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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

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

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