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Ci -- the cosine integral function

Introduction

Ci(x) represents the cosine integral EULER + ln(x) + int((cos(t)-1)/t, t=0..x).

Call(s)

Ci(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

Ei, int, Si, cos

Details

Example 1

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

Example 2

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

Example 3

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

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