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Shifted sine integral function

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Ssi(x) represents the shifted sine integral Si(x)π2.

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.

Environment Interactions

When called with a floating-point argument, the functions are sensitive to the environment variable DIGITS which determines the numerical working precision.


Example 1

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 float:

float(Ssi(1)), float(Ssi(sqrt(2)))

Alternatively, use a floating-point value as an argument:

Ssi(-5.0), Ssi(1.0), Ssi(2.0 + 10.0*I)

Example 2

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)

Example 3

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)



An arithmetical expression

Return Values

Arithmetical expression.

Overloaded By



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).

See Also

MuPAD Functions

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