Giving something just the right amount of thought and consideration.
Thoughtlessness is where I don't think about something enough, over thinking is where I think about it so much it's paralyzing. To analysize is to give something the perfect amount of thought.
As a well-established branch of pure mathematics, real analysis is known for being notoriously hard. This is due to hoardes of ill-prepared students not realizing that they've only studied baby mathematicspreviously.
Charles: Man, I thought calculus in high school was legit until I studied real analysis for the first time.
James: Yeah, it's the point in my life where I understood that my analysis of math just got real.