Add more DMLF links and example plots (#1373)

Author: rocky

mathics/builtin/specialfns/__init__.py

@@ -5,7 +5,11 @@
 
 There are a number of functions found in mathematical physics and found in standard handbooks.
 
-One thing to note is that the technical literature often contains several conflicting definitions. So beware and check for conformance with the Mathics documentation.
+One good source is <url>
+:The NIST Digital Library of Mathematical Functions:
+https://dlmf.nist.gov/</url>.
+
+The technical literature often contains several conflicting definitions. So beware, and check for conformance with the Mathics3 documentation.
 
 A number of special functions can be evaluated for arbitrary complex values of their arguments. However defining relations may apply only for some special choices of arguments. Here, the full function corresponds to an extension or "analytic continuation" of the defining relation.
 

mathics/builtin/specialfns/expintegral.py

@@ -2,6 +2,10 @@
 
 """
 Exponential Integral and Special Functions
+
+See also <url>
+:Chapters 4.2-4.13 Logarithm, Exponential, Powers in the Digital Libary of Mathematical Functions:
+https://dlmf.nist.gov/4#PT2</url>.
 """
 
 

mathics/builtin/specialfns/orthogonal.py

@@ -1,5 +1,9 @@
 """
 Orthogonal Polynomials
+
+See also <url>
+:Chapters 18.3 Classical Orthogonal Polynomials in the Digital Libary of Mathematical Functions:
+https://dlmf.nist.gov/18.3</url>.
 """
 
 

mathics/builtin/specialfns/zeta.py

@@ -2,6 +2,10 @@
 
 """
 Zeta Functions and Polylogarithms
+
+See also <url>
+:Chapters 25 Zeta and Related Functions in the Digital Libary of Mathematical Functions:
+https://dlmf.nist.gov/25</url>.
 """
 
 import mpmath
@@ -34,8 +38,13 @@ class LerchPhi(MPMathFunction):
     >> LerchPhi[2, 3, -1.5]
      = 19.3893 - 2.1346 I
 
-    >> LerchPhi[1, 2, 1/4]
-     = 17.1973
+    >> LerchPhi[1, 2, 1/4] == 8 Catalan + Pi^2
+     = True
+
+    Plot between between -1 and 1:
+    >> Plot[LerchPhi[x, 1, 2], {x, -1, 1}]
+     = -Graphics-
+
     """
 
     attributes = A_LISTABLE | A_NUMERIC_FUNCTION | A_PROTECTED | A_READ_PROTECTED
@@ -72,6 +81,10 @@ class PolyLog(MPMathFunction):
      = Zeta[s]
     >> PolyLog[-7, I] //Chop
      = 136.
+
+    Dilogarithm function $Li_2(x)$:
+    >> Plot[PolyLog[2,x], {x, -20, 1}]
+     = -Graphics-
     """
 
     attributes = A_LISTABLE | A_NUMERIC_FUNCTION | A_PROTECTED | A_READ_PROTECTED
@@ -103,6 +116,9 @@ class Zeta(MPMathFunction):
 
     >> Zeta[-2.5 + I]
      = 0.0235936 + 0.0014078 I
+
+    >> Plot[Zeta[z], {z, -20, 10}]
+     = -Graphics-
     """
 
     attributes = A_LISTABLE | A_NUMERIC_FUNCTION | A_PROTECTED | A_READ_PROTECTED