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Sine integral function

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Si(x) represents the sine integral 0xsin(t)tdt.

If x is a floating-point number, then Si(x) returns floating-point 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, floating-point evaluation is fast.

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:

Si(0), Si(1), Si(sqrt(2)), Si(x + 1), Si(infinity)

To approximate exact results with floating-point numbers, use float:

float(Si(1)), float(Si(sqrt(2)))

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

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

Example 2

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)

Example 3

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)



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.

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).

See Also

MuPAD Functions

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