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Symbolics functions

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Kwankyu Lee edited this page Feb 12, 2023 · 1 revision

Sage is missing symbolic definitions for many special functions that we are capable of evaluating numerically. Some of these are even provided in symbolic form by Maxima.

Tips for implementing new functions

Tickets related to improving an existing function or implementing a new one

  • #6466 Implement functional derivative and Euler-Lagrange equation
  • #8383 make symbolic versions of moebius, sigma and euler_phi
  • #9874 derivative of ceil and floor
  • #9935 Make a symbolic mod function
  • #10050 wrap the polylogarithm functions from pynac
  • #10071 make a few held functions able to evaluate
  • #10636 Make Bessel zeros available as symbolic entities
  • #12074 real nth root function
  • #12179 Binomial of integer (mod n) returns integer
  • #13050 allow different algorithms for evaluating erf
  • #13869 Gamma of complex numbers incorrectly simplifies to factorial
  • #14897 binomial(x, m) gives incorrect answer when m is float
  • #15344 asin(2.0) should not return NaN
  • #15354 Make elliptic_j function symbolic
  • #15497 Make lcm() symbolic
  • #16202 agm(x,y)
  • #16670 make all orthogonal polynomials symbolic
  • #16816 symbolic sums of roots
  • #17722 lower incomplete gamma as gamma(a,0,x)
  • #17970 implement Meijer G function
  • #18141 special values of transcendental functions
  • #18956 incomplete gamma identities
  • #19032 qgamma function
  • #19461 Pochhammer symbols
  • #20615 derivative of Bessel with respect to order
  • #21215 Periodic piecewise functions
  • #21274 frac(x) immediate simplifications
  • #21560 return Infinity from factorial of negative integer
  • #21639 Implement derivative of gegenbauer(n,a,x) wrt to a
  • #21655 Return (d/dn)f with f(n,x).diff(n) instead of runtime error
  • #21945 Symbolic min/max
  • #22028 Symbolic catalan number
  • #22146 Symbolic eta function
  • #22399 piecewise and hypergeometric functions fail their TestSuite: _test_category, _test_pickling
  • #22569 Symbolic fibonacci
  • #22651 symbolic AppellF1
  • #22713 multiple zeta
  • #22873 No evaluation with gamma of ball arguments
  • #24171 Formal set membership function
  • #24176 Implement formal Set comprehension
  • #24365 Nonnumeric integer expressions not handled by floor/ceil
  • #24554 Refactoring in Chebyshev functions
  • #24603 chebyshev_T/U fail with float/complex argument
  • #24604 No evaluation with some functions
  • #24861 Formal diff (Option to hold for derivative)

other symbolic function tickets

  • #12449 - symbolic functions on basic types improvements
  • #14270 - remove deprecated function-call syntax
  • #14608 - Symbolic functions break the __hash__ contract
  • #15021 - Return unevaluated derivative from BuiltinFunction
  • #15025 - automatically injected function does not work with desolve
  • #15200 - _evalf_ handling of backends
  • #17547 - BuiltinFunction overriding GiNaC function is allowed
  • #17701 - SR(f) or diff(f,t) should work even with NewSymbolicFunction
  • #18259 - comparison of symbolic functions
  • #20812 - derivative of integer wrt to variable in polynomial ring should belong to that ring, not symbolic ring
  • #24398 - Document function initialization parameters
  • #24832 - Extend function domain with some symbolic function calls

Tickets of this type closed

  • #24212 Fresnel integrals
  • #24411 Move gamma functions into their own file
  • #17790 BuiltinFunction doesn't pass non-SR-coercible arguments to function code
  • #22024 symbolic placeholder for complex root
  • #22079 Infinite loop in ceil() function
  • #23224 wrong symbolic comparison of log
  • #18386 polylog quirks
  • #20191 implement ExprCondPair equivalent
  • #19906 dilog(RR) should return an element of RR
  • #11349 Implement Inverse Erf function
  • #17505 symbolic product
  • #21819 Rewrite error functions and documentation
  • #22209 Differentiation of conj/imag/real/abs functions
  • #22844 symbolic limit
  • #10070 make heaviside and step play nicely together.
  • #19439 Different infinities returned by zeta/polylog
  • #21657 Import abs in functions/all.py
  • #20939 Remove pexpect-Maxima usage in Y(m,n)
  • #21906 Bug in bessel_K
  • #22004 Allow algorithm='sympy' in symbolic_sum function
  • #16813 Legendre functions/polynomials
  • #21365 cot(float) returns complex
  • #21614 Make atan2(0,0) return NaN
  • #21645 Full symbolic sum function
  • #16671 implement harmonic number function H_n
  • #17678 special values of Bessel functions
  • #20139 implement trigonometric symmetry simplifications
  • #18832 - non-numeric non-symbolic BuiltinFunction arguments?
  • #16587 - f(expr).n() fails for all generalized functions
  • #12521 evaluate log gamma for complex input
  • #14896 Symbolic hypergeometric confluent
  • #15024 More Hankel functions available
  • #16697 implement symbolic lower incomplete gamma function
  • #15046 Symbolic elliptic integrals
  • #17770 Euler numbers revamp
  • #19464 floor/ceil don't accept hold
  • #20297 ECL crash with Hermite polynomials
  • #20428 crash with ultraspherical polynomials
  • #20098 Re/Im(tanh) wrong formula
  • #16221 add Struve functions
  • #19834 implement symbolic Stieltjes constants
  • #19836 expansion of zeta using stieltjes-constants
  • #19425 - Order function in symbolic ring: error calling operator
  • #17447 Clarify and complete documentation of function()
  • #19336 bug in lambert_w.print_latex()
  • #12588 abs(pi*I) should return pi
  • #18954 special values of trig. functions with arguments (m/n)*pi
  • #17151 symbolic Laguerre / associated Laguerre polynomials
  • #17953 symbolic function args prevent forced conversion of result to numeric
  • #18091 symbolic floor,ceil,factorial need _evalf_ too
  • #10074 Improve less-used hyperbolic functions
  • #12455 Make Airy functions symbolic
  • #15017 Symbolic spherical harmonic
  • #2516 hypergeometric function
  • #12596 improve incomplete elliptic integrals docs
  • #9130 Access to beta function
  • #3401 extend li to work with complex arguments
  • #7357 add non offset logarithm
  • #8983 erf(0) should return 0
  • #4498 symbolic arg function
  • #11143 exponential integral
  • #10075 Make log gamma symbolic
  • #11155 abs(pi+I) = pi+I (new _eval_ method for abs())
  • #11423 Make atan2(0,0) consistent
  • #14996 Lots more elliptic functions
  • #1173 implement numerical evaluation of erf at complex arguments
  • #3426 bessel_K function is broken
  • #4102 make bessel_J symbolic
  • #4230 implement arbitrary precision Bessel Y
  • #9424 make symbolic summation numerically evaluable
  • #20312 - parent of argument lost with GinacFunctions

Tickets to make

  • implement parabolic_cylinder_d
  • add maxima conversion for elliptic_kc (#15046)
  • Associated Legendre functions

Special functions defined in Maxima

Notes from Benjamin Jones (#11143) (http://maxima.sourceforge.net/docs/manual/en/maxima_16.html#SEC56)

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. #15024 adds Hankel. #16221 is for Struve.
assoc_legendre_p[v,u] (z)      Associated Legendre function of degree v and order u 
assoc_legendre_q[v,u] (z)      Associated Legendre function, 2nd kind
  • These are not Maxima's legendre_p(n,x) and legendre_q(n,x) functions, which correspond to legendre_P(n,x) and legendre_Q(n,x) in Sage.
%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 (see #2516). 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: This was done in #11143!
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 (done in #15046) and parabolic_cylinder_d (v,z) does not seem to be exposed at top level. It can be evaluated by mpmath.
inverse_jacobi_cd
inverse_jacobi_cn
inverse_jacobi_cs
inverse_jacobi_dc
inverse_jacobi_dn
inverse_jacobi_ds
inverse_jacobi_nc
inverse_jacobi_nd
inverse_jacobi_ns
inverse_jacobi_sc
inverse_jacobi_sd
inverse_jacobi_sn
jacobi_cd
jacobi_cn
jacobi_cs
jacobi_dc
jacobi_dn
jacobi_ds
jacobi_nc
jacobi_nd
jacobi_ns
jacobi_sc
jacobi_sd
jacobi_sn
  • It turns out there are a slew of elliptic functions that we only have thinly wrapped and could make better - see this Maxima page. Basically, all of these are in Sage, but could be made more native. This was done in #14996.
dgauss_a
dgauss_b
dkummer_m
dkummer_u
gauss_a
gauss_b
kummer_m
kummer_u
  • These are some specific functions defined in Maxima's contrib_ode package, some of which we may have. Some hypergeometric function returned by certain ODE solvers in Maxima and which mpmath can evaluate are out there too (maybe same ones?). See #2516, for instance.

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