
are implemented.
If x
is a negative integer or a negative
rational number, then Ssi(x) = Ssi(x)  π
.
The Ssi
function also uses this reflection rule
when argument is a symbolic product involving such a factor. See Example 2.
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:
Ssi(0), Ssi(1), Ssi(sqrt(2)), Ssi(x + 1), Ssi(infinity)
To approximate exact results with floatingpoint numbers, use float
:
float(Ssi(1)), float(Ssi(sqrt(2)))
Alternatively, use a floatingpoint value as an argument:
Ssi(5.0), Ssi(1.0), Ssi(2.0 + 10.0*I)
For negative real numbers and products involving such numbers, Ssi
applies
the reflection rule Ssi(x) =  Ssi(x)  π
:
Ssi(3), Ssi(3/7), Ssi(sqrt(2)), Ssi(x/7), Ssi(0.3*x)
No such “normalization” occurs for complex numbers or arguments that are not products:
Ssi( 3  I), Ssi(3 + I), Ssi(x  1), Ssi(1  x)
diff
, float
, limit
, series
, and other functions
handle expressions involving Ssi
:
diff(Ssi(x), x, x, x), float(ln(3 + Ssi(sqrt(PI))))
limit(Ssi(2*x^2/(1+x)), x = infinity)
series(Ssi(x), x = 0)
series(Ssi(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.
Reference: M. Abramowitz and I. Stegun, “Handbook of Mathematical Functions”, Dover Publications Inc., New York (1965).