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The Exponential Integral and related functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Chapter 5.
The Exponential Integral E1(z) defined as
with \(\left| \arg z \right| < \pi\) . (A&S eqn 5.1.1) and (DLMF 6.2E2)
The Exponential Integral Ei(x) defined as
with \(x\) real and \(x > 0\). (A&S eqn 5.1.2) and (DLMF 6.2E5)
The Exponential Integral li(x) defined as
with \(x\) real and \(x > 1\). (A&S eqn 5.1.3) and (DLMF 6.2E8)
The Exponential Integral En(z) (A&S eqn 5.1.4) defined as
with \({\rm Re}(z) > 1\) and \(n\) a non-negative integer.
The Exponential Integral Si(z) (A&S eqn 5.2.1) defined as
The Exponential Integral Ci(z) (A&S eqn 5.2.2) defined as
with \(|\arg z| < \pi\) .
The Exponential Integral Shi(z) (A&S eqn 5.2.3) defined as
The Exponential Integral Chi(z) (A&S eqn 5.2.4) defined as
with \(|\arg z| < \pi\) .
Default value: false
Change the representation of one of the exponential integrals,
expintegral_e(m, z)
, expintegral_e1
, or
expintegral_ei
to an equivalent form if possible.
Possible values for expintrep
are false
,
gamma_incomplete
, expintegral_e1
, expintegral_ei
,
expintegral_li
, expintegral_trig
, or
expintegral_hyp
.
false
means that the representation is not changed. Other
values indicate the representation is to be changed to use the
function specified where expintegral_trig
means
expintegral_si
, expintegral_ci
, and expintegral_hyp
means expintegral_shi
or expintegral_chi
.
Default value: false
Expand expintegral_e(n,z)
for half
integral values in terms of erfc
or erf
and
for positive integers in terms of expintegral_ei
.
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