Ci
-- the cosine integral
functionCi
(x)
represents the cosine integral
EULER + ln(x) + int((cos(t)-1)/t, t=0..x).
Ci(x)
x |
- | an arithmetical expression |
an arithmetical expression.
x
When called with a floating point argument, the function is
sensitive to the environment variable DIGITS
which determines the numerical
working precision.
x
is a floating point number, then
Ci
(x)
returns the numerical value of the
cosine integral. The special values Ci(infinity)=0 and
Ci(-infinity)=I*PI are implemented. For all other arguments,
Ci
returns a symbolic function call.Ci
is a kernel function, i.e.,
floating point evaluation is fast.We demonstrate some calls with exact and symbolic input data:
>> Ci(1), Ci(sqrt(2)), Ci(x + 1), Ci(infinity), Ci(-infinity)
1/2 Ci(1), Ci(2 ), Ci(x + 1), 0, I PI
Floating point values are computed for floating point arguments:
>> Ci(1.0), Ci(2.0 + 10.0*I)
0.3374039229, - 242.5252694 - 1185.8387 I
Ci
is singular at the origin:
>> Ci(0)
Error: singularity [Ci]
The negative real axis is a branch cut of
Ci
. A jump of height I*2*PI occurs when
crossing this cut:
>> Ci(-1.0), Ci(-1.0 + 10^(-10)*I), Ci(-1.0 - 10^(-10)*I)
0.3374039229 + 3.141592654 I, 0.3374039229 + 3.141592654 I, 0.3374039229 - 3.141592654 I
The functions diff
and float
handle expressions involving
Ci
:
>> diff(Ci(x), x, x, x), float(ln(3 + Ci(sqrt(PI))))
2 cos(x) cos(x) 2 sin(x) -------- - ------ + --------, 1.241299561 3 x 2 x x
Ci
(x)-ln(x)
is an entire
function. Ci
has a logarithmic singularity at the origin
and a branch cut along the negative real axis. The values on the
negative real axis coincide with the limit ``from above'':
Ci(x)=limit(Ci(x+eps*I), eps=0, Right).