## `Si` -- the sine integral function

#### Introduction

`Si``(x)` represents the sine integral int(sin(t)/t, t=0..x).

#### Call(s)

`Si(x)`

#### Parameters

`x` | - | an arithmetical expression

#### Returns

an arithmetical expression.

### Overloadable:

`x`

#### Side Effects

When called with a floating point argument, the function is sensitive to the environment variable `DIGITS` which determines the numerical working precision.

#### Related Functions

`Ci`, `Ei`, `int`, `sin`

#### Details

- If `x` is a floating point number, then `Si``(x)` returns the numerical value of the sine integral. The special values Si(0)=0 and Si(infinity)=PI/2, Si(-infinity)=-PI/2 are implemented. For all other arguments, `Si` returns a symbolic function call.
- The reflection rule Si(x)=-Si(-x) is used if the argument is a negative integer or a negative rational number. It is also used if the argument is a symbolic product involving such a factor. Cf. example 2.
- The float attribute of `Si` is a kernel function, i.e., floating point evaluation is fast.

#### Example 1

We demonstrate some calls with exact and symbolic input data:

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

                                       1/2              PI
                         0, Si(1), Si(2   ), Si(x + 1), --
                                                        2

Floating point values are computed for floating point arguments:

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

              -1.549931245, 0.9460830704, 1187.409493 - 242.5252717 I

#### Example 2

The reflection rule Si(-x)=-Si(x) is implemented for negative real numbers and products involving such numbers:

    >> Si(-3), Si(-3/7), Si(-sqrt(2)), Si(-x/7), Si(-0.3*x)

                                        1/2       / x \
                -Si(3), -Si(3/7), - Si(2   ), - Si| - |, -Si(0.3 x)
                                                  \ 7 /

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)

                   Si(- 3 - I), Si(3 + I), Si(x - 1), Si(1 - x)

#### Example 3

The functions `diff`, `float`, `limit`, and `series` handle expressions involving `Si`:

    >> diff(Si(x), x, x, x), float(ln(3 + Si(sqrt(PI))))

                     2 sin(x)   sin(x)   2 cos(x)
                     -------- - ------ - --------, 1.502020149
                         3        x          2
                        x                   x

    >> limit(Si(2*x^2/(1+x)), x = infinity)

                                        PI
                                        --
                                        2

    >> series(Si(x), x = 0), series(Si(x), x = infinity, 3)

                    3    5
                   x    x        6   PI   cos(x)   sin(x)    / 1  \
               x - -- + --- + O(x ), -- - ------ - ------ + O| -- |
                   18   600          2      x         2      |  3 |
                                                     x       \ x  /

#### Background

- Si is an entire function.
- Reference: M. Abramowitz and I. Stegun, ``Handbook of Mathematical Functions'', Dover Publications Inc., New York (1965).

#### Changes

- no changes

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