Repository: scipy/scipy

Issue:
BUG: special.betainc documentation should say "regularized incomplete beta function"
### Describe your issue.

The documentation for [scipy.special.betainc](https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.betainc.html) should say "_regularized_ incomplete beta function" instead of "incomplete beta function" everywhere, especially up front (as opposed to burying it in the notes). The latter almost universally refers to the unregularized version (see e.g. [here](https://mathworld.wolfram.com/IncompleteBetaFunction.html)). Not emphasizing this up front can cause confusion and bugs. (I had to chase down a bug caused by this.) See #1350 and #14744 for previous related issues.

### Reproducing Code Example

```python
N/A
```


### Error message

```shell
N/A
```


### SciPy/NumPy/Python version information

N/A


Discussion:
This is mentioned in the Notes section of the [docstring](https://scipy.github.io/devdocs/reference/generated/scipy.special.betainc.html).  But I agree that it would be even better to mention this earlier.  As suggested, the first line of the docstring could be "Regularized incomplete beta function" (with the Notes updated accordingly).  Similarly, the first line of the docstring for [`betaincinv`](https://scipy.github.io/devdocs/reference/generated/scipy.special.betaincinv.html) could be "Inverse of the regularized incomplete beta function."

Resolving patch:
diff --git a/scipy/special/_add_newdocs.py b/scipy/special/_add_newdocs.py
--- a/scipy/special/_add_newdocs.py
+++ b/scipy/special/_add_newdocs.py
@@ -1210,7 +1210,7 @@ def add_newdoc(name, doc):
     See Also
     --------
     gamma : the gamma function
-    betainc :  the incomplete beta function
+    betainc :  the regularized incomplete beta function
     betaln : the natural logarithm of the absolute
              value of the beta function
 
@@ -1250,9 +1250,9 @@ def add_newdoc(name, doc):
     r"""
     betainc(a, b, x, out=None)
 
-    Incomplete beta function.
+    Regularized incomplete beta function.
 
-    Computes the incomplete beta function, defined as [1]_:
+    Computes the regularized incomplete beta function, defined as [1]_:
 
     .. math::
 
@@ -1274,21 +1274,23 @@ def add_newdoc(name, doc):
     Returns
     -------
     scalar or ndarray
-        Value of the incomplete beta function
+        Value of the regularized incomplete beta function
 
     See Also
     --------
     beta : beta function
-    betaincinv : inverse of the incomplete beta function
+    betaincinv : inverse of the regularized incomplete beta function
 
     Notes
     -----
-    The incomplete beta function is also sometimes defined
-    without the `gamma` terms, in which case the above
-    definition is the so-called regularized incomplete beta
-    function. Under this definition, you can get the incomplete
-    beta function by multiplying the result of the SciPy
-    function by `beta`.
+    The term *regularized* in the name of this function refers to the
+    scaling of the function by the gamma function terms shown in the
+    formula.  When not qualified as *regularized*, the name *incomplete
+    beta function* often refers to just the integral expression,
+    without the gamma terms.  One can use the function `beta` from
+    `scipy.special` to get this "nonregularized" incomplete beta
+    function by multiplying the result of ``betainc(a, b, x)`` by
+    ``beta(a, b)``.
 
     References
     ----------
@@ -1334,7 +1336,7 @@ def add_newdoc(name, doc):
     r"""
     betaincinv(a, b, y, out=None)
 
-    Inverse of the incomplete beta function.
+    Inverse of the regularized incomplete beta function.
 
     Computes :math:`x` such that:
 
@@ -1359,11 +1361,11 @@ def add_newdoc(name, doc):
     Returns
     -------
     scalar or ndarray
-        Value of the inverse of the incomplete beta function
+        Value of the inverse of the regularized incomplete beta function
 
     See Also
     --------
-    betainc : incomplete beta function
+    betainc : regularized incomplete beta function
     gamma : gamma function
 
     References
@@ -1413,7 +1415,7 @@ def add_newdoc(name, doc):
     See Also
     --------
     gamma : the gamma function
-    betainc :  the incomplete beta function
+    betainc :  the regularized incomplete beta function
     beta : the beta function
 
     Examples


Test patch: