#11143: define symbolic functions for exponential integrals
--------------------------------------------------+-------------------------
Reporter: kcrisman | Owner:
benjaminfjones
Type: defect | Status: new
Priority: major | Milestone: sage-4.7.2
Component: symbolics | Keywords: ei Ei
special function maxima sd32
Work_issues: | Upstream: N/A
Reviewer: Burcin Erocal, Karl-Dieter Crisman | Author: Benjamin
Jones
Merged: | Dependencies: #11513
--------------------------------------------------+-------------------------
Description changed by benjaminfjones:
Old description:
> We're missing some conversions from Maxima. Like exponential integrals
> of various kinds.
> {{{
> sage: f(x) = e^(-x) * log(x+1)
> sage: uu = integral(f,x,0,oo)
> sage: uu
> x |--> e*expintegral_e(1, 1)
> }}}
> See [http://ask.sagemath.org/question/488/calculating-integral this
> ask.sagemath post] for some details.
>
> == Current symbol conversion table ==
> From `sage.symbolic.pynac.symbol_table['maxima']` as of Sage-4.7
> {{{
> Maxima ---> Sage
>
> %gamma ---> euler_gamma
> %pi ---> pi
> (1+sqrt(5))/2 ---> golden_ratio
> acos ---> arccos
> acosh ---> arccosh
> acot ---> arccot
> acoth ---> arccoth
> acsc ---> arccsc
> acsch ---> arccsch
> asec ---> arcsec
> asech ---> arcsech
> asin ---> arcsin
> asinh ---> arcsinh
> atan ---> arctan
> atan2 ---> arctan2
> atanh ---> arctanh
> binomial ---> binomial
> brun ---> brun
> catalan ---> catalan
> ceiling ---> ceil
> cos ---> cos
> delta ---> dirac_delta
> elliptic_e ---> elliptic_e
> elliptic_ec ---> elliptic_ec
> elliptic_eu ---> elliptic_eu
> elliptic_f ---> elliptic_f
> elliptic_kc ---> elliptic_kc
> elliptic_pi ---> elliptic_pi
> exp ---> exp
> expintegral_e ---> En
> factorial ---> factorial
> gamma_incomplete ---> gamma
> glaisher ---> glaisher
> imagpart ---> imag_part
> inf ---> +Infinity
> infinity ---> Infinity
> khinchin ---> khinchin
> kron_delta ---> kronecker_delta
> li[2] ---> dilog
> log ---> log
> log(2) ---> log2
> mertens ---> mertens
> minf ---> -Infinity
> psi[0] ---> psi
> realpart ---> real_part
> signum ---> sgn
> sin ---> sin
> twinprime ---> twinprime
> }}}
>
> = Summary of missing conversions =
>
> == Special functions defined in Maxima ==
> (http://maxima.sourceforge.net/docs/manual/en/maxima_16.html#SEC56)
>
> {{{
> bessel_j (index, expr) Bessel function, 1st kind
> bessel_y (index, expr) Bessel function, 2nd kind
> bessel_i (index, expr) Modified Bessel function, 1st kind
> bessel_k (index, expr) Modified Bessel function, 2nd kind
> }}}
>
> * Notes: bessel_I, bessel_J, etc. are functions in Sage for numerical
> evaluation. There is also the `Bessel` class, but no conversions from
> Maxima's bessel_i etc. to Sage.
>
> {{{
> hankel_1 (v,z) Hankel function of the 1st kind
> hankel_2 (v,z) Hankel function of the 2nd kind
> struve_h (v,z) Struve H function
> struve_l (v,z) Struve L function
> }}}
>
> * Notes: None of these functions are currently exposed at the top level
> in Sage. Evaluation is possible using mpmath.
>
> {{{
> assoc_legendre_p[v,u] (z) Legendre function of degree v and order u
> assoc_legendre_q[v,u] (z) Legendre function, 2nd kind
> }}}
>
> * Notes: In Sage we have `legendre_P(n, x)` and `legendre_Q(n, x)` both
> described as Legendre functions. It's not clear to me how there are
> related to Maxima's versions since the number of arguments differs.
>
> {{{
> %f[p,q] ([], [], expr) Generalized Hypergeometric function
> hypergeometric(l1, l2, z) Hypergeometric function
> slommel
> %m[u,k] (z) Whittaker function, 1st kind
> %w[u,k] (z) Whittaker function, 2nd kind
> }}}
>
> * Notes: `hypergeometric(l1, l2, z)` needs a conversion to Sage's
> `hypergeometric_U`. The others can be evaluated using mpmath. `slommel`
> is presumably mpmath's `lommels1()` or `lommels2()` (or both?). This
> isn't well documented in Maxima.
>
> {{{
> expintegral_e (v,z) Exponential integral E
> expintegral_e1 (z) Exponential integral E1
> expintegral_ei (z) Exponential integral Ei
> expintegral_li (z) Logarithmic integral Li
> expintegral_si (z) Exponential integral Si
> expintegral_ci (z) Exponential integral Ci
> expintegral_shi (z) Exponential integral Shi
> expintegral_chi (z) Exponential integral Chi
> erfc (z) Complement of the erf function
> }}}
>
> * Notes: The exponential integral functions `expintegral_e1` and
> `expintegral_ei (z)` are called `exponential_integral_1` and `Ei` resp.
> in Sage. They both need conversions. The rest need `BuiltinFunction`
> classes defined for them with evaluation handled by mpmath and the symbol
> table conversion added. Also, `erfc` is called `error_fcn`, so also needs
> a conversion.
>
> {{{
> kelliptic (z) Complete elliptic integral of the first
> kind (K)
> parabolic_cylinder_d (v,z) Parabolic cylinder D function
> }}}
>
> * Notes: `kelliptic(z)` needs a conversion to `elliptic_kc` in Sage and
> `parabolic_cylinder_d (v,z)` does not seem to be exposed at top level. It
> can be evaluated by mpmath.
New description:
We're missing some conversions from Maxima. Like exponential integrals of
various kinds.
{{{
sage: f(x) = e^(-x) * log(x+1)
sage: uu = integral(f,x,0,oo)
sage: uu
x |--> e*expintegral_e(1, 1)
}}}
See [http://ask.sagemath.org/question/488/calculating-integral this
ask.sagemath post] for some details.
== Current symbol conversion table ==
From `sage.symbolic.pynac.symbol_table['maxima']` as of Sage-4.7
{{{
Maxima ---> Sage
%gamma ---> euler_gamma
%pi ---> pi
(1+sqrt(5))/2 ---> golden_ratio
acos ---> arccos
acosh ---> arccosh
acot ---> arccot
acoth ---> arccoth
acsc ---> arccsc
acsch ---> arccsch
asec ---> arcsec
asech ---> arcsech
asin ---> arcsin
asinh ---> arcsinh
atan ---> arctan
atan2 ---> arctan2
atanh ---> arctanh
binomial ---> binomial
brun ---> brun
catalan ---> catalan
ceiling ---> ceil
cos ---> cos
delta ---> dirac_delta
elliptic_e ---> elliptic_e
elliptic_ec ---> elliptic_ec
elliptic_eu ---> elliptic_eu
elliptic_f ---> elliptic_f
elliptic_kc ---> elliptic_kc
elliptic_pi ---> elliptic_pi
exp ---> exp
expintegral_e ---> En
factorial ---> factorial
gamma_incomplete ---> gamma
glaisher ---> glaisher
imagpart ---> imag_part
inf ---> +Infinity
infinity ---> Infinity
khinchin ---> khinchin
kron_delta ---> kronecker_delta
li[2] ---> dilog
log ---> log
log(2) ---> log2
mertens ---> mertens
minf ---> -Infinity
psi[0] ---> psi
realpart ---> real_part
signum ---> sgn
sin ---> sin
twinprime ---> twinprime
}}}
= Summary of missing conversions =
== Special functions defined in Maxima ==
(http://maxima.sourceforge.net/docs/manual/en/maxima_16.html#SEC56)
{{{
bessel_j (index, expr) Bessel function, 1st kind
bessel_y (index, expr) Bessel function, 2nd kind
bessel_i (index, expr) Modified Bessel function, 1st kind
bessel_k (index, expr) Modified Bessel function, 2nd kind
}}}
* Notes: bessel_I, bessel_J, etc. are functions in Sage for numerical
evaluation. There is also the `Bessel` class, but no conversions from
Maxima's bessel_i etc. to Sage.
{{{
hankel_1 (v,z) Hankel function of the 1st kind
hankel_2 (v,z) Hankel function of the 2nd kind
struve_h (v,z) Struve H function
struve_l (v,z) Struve L function
}}}
* Notes: None of these functions are currently exposed at the top level
in Sage. Evaluation is possible using mpmath.
{{{
assoc_legendre_p[v,u] (z) Legendre function of degree v and order u
assoc_legendre_q[v,u] (z) Legendre function, 2nd kind
}}}
* Notes: In Sage we have `legendre_P(n, x)` and `legendre_Q(n, x)` both
described as Legendre functions. It's not clear to me how there are
related to Maxima's versions since the number of arguments differs.
{{{
%f[p,q] ([], [], expr) Generalized Hypergeometric function
hypergeometric(l1, l2, z) Hypergeometric function
slommel
%m[u,k] (z) Whittaker function, 1st kind
%w[u,k] (z) Whittaker function, 2nd kind
}}}
* Notes: `hypergeometric(l1, l2, z)` needs a conversion to Sage's
`hypergeometric_U`. The others can be evaluated using mpmath. `slommel` is
presumably mpmath's `lommels1()` or `lommels2()` (or both?). This isn't
well documented in Maxima.
{{{
expintegral_e (v,z) Exponential integral E
expintegral_e1 (z) Exponential integral E1
expintegral_ei (z) Exponential integral Ei
expintegral_li (z) Logarithmic integral Li
expintegral_si (z) Exponential integral Si
expintegral_ci (z) Exponential integral Ci
expintegral_shi (z) Exponential integral Shi
expintegral_chi (z) Exponential integral Chi
erfc (z) Complement of the erf function
}}}
* Notes: The exponential integral functions `expintegral_e1` and
`expintegral_ei (z)` are called `exponential_integral_1` and `Ei` resp. in
Sage. They both need conversions. The rest need `BuiltinFunction` classes
defined for them with evaluation handled by mpmath and the symbol table
conversion added. Also, `erfc` is called `error_fcn`, so also needs a
conversion.
{{{
kelliptic (z) Complete elliptic integral of the first
kind (K)
parabolic_cylinder_d (v,z) Parabolic cylinder D function
}}}
* Notes: `kelliptic(z)` needs a conversion to `elliptic_kc` in Sage and
`parabolic_cylinder_d (v,z)` does not seem to be exposed at top level. It
can be evaluated by mpmath.
----
Depends on #11513 and #11885.
Apply [attachment:trac_11143.patch] to the Sage library.
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11143#comment:27>
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