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SinIntegral

Status: Stable

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

Description

SinIntegral[z]
    gives the sine integral Si(z) = Integral_0^z Sin[t]/t dt.
An entire, odd function with no branch cuts. SinIntegral[0] = 0,
SinIntegral[+-Infinity] = +-Pi/2, SinIntegral[+-I Infinity] = +-I Infinity.
Real and complex inputs evaluate numerically at machine or arbitrary (MPFR)
precision; D[SinIntegral[z], z] = Sinc[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]:= SinIntegral[2.8]
Out[1]= 1.8321

In[1]:= N[SinIntegral[2], 50]
Out[1]= 1.6054129768026948485767201481985889408485834223285

In[1]:= SinIntegral[{-Infinity, Infinity, -I Infinity, I Infinity}]
Out[1]= {-1/2 Pi, 1/2 Pi, -I Infinity, I Infinity}

In[1]:= SinIntegral[2.5 + I]
Out[1]= 1.99549 + 0.222995 I

In[1]:= D[SinIntegral[x], x]
Out[1]= Sinc[x]

In[1]:= Series[SinIntegral[x], {x, 0, 7}]
Out[1]= x - 1/18 x^3 + 1/600 x^5 - 1/35280 x^7 + O[x]^8

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

Notes

SinIntegral[z] is the sine integral Si(z) = Integral_0^z Sin[t]/t dt, an entire, odd function with no branch cuts. Its derivative is Sinc, the cardinal sine Sin[z]/z. On the imaginary axis Si(I y) = I Shi(y) in terms of the hyperbolic sine integral, and as x -> ±Infinity, Si(x) -> ±Pi/2. A leading negative is pulled out by odd symmetry (SinIntegral[-x] = -SinIntegral[x]). Numeric evaluation uses a convergent Maclaurin series near the origin and an asymptotic expansion for large |z|, at machine or arbitrary (MPFR) precision. Listable. See also CosIntegral.