Shifted sine integral function
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Ssi(x) represents the shifted sine integral .
The special values
Ssi(0) = -π/2,
Ssi(- ∞) = -π are implemented.
x is a negative integer or a negative
rational number, then
Ssi(x) = -Ssi(-x) - π.
Ssi function also uses this reflection rule
when argument is a symbolic product involving such a factor. See Example 2.
When called with a floating-point 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 floating-point numbers, use
Alternatively, use a floating-point 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,
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(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)
Ssi(x) = Si(x) - π for all
the complex plane.
Reference: M. Abramowitz and I. Stegun, “Handbook of Mathematical Functions”, Dover Publications Inc., New York (1965).