How do I calculate the normal component of the velocity at the midpoint of a source straight panel?
You can find a mathematical derivation of the $\lambda_i/2$ result in Katz and Plotkin, chapter 10. In this PDF, the result you are looking for is on page 234 (250 of the file itself). Anderson approaches things a bit differently, but you should be able to see at a high level how the approaches are analogous. The math gets a bit complicated and it's really not necessary to comprehend every step in detail to understand the application to aerodynamics. However, a good exercise if you really want to dive into the math is to translate Katz and Plotkin's approach into Anderson's notation, or vice versa. For me, though, it's good enough to see that it can be done and move on with the more interesting applications.