If a product is invoiced on January 31st what month should be recorded as revenue?

There are a couple scenariOS that can be thought of, your question is a bit obscure: You state the product is "invoiced" on January 31st, but you don't say when the product was sent out.

1. If the product was sent out prior to Invoice then sending the invoice makes absolutely no changes to the books as the transaction was recorded upon shipping. For example your company sold a computer on January 15th and is now sending an invoice for it, the entry would have been made on January 15th, not additional entry required until "after" payment is received.

2. If the product is being shipped with the invoice then it should be recorded the date the transaction occurs. January 31st being that date. It appear that you haven't received payment as of yet, hence the "invoice".

Either way on the above, the Journal Entry is as follows on the date the product is sent, whether it is invoiced or not.

Accounts Receivable (debit) $$$$

Revenue (credit) $$$$

Whether its January 1st, or 31st. The journal entry should be made the same day.

The exception to this rule is if you are using Cash Basis Accounting, then no entry is made until Cash is received for the product, regardless of date.