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Can I set up the company first and immediately buy the laptop as a business expense?
Yes
Also if you are going for flat rate VAT, you get first quarter where you can reclaim VAT on business purchases (this was the case when I set up 6 years ago, do check).
could transfer from my personal account for the short term until my first invoices are paid?
You can or you can pay from your personal card and record it as that, so the company owes you (the director) that money.
I'm about to jump into the world of freelance again. I need to set up a ltd co and buy a new laptop. Can I set up the company first and immediately buy the laptop as a business expense? As I won't yet have any cash in the business bank account, could transfer from my personal account for the short term until my first invoices are paid?