DOC: User Guide Page on user-defined functions
- [x] closes #61126 
- [ ] ~[Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature~
- [ ] ~All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).~
- [ ] ~Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.~
- [ ] ~Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.~

Currently writing this, so I would appreciate any feedback on it! Hi @rhshadrach thanks for the feedback! I agree with you and will push updates soon /preview Website preview of this PR available at: https://pandas.pydata.org/preview/pandas-dev/pandas/61195/ @arthurlw I generated a preview of the rendered docs in this PR. If you want to have a look to see if everything looks as expected: https://pandas.pydata.org/preview/pandas-dev/pandas/61195/docs/user_guide/user_defined_functions.html

@rhshadrach can you have a look and see if this can be merged? Even if we want to make some improvements and extend this in the future, I think this PR is already a great first version. So, whenever there is no blocker, probably easier to get this merged and iterate in follow up PRs if needed. Thanks for the preview! I think adding an example that combines groupby and filter (which takes a UDF) could be beneficial. That said, I do think it might duplicate some existing docs, so not sure if it's worth including here. Just noting this is still on my radar, should be able to get to it in the next 3 days. Thanks @arthurlw for all the work here! Thanks for the guidance on this UDF User Guide @datapythonista @rhshadrach!

I'm interested in diving deeper into this area of pandas. Are there any related issues, features, or improvements you think could be tackled next?

Thanks in advance! @arthurlw I created #61458 and assigned you to it. I think it's a good one to work in pandas udf. Good to learn more about the status quo, with reasonable complexity, and very useful to the project since it will make the code much clearer for future changes. Also, if you work on this issue, I think you'll realize of inconsistencies, missing documentation, and other related tasks that may be good to work on too. For example, it could be useful to document the executor interface in the documentation about extending pandas. So, third-party library authors can easily learn how to create an execution engine for pandas map/apply.

Please let me know if this is not what you're looking for, I can try to think of something else. Or if you have any question (better to ask them in the other issue).