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CosIntegral

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

CosIntegral[z]
    gives the cosine integral Ci(z) = -Integral_z^Infinity Cos[t]/t dt.
Has a logarithmic singularity at 0 and a branch cut on (-Infinity, 0].
CosIntegral[0] = -Infinity, CosIntegral[Infinity] = 0,
CosIntegral[-Infinity] = I Pi, CosIntegral[+-I Infinity] = Infinity.
Real and complex inputs evaluate numerically at machine or arbitrary (MPFR)
precision; D[CosIntegral[z], z] = Cos[z]/z. Listable.

Examples

No verified examples yet for this function.

Implementation notes

Attributes: Listable, NumericFunction, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= CosIntegral[2.8]
Out[1]= 0.186488

In[1]:= N[CosIntegral[2], 50]
Out[1]= 0.42298082877486499569856515319825589413573775630619

In[1]:= CosIntegral[{-Infinity, Infinity, -I Infinity, I Infinity}]
Out[1]= {I Pi, 0, Infinity, Infinity}

In[1]:= CosIntegral[-2.]
Out[1]= 0.422981 + 3.14159 I

In[1]:= CosIntegral[3. I]
Out[1]= 4.96039 + 1.5708 I

In[1]:= D[CosIntegral[x], x]
Out[1]= Cos[x]/x

In[1]:= Series[CosIntegral[x], {x, 0, 6}]
Out[1]= EulerGamma + Log[x] - 1/4 x^2 + 1/96 x^4 - 1/4320 x^6 + O[x]^7

In[1]:= Normal[Series[CosIntegral[x], {x, Infinity, 3}]]
Out[1]= -Cos[x]/x^2 + Sin[x] (1/x - 2/x^3)

Notes

CosIntegral[z] is the cosine integral Ci(z) = -Integral_z^Infinity Cos[t]/t dt. Unlike its sibling SinIntegral — which is entire and odd — Ci has a logarithmic singularity at the origin (CosIntegral[0] = -Infinity) and a branch cut running along the negative real axis (-Infinity, 0]. On the cut it takes the from-above value, so for a negative real x the result is complex: Ci(x) = Ci(|x|) + I Pi, matching CosIntegral[-Infinity] = I Pi. On the imaginary axis Ci(I y) = Chi(y) + I Pi/2 in terms of the hyperbolic cosine integral. Its derivative is Cos[z]/z. Numeric evaluation uses a convergent series near the origin and a trig-prefactored asymptotic expansion for large |z|, at machine or arbitrary (MPFR) precision. Listable.