How do I calculate the normal component of the velocity at the midpoint of a source straight panel? I am reading "Fundamental of Aerodynamics " by J.D.Anderson Fifth edition. Now I am at the chapter 3. If you got the book, please go to the page 286, the part that says: "when j = i, the contribution to the derivative is simply $\lambda/2$ ". How could it be $\lambda/2$ ? I thought it would be zero.

I will summarize the problem for those who don't have the book:

We have a uniform flow and a source sheet cover the the surface of a known body. We have to find out the $\lambda=\lambda(s)$ along the source sheet that when we superimpose the uniform flow with this source sheet, we obtain the streamline over surface of the body. 



A source sheet is an infinite number of line sources side by side,
where the strength of each line source is infinitesimally small



We approximate the source sheet by a series of straight panel over the body surface (if we take n panels, then we have to find n values of $\lambda$)  and calculate the velocity at each panel and set the normal component of it equal zero.



Take a point $P$ at the middle of $i$th panel and calculate the velocity at $P$ induced by the panels, or just calculate the normal component of the velocity at P. The book said that the normal component of the velocity contributed by only $i$th panel at the mid point $P$ of this panel is $\lambda_i/2$. I don't know how to calculate this, I have tried and thought it would be zero.


@Peter: Look at the image, you can see the velocity induced at P by the panel i is in the ab-line direction (I mean the horizontal direction). Then, the normal component (vertical direction) of it will be zero
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.