Trac #21819: Rewrite error functions and documentation

This ticket should accomplish:
 * implement inverse error function `erfinv` (#11349)
 * implement imaginary error function `erfi`.
 * People need to manipulate statistical distribution functions as in
https://ask.sagemath.org/question/35407/use-cumulative-distribution-
functions/. To provide the means a symbolical version of `erfc` is
necessary. The current `error_fcn()` is only numerical.
 * put it all in a separate file `functions/error.py` with central
documentation

http://dlmf.nist.gov/7

URL: https://trac.sagemath.org/21819
Reported by: rws
Ticket author(s): Ralf Stephan
Reviewer(s): Travis Scrimshaw

src/doc/en/reference/functions/index.rst

@@ -8,6 +8,7 @@ Functions
    sage/functions/trig
    sage/functions/hyperbolic
    sage/functions/transcendental
+   sage/functions/error
    sage/functions/piecewise
    sage/functions/spike_function
    sage/functions/orthogonal_polys

src/doc/en/reference/references/index.rst

@@ -563,6 +563,10 @@ REFERENCES:
                  Functions, in NIST Digital Library of Mathematical
                  Functions. http://dlmf.nist.gov/10
 
+.. [DLMF-Error] \N. M. Temme: 7. Error Functions, Dawson’s and Fresnel
+                 Integrals, in NIST Digital Library of Mathematical
+                 Functions. http://dlmf.nist.gov/7
+
 .. [DLMF-Struve] \R. B. Paris: 11. Struve and Related Functions, in
                  NIST Digital Library of Mathematical
                  Functions. http://dlmf.nist.gov/11
@@ -1694,6 +1698,8 @@ REFERENCES:
 
 .. [WP-Bessel] :wikipedia:`Bessel_function`
 
+.. [WP-Error] :wikipedia:`Error_function`
+
 .. [WP-Struve] :wikipedia:`Struve_function`
 
 .. _ref-X:

src/sage/functions/all.py

@@ -4,6 +4,7 @@
 
 lazy_import('sage.functions.piecewise_old', 'Piecewise')   # deprecated
 lazy_import('sage.functions.piecewise', 'piecewise')
+lazy_import('sage.functions.error', ['erf', 'erfc', 'erfi', 'erfinv'])
 
 from .trig import ( sin, cos, sec, csc, cot, tan,
                    asin, acos, atan,
@@ -21,7 +22,7 @@
 
 
 from .other import ( ceil, floor, gamma, psi, factorial, beta, binomial,
-                    abs_symbolic, erf, sqrt, log_gamma,
+                    abs_symbolic, sqrt, log_gamma,
                     gamma_inc, incomplete_gamma, gamma_inc_lower,
                     arg, real_part, real, frac,
                     imag_part, imag, imaginary, conjugate)
@@ -37,10 +38,9 @@
                                    spherical_bessel_J, spherical_bessel_Y,
                                    spherical_hankel1, spherical_hankel2)
 
-from .special import (spherical_harmonic,
-                     error_fcn, elliptic_e,
+from .special import (spherical_harmonic, elliptic_e,
                      elliptic_f, elliptic_ec, elliptic_eu,
-                     elliptic_kc, elliptic_pi, elliptic_j)
+                     elliptic_kc, elliptic_pi, elliptic_j, error_fcn)
 
 from .jacobi import (jacobi, inverse_jacobi, jacobi_nd, jacobi_ns, jacobi_nc,
                     jacobi_dn, jacobi_ds, jacobi_dc, jacobi_sn, jacobi_sd,
@@ -84,3 +84,4 @@
                           exponential_integral_1, Ei, exp_integral_ei)
 
 from .hypergeometric import hypergeometric, hypergeometric_M, hypergeometric_U
+