Go over docs for gamma and no-meaning infix ops (#1394)

Go over docs for `Beta`, `Gamma`, `Product`, and infix operators with no meaning

Author: rocky

mathics/builtin/arithmetic.py

@@ -713,14 +713,14 @@ class Product(IterationFunction, SympyFunction):
       outermost-to-innermost order.
     </dl>
 
-    >> Product[k, {k, 1, 10}]
-     = 3628800
-    >> 10!
-     = 3628800
-    >> Product[x^k, {k, 2, 20, 2}]
-     = x ^ 110
+    The product of the first 10 integers is its factorial:
+    >> Product[k, {k, 1, 10}] == 10!
+     = True
+
+    'Product' can be used symbolically:
     >> Product[2 ^ i, {i, 1, n}]
      = 2 ^ (n / 2 + n ^ 2 / 2)
+
     >> Product[f[i], {i, 1, 7}]
      = f[1] f[2] f[3] f[4] f[5] f[6] f[7]
 

mathics/builtin/no_meaning/infix.py

@@ -29,7 +29,7 @@ def create_class_function(
                 operator_name=operator_name, operator_string=operator_string
             ),
             "operator": operator_string,
-            "summary_text": f"""{operator_name} postfix operator "{operator_string}" (no pre-set meaning attached)""",
+            "summary_text": f"""{operator_name} infix operator "{operator_string}" (no pre-set meaning attached)""",
             "formats": {
                 (
                     ("InputForm", "OutputForm", "StandardForm"),

mathics/builtin/specialfns/gamma.py

@@ -48,28 +48,28 @@
 
 
 class Beta(MPMathMultiFunction):
-    """
-        <url>
-        :Euler beta function:
-        https://en.wikipedia.org/wiki/Beta_function</url> (<url>:SymPy:
-        https://docs.sympy.org/latest/modules/functions/
+    r"""
+    <url>
+    :Euler beta function:
+    https://en.wikipedia.org/wiki/Beta_function</url> (<url>:SymPy:
+    https://docs.sympy.org/latest/modules/functions/
     special.html#sympy.functions.special.beta_functions.beta</url>, <url>
-        :WMA:
-        https://reference.wolfram.com/language/ref/Beta.html</url>)
-
-        <dl>
-          <dt>'Beta'[$a$, $b$]
-          <dd>is the Euler's Beta function.
-          <dt>'Beta'[$z$, $a$, $b$]
-          <dd>gives the incomplete Beta function.
-        </dl>
-
-        The Beta function satisfies the property
-        Beta[x, y] = Integrate[t^(x-1)(1-t)^(y-1),{t,0,1}] = Gamma[a] Gamma[b] / Gamma[a + b]
-        >> Beta[2, 3]
-         = 1 / 12
-        >> 12* Beta[1., 2, 3]
-         = 1.
+    :WMA:
+    https://reference.wolfram.com/language/ref/Beta.html</url>)
+
+    <dl>
+      <dt>'Beta'[$a$, $b$]
+      <dd>is the Euler's Beta function $\mathrm{B}(a, b)$.
+      <dt>'Beta'[$z$, $a$, $b$]
+      <dd>gives the incomplete Beta function $\mathrm{B}_z(a, b)$.
+    </dl>
+
+    The Beta function satisfies the property:
+    $\mathrm{B} = \int_0^t t^{(x-1)(1-t)^(y-1)} = \Gamma(a) \Gamma(b) / \Gamma(a + b)$
+    >> Beta[2, 3]
+     = 1 / 12
+    >> 12* Beta[1., 2, 3]
+     = 1.
     """
 
     attributes = A_LISTABLE | A_NUMERIC_FUNCTION | A_PROTECTED
@@ -272,7 +272,7 @@ def fact2_generic(x):
 
 
 class Gamma(MPMathMultiFunction):
-    """
+    r"""
     <url>:Gamma function:
     https://en.wikipedia.org/wiki/Gamma_function</url> (<url>
     :SymPy:https://docs.sympy.org/latest/modules/functions
@@ -287,35 +287,39 @@ class Gamma(MPMathMultiFunction):
 
     <dl>
       <dt>'Gamma'[$z$]
-      <dd>is the gamma function on the complex number $z$.
+      <dd>is the Euler gamma function, $\Gamma(z)$ on the complex number $z$.
 
-      <dt>'Gamma'[$z$, $x$]
-      <dd>is the upper incomplete gamma function.
+      <dt>'Gamma'[$a$, $z$]
+      <dd>is the upper incomplete gamma function, $\Gamma(a, z)$.
 
-      <dt>'Gamma'[$z$, $x_0$, $x_1$]
-      <dd>is equivalent to 'Gamma[$z$, $x_0$] - Gamma[$z$, $x_1$]'.
+      <dt>'Gamma'[$a$, $z_0$, $z_1$]
+      <dd>is the generalized incomplete gamma function $\Gamma[a, z_0] - \Gamma(a, z_1)$.
     </dl>
 
-    'Gamma[$z$]' is equivalent to '($z$ - 1)!':
+    'Gamma[$z$]' is equivalent to $(z - 1)!$:
     >> Simplify[Gamma[z] - (z - 1)!]
      = 0
 
-    Exact arguments:
+    Examples of using 'Gamma' with exact numeric arguments:
     >> Gamma[8]
      = 5040
+
     >> Gamma[1/2]
      = Sqrt[Pi]
-    >> Gamma[1, x]
-     = E ^ (-x)
-    >> Gamma[0, x]
-     = ExpIntegralE[1, x]
 
-    Numeric arguments:
     >> Gamma[123.78]
      = 4.21078×10^204
+
     >> Gamma[1. + I]
      = 0.498016 - 0.15495 I
 
+    Examples of 'Gamma' with symbolic arguments:
+
+   >> Gamma[1, x]
+     = E ^ (-x)
+    >> Gamma[0, x]
+     = ExpIntegralE[1, x]
+
     Both 'Gamma' and 'Factorial' functions are continuous:
     >> Plot[{Gamma[x], x!}, {x, 0, 4}]
      = -Graphics-

mathics/core/builtin.py

@@ -1279,7 +1279,7 @@ class NoMeaningInfixOperator(InfixOperator):
     https://reference.wolfram.com/language/ref/{operator_name}.html</url>
 
     <dl>
-      <dt>'{operator_name}[$x$, $y$, ...]'
+      <dt>'{operator_name}['$x$, $y$, ...']'
       <dd>displays $x$ {operator_string} $y$ {operator_string} ...
     </dl>
 
@@ -1379,7 +1379,7 @@ class NoMeaningPostfixOperator(PostfixOperator):
     https://reference.wolfram.com/language/ref/{operator_name}.html</url>
 
     <dl>
-      <dt>'{operator_name}[$x$]'
+      <dt>'{operator_name}['$x$']'
       <dd>displays $x$ {operator_string}
     </dl>
 
@@ -1421,7 +1421,7 @@ class NoMeaningPrefixOperator(PrefixOperator):
     https://reference.wolfram.com/language/ref/{operator_name}.html</url>
 
     <dl>
-      <dt>'{operator_name}[$x$]'
+      <dt>'{operator_name}['$x$']'
       <dd>displays {operator_string} $x$
     </dl>