Special functions, first tranche: the error functions and the exponential, logarithmic, trigonometric and hyperbolic integrals

[Rafael-SOWNet]

Agreed on #1469. Rubi's family 8 is 1949 problems that the corpus harness doesn't run. This first tranche is 791 of them:

functions	Rubi file	problems
erf, erfc, erfi	8.1 Error functions	311
Ei, li	8.3 Exponential integrals	208
Si, Ci	8.4 Trig integrals	136
Shi, Chi	8.5 Hyperbolic integrals	136

(On #1469 I wrote about 655. That count left out 8.5.)

The work goes in this order:

1. Family 8 in the corpus harness, so the number is visible before anything lands.

2. Each function as a node, with four parts:

   • its definition and branch convention written down: Ei(x) as the principal value for x > 0, li(x) = Ei(ln x) with the same cut, erfi(x) = -i erf(i x) exactly;
   • a numerical evaluation that agrees with the definition;
   • its derivative;
   • its LaTeX and SymPy forms.

   The derivative is what lets the checks verify every answer that uses the function.

3. Integration rules, each naming the identity it uses. A special function is never a fallback for an elementary search that failed.

erf is refused by name today (#750). The refusal goes when its node lands.

8.10's 97 formal-derivative problems aren't part of this tranche. They integrate undefined functions, such as Derivative(1)(f)(x)/f(x) to log(f(x)), so they need f(x) as a function in its own right, not any special function.

[Rafael-SOWNet]

Step 1 is done. The corpus harness now counts every problem it excludes under each function it needs, and under the function that alone would make it statable. Family 8, on 856831cf:

function	problems that use it	problems that need only it
ProductLog	398	398
GAMMA	208	207
polylog	198	198
Ei	195	195
FresnelC, FresnelS	109 each	109 each
erfi, erf, erfc	105, 103, 103	the same
Si, Ci, Shi, Chi	68 each	68 each
Psi, Li, Zeta, lnGAMMA	18, 13, 14, 9	15, 13, 13, 9

Almost every problem needs exactly one function, so each function lands with a number attached. The error functions come next.

[Rafael-SOWNet]

Step 3 is #1512: the Gaussian, its even moments and the error functions integrated. It solves 473 of the 811 Rubi problems whose answer uses erf, erfc or erfi, against 0 on master, with 0 wrong.