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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)
This can be written in terms of other functions. See expintrep for examples.
The Exponential Integral Ei(x) defined as
with \(x\) real and \(x > 0\). (A&S eqn 5.1.2) and (DLMF 6.2E5)
This can be written in terms of other functions. See expintrep for examples.
The Exponential Integral li(x) defined as
with \(x\) real and \(x > 1\). (A&S eqn 5.1.3) and (DLMF 6.2E8)
This can be written in terms of other functions. See expintrep for examples.
The Exponential Integral En(z) (A&S eqn 5.1.4) defined as
with \({\rm Re}(z) > 1\) and \(n\) a non-negative integer.
For half-integral orders, this can be written in terms of erfc
or erf
. See expintexpand for examples.
The Exponential Integral Si(z) (A&S eqn 5.2.1 and DLMF 6.2#E9) defined as
This can be written in terms of other functions. See expintrep for examples.
The Exponential Integral Ci(z) (A&S eqn 5.2.2 and DLMF 6.2#E13) defined as
with \(|\arg z| < \pi\) .
This can be written in terms of other functions. See expintrep for examples.
The Exponential Integral Shi(z) (A&S eqn 5.2.3 and DLMF 6.2#E15) defined as
This can be written in terms of other functions. See expintrep for examples.
The Exponential Integral Chi(z) (A&S eqn 5.2.4 and DLMF 6.2#E16) defined as
with \(|\arg z| < \pi\) .
This can be written in terms of other functions. See expintrep for examples.
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
.
Here are some examples for expintrep
set to gamma_incomplete
:
(%i1) expintrep:'gamma_incomplete; (%o1) gamma_incomplete (%i2) expintegral_e1(z); (%o2) gamma_incomplete(0, z) (%i3) expintegral_ei(z); (%o3) log(z) - log(- z) - gamma_incomplete(0, - z) (%i4) expintegral_li(z); (%o4) log(log(z)) - log(- log(z)) - gamma_incomplete(0, - log(z)) (%i5) expintegral_e(n,z); n - 1 (%o5) gamma_incomplete(1 - n, z) z (%i6) expintegral_si(z); (%o6) (%i ((- log(%i z)) + log(- %i z) - gamma_incomplete(0, %i z) + gamma_incomplete(0, - %i z)))/2 (%i7) expintegral_ci(z); (%o7) log(z) - (log(%i z) + log(- %i z) + gamma_incomplete(0, %i z) + gamma_incomplete(0, - %i z))/2 (%i8) expintegral_shi(z); log(z) - log(- z) + gamma_incomplete(0, z) - gamma_incomplete(0, - z) (%o8) --------------------------------------------------------------------- 2 (%i9) expintegral_chi(z); (%o9) (- log(z)) + log(- z) + gamma_incomplete(0, z) + gamma_incomplete(0, - z) - ------------------------------------------------------------------------- 2
For expintrep
set to expintegral_e1
:
(%i1) expintrep:'expintegral_e1; (%o1) expintegral_e1 (%i2) expintegral_ei(z); (%o2) log(z) - log(- z) - expintegral_e1(- z) (%i3) expintegral_li(z); (%o3) log(log(z)) - log(- log(z)) - expintegral_e1(- log(z)) (%i4) expintegral_e(n,z); (%o4) expintegral_e(n, z) (%i5) expintegral_si(z); (%o5) (%i ((- log(%i z)) - expintegral_e1(%i z) + log(- %i z) + expintegral_e1(- %i z)))/2 (%i6) expintegral_ci(z); (%o6) log(z) log(- %i z) (expintegral_e1(%i z) + expintegral_e1(- %i z)) log(%i z) - --------------------------------------------------------------------- 2 (%i7) expintegral_shi(z); log(z) + expintegral_e1(z) - log(- z) - expintegral_e1(- z) (%o7) ----------------------------------------------------------- 2 (%i8) expintegral_chi(z); (- log(z)) + expintegral_e1(z) + log(- z) + expintegral_e1(- z) (%o8) - --------------------------------------------------------------- 2
For expintrep
set to expintegral_ei
:
(%i1) expintrep:'expintegral_ei; (%o1) expintegral_ei (%i2) expintegral_e1(z); 1 log(- z) - log(- -) z (%o2) (- log(z)) + ------------------- - expintegral_ei(- z) 2 (%i3) expintegral_ei(z); (%o3) expintegral_ei(z) (%i4) expintegral_li(z); (%o4) expintegral_ei(log(z)) (%i5) expintegral_e(n,z); (%o5) expintegral_e(n, z) (%i6) expintegral_si(z); (%o6) (%i (log(%i z) + 2 (expintegral_ei(- %i z) - expintegral_ei(%i z)) %i %i - log(- %i z) + log(--) - log(- --)))/4 z z (%i7) expintegral_ci(z); (%o7) ((- log(%i z)) + 2 (expintegral_ei(%i z) + expintegral_ei(- %i z)) %i %i - log(- %i z) + log(--) + log(- --))/4 + log(z) z z (%i8) expintegral_shi(z); (%o8) ((- 2 log(z)) + 2 (expintegral_ei(z) - expintegral_ei(- z)) + log(- z) 1 - log(- -))/4 z (%i9) expintegral_chi(z); (%o9) 1 2 log(z) + 2 (expintegral_ei(z) + expintegral_ei(- z)) - log(- z) + log(- -) z ---------------------------------------------------------------------------- 4
For expintrep
set to expintegral_li
:
(%i1) expintrep:'expintegral_li; (%o1) expintegral_li (%i2) expintegral_e1(z); 1 log(- z) - log(- -) - z z (%o2) (- expintegral_li(%e )) - log(z) + ------------------- 2 (%i3) expintegral_ei(z); z (%o3) expintegral_li(%e ) (%i4) expintegral_li(z); (%o4) expintegral_li(z) (%i5) expintegral_e(n,z); (%o5) expintegral_e(n, z) (%i6) expintegral_si(z); %i z - %e z %pi signum(z) %i (expintegral_li(%e ) - expintegral_li(%e ) - -------------) 2 (%o6) - ---------------------------------------------------------------------- 2 (%i7) expintegral_ci(z); %i z - %i z expintegral_li(%e ) + expintegral_li(%e ) (%o7) ------------------------------------------------- - signum(z) + 1 2 (%i8) expintegral_shi(z); z - z expintegral_li(%e ) - expintegral_li(%e ) (%o8) ------------------------------------------- 2 (%i9) expintegral_chi(z); z - z expintegral_li(%e ) + expintegral_li(%e ) (%o9) ------------------------------------------- 2
For expintrep
set to expintegral_trig
:
(%i1) expintrep:'expintegral_trig; (%o1) expintegral_trig (%i2) expintegral_e1(z); (%o2) log(%i z) - %i expintegral_si(%i z) - expintegral_ci(%i z) - log(z) (%i3) expintegral_ei(z); (%o3) (- log(%i z)) - %i expintegral_si(%i z) + expintegral_ci(%i z) + log(z) (%i4) expintegral_li(z); (%o4) (- log(%i log(z))) - %i expintegral_si(%i log(z)) + expintegral_ci(%i log(z)) + log(log(z)) (%i5) expintegral_e(n,z); (%o5) expintegral_e(n, z) (%i6) expintegral_si(z); (%o6) expintegral_si(z) (%i7) expintegral_ci(z); (%o7) expintegral_ci(z) (%i8) expintegral_shi(z); (%o8) - %i expintegral_si(%i z) (%i9) expintegral_chi(z); (%o9) (- log(%i z)) + expintegral_ci(%i z) + log(z)
For expintrep
set to expintegral_hyp
:
(%i1) expintrep:'expintegral_hyp; (%o1) expintegral_hyp (%i2) expintegral_e1(z); (%o2) expintegral_shi(z) - expintegral_chi(z) (%i3) expintegral_ei(z); (%o3) expintegral_shi(z) + expintegral_chi(z) (%i4) expintegral_li(z); (%o4) expintegral_shi(log(z)) + expintegral_chi(log(z)) (%i5) expintegral_e(n,z); (%o5) expintegral_e(n, z) (%i6) expintegral_si(z); (%o6) - %i expintegral_shi(%i z) (%i7) expintegral_ci(z); (%o7) (- log(%i z)) + expintegral_chi(%i z) + log(z) (%i8) expintegral_shi(z); (%o8) expintegral_shi(z) (%i9) expintegral_chi(z); (%o9) expintegral_chi(z)
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
.
(%i1) expintegral_e(1/2,z); 1 (%o1) expintegral_e(-, z) 2 (%i2) expintegral_e(1,z); (%o2) expintegral_e(1, z) (%i3) expintexpand:true; (%o3) true (%i4) expintegral_e(1/2,z); sqrt(%pi) erfc(sqrt(z)) (%o4) ----------------------- sqrt(z) (%i5) expintegral_e(1,z); 1 log(- -) - log(- z) z (%o5) (- log(z)) - ------------------- - expintegral_ei(- z) 2
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