Si -- the sine integral function
IntroductionSi(x) represents the sine integral
int(sin(t)/t, t=0..x).
Call(s)Si(x)
Parametersx |
- | an arithmetical expression |
Returnsan arithmetical expression.
x
Side
EffectsWhen called with a floating point argument, the function is
sensitive to the environment variable DIGITS which determines the numerical
working precision.
Related
Functions
Detailsx 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.Si is a kernel function, i.e.,
floating point evaluation is fast.
Example
1We 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
2The 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
3The 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