ExpIntegralEi¶

Status: Stable

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

Description¶

ExpIntegralEi[z]
    gives the exponential integral Ei(z), the principal value of
    -Integral_{-z}^Infinity e^-t/t dt, with a branch cut on (-Infinity, 0).
ExpIntegralEi[0] = -Infinity, ExpIntegralEi[Infinity] = Infinity,
ExpIntegralEi[-Infinity] = 0, ExpIntegralEi[+-I Infinity] = +-I Pi. Real and
complex inputs evaluate numerically at machine or arbitrary (MPFR) precision;
D[ExpIntegralEi[z], z] = E^z/z. Listable.

Examples¶

No verified examples yet for this function.

Implementation notes¶

Exact special values: ExpIntegralEi[0] = -Infinity,

Attributes: Listable, NumericFunction, Protected.

Implementation status¶

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

References¶

Source: src/info.c
Specification: docs/spec/builtins/special-functions.md

Notes & additional examples¶

Worked examples¶

In[1]:= ExpIntegralEi[0]
Out[1]= -Infinity

In[1]:= D[ExpIntegralEi[z], z]
Out[1]= E^z/z

In[1]:= N[ExpIntegralEi[1], 40]
Out[1]= 1.8951178163559367554665209343316342690171

In[1]:= N[ExpIntegralEi[I], 30]
Out[1]= 0.3374039229009681346626462038893 + 2.516879397162079634172675005462*I

Notes¶

ExpIntegralEi[z] is the exponential integral Ei(z), with a branch cut on (-Infinity, 0) and derivative E^z/z. On the imaginary axis it ties to the cosine/sine integrals via Ei(I) = Ci(1) + I (Pi/2 + Si(1)). Real and complex arguments evaluate at machine or arbitrary (MPFR) precision. Listable.