[SPARK-50044][PYTHON] Refine the docstring of multiple math functions

### What changes were proposed in this pull request?
Refine the docstring of multiple math functions

### Why are the changes needed?
1, make them copy-pasteable;
2, show the projection: input -> output

### Does this PR introduce _any_ user-facing change?
doc changes

### How was this patch tested?
updated doctests

### Was this patch authored or co-authored using generative AI tooling?
no

Closes apache#48567 from zhengruifeng/doc_refine_ln.

Authored-by: Ruifeng Zheng <ruifengz@apache.org>
Signed-off-by: Max Gekk <max.gekk@gmail.com>

python/pyspark/sql/functions/builtin.py

@@ -7157,27 +7157,46 @@ def log(arg1: Union["ColumnOrName", float], arg2: Optional["ColumnOrName"] = Non
 
     Examples
     --------
+    Example 1: Specify both base number and the input value
+
     >>> from pyspark.sql import functions as sf
     >>> df = spark.sql("SELECT * FROM VALUES (1), (2), (4) AS t(value)")
-    >>> df.select(sf.log(2.0, df.value).alias('log2_value')).show()
-    +----------+
-    |log2_value|
-    +----------+
-    |       0.0|
-    |       1.0|
-    |       2.0|
-    +----------+
+    >>> df.select("*", sf.log(2.0, df.value)).show()
+    +-----+---------------+
+    |value|LOG(2.0, value)|
+    +-----+---------------+
+    |    1|            0.0|
+    |    2|            1.0|
+    |    4|            2.0|
+    +-----+---------------+
 
-    And Natural logarithm
+    Example 2: Return NULL for invalid input values
 
-    >>> df.select(sf.log(df.value).alias('ln_value')).show()
-    +------------------+
-    |          ln_value|
-    +------------------+
-    |               0.0|
-    |0.6931471805599453|
-    |1.3862943611198906|
-    +------------------+
+    >>> from pyspark.sql import functions as sf
+    >>> df = spark.sql("SELECT * FROM VALUES (1), (2), (0), (-1), (NULL) AS t(value)")
+    >>> df.select("*", sf.log(3.0, df.value)).show()
+    +-----+------------------+
+    |value|   LOG(3.0, value)|
+    +-----+------------------+
+    |    1|               0.0|
+    |    2|0.6309297535714...|
+    |    0|              NULL|
+    |   -1|              NULL|
+    | NULL|              NULL|
+    +-----+------------------+
+
+    Example 3: Specify only the input value (Natural logarithm)
+
+    >>> from pyspark.sql import functions as sf
+    >>> df = spark.sql("SELECT * FROM VALUES (1), (2), (4) AS t(value)")
+    >>> df.select("*", sf.log(df.value)).show()
+    +-----+------------------+
+    |value|         ln(value)|
+    +-----+------------------+
+    |    1|               0.0|
+    |    2|0.6931471805599...|
+    |    4|1.3862943611198...|
+    +-----+------------------+
     """
     from pyspark.sql.classic.column import _to_java_column
 
@@ -7205,13 +7224,22 @@ def ln(col: "ColumnOrName") -> Column:
 
     Examples
     --------
-    >>> df = spark.createDataFrame([(4,)], ['a'])
-    >>> df.select(ln('a')).show()
-    +------------------+
-    |             ln(a)|
-    +------------------+
-    |1.3862943611198906|
-    +------------------+
+    >>> from pyspark.sql import functions as sf
+    >>> spark.range(10).select("*", sf.ln('id')).show()
+    +---+------------------+
+    | id|            ln(id)|
+    +---+------------------+
+    |  0|              NULL|
+    |  1|               0.0|
+    |  2|0.6931471805599...|
+    |  3|1.0986122886681...|
+    |  4|1.3862943611198...|
+    |  5|1.6094379124341...|
+    |  6| 1.791759469228...|
+    |  7|1.9459101490553...|
+    |  8|2.0794415416798...|
+    |  9|2.1972245773362...|
+    +---+------------------+
     """
     return _invoke_function_over_columns("ln", col)
 
@@ -7237,13 +7265,22 @@ def log2(col: "ColumnOrName") -> Column:
 
     Examples
     --------
-    >>> df = spark.createDataFrame([(4,)], ['a'])
-    >>> df.select(log2('a').alias('log2')).show()
-    +----+
-    |log2|
-    +----+
-    | 2.0|
-    +----+
+    >>> from pyspark.sql import functions as sf
+    >>> spark.range(10).select("*", sf.log2('id')).show()
+    +---+------------------+
+    | id|          LOG2(id)|
+    +---+------------------+
+    |  0|              NULL|
+    |  1|               0.0|
+    |  2|               1.0|
+    |  3| 1.584962500721...|
+    |  4|               2.0|
+    |  5| 2.321928094887...|
+    |  6| 2.584962500721...|
+    |  7| 2.807354922057...|
+    |  8|               3.0|
+    |  9|3.1699250014423...|
+    +---+------------------+
     """
     return _invoke_function_over_columns("log2", col)
 
@@ -7274,9 +7311,16 @@ def conv(col: "ColumnOrName", fromBase: int, toBase: int) -> Column:
 
     Examples
     --------
-    >>> df = spark.createDataFrame([("010101",)], ['n'])
-    >>> df.select(conv(df.n, 2, 16).alias('hex')).collect()
-    [Row(hex='15')]
+    >>> from pyspark.sql import functions as sf
+    >>> df = spark.createDataFrame([("010101",), ( "101",), ("001",)], ['n'])
+    >>> df.select("*", sf.conv(df.n, 2, 16)).show()
+    +------+--------------+
+    |     n|conv(n, 2, 16)|
+    +------+--------------+
+    |010101|            15|
+    |   101|             5|
+    |   001|             1|
+    +------+--------------+
     """
     from pyspark.sql.classic.column import _to_java_column
 
@@ -7307,9 +7351,22 @@ def factorial(col: "ColumnOrName") -> Column:
 
     Examples
     --------
-    >>> df = spark.createDataFrame([(5,)], ['n'])
-    >>> df.select(factorial(df.n).alias('f')).collect()
-    [Row(f=120)]
+    >>> from pyspark.sql import functions as sf
+    >>> spark.range(10).select("*", sf.factorial('id')).show()
+    +---+-------------+
+    | id|factorial(id)|
+    +---+-------------+
+    |  0|            1|
+    |  1|            1|
+    |  2|            2|
+    |  3|            6|
+    |  4|           24|
+    |  5|          120|
+    |  6|          720|
+    |  7|         5040|
+    |  8|        40320|
+    |  9|       362880|
+    +---+-------------+
     """
     return _invoke_function_over_columns("factorial", col)