[jacobolus]

> i would say that analysis is not necessary if your interest is actually applied math

If you start reading research papers in applied math, there’s a ton of measure theory and functional analysis there.

More generally, both introductory real analysis and introductory complex analysis are assumed basic foundational background for pretty much any kind of research mathematics, applied or otherwise.

I’m also not sure I would recommend trying to self-study them though. Some expert guidance/feedback is pretty helpful for someone starting out.