# What is the approximation used here?

• I
• Unconscious
In summary, the author provides an approximation for sin(u_0) using a series of terms that are K/(K+p).

#### Unconscious

I'm reading a paper (Beamwidth and directivity of large scanning arrays, R. S. Elliott, Appendix A) in which the author starts from this expression:

##\frac{\sin\left [ \left ( 2N+1 \right ) u_0\right ]}{\sin(u_0)}\sum_{p=-P}^Pa_p\cos(p\pi)\left [\frac{\sin(u_0)}{\sin(u_p)} -1+1 \right ]##

where ##a_p=a_{-p}## and ##u_p=\frac{\pi L}{(N+1)\lambda}\left(\cos \theta'-\cos\theta_0+\frac{p\lambda}{L}\right)##, and says that we can approximate it in the following way:

##( 2N+1)\frac{\sin\left [ \left ( 2N+1 \right ) u_0\right ]}{( 2N+1)u_0}\left \{ a_0+\sum_{p=1}^P2a_p\cos(p\pi) \right \} - ( 2N+1)\frac{\sin\left [ \left ( 2N+1 \right ) u_0\right ]}{( 2N+1)u_0}\sum_{p=1}^P2a_p\cos(p\pi)\frac{p^2}{p^2-K^2}##

where ##K=\frac{L}{\lambda}\left(\cos\theta'-\cos\theta_0\right)##, because "##u_0## and ##u_p## are small and ##P## is a small integer".

I simply can't understand why. My attempt, using ##\sin u_p\approx u_p##, gives the same approximate expression, but with the difference that I have a factor ##\frac{p}{p+K}## instead of the factor ##\frac{p^2}{p^2-K^2}##.

What am I missing?

The first line looks funny. -1+1 at end?? I went no further.

I didn't work out every single detail, but I got a term that is ## K/(K+p) ##. When the negative ## p's ## are summed, this will change that to a ## K/(K-p) ##. (Notice the final sum is over positive ## p ##). The common denominator becomes ## K^2-p^2 ##. It's a lot of algebra, but if the author was careful, he might have got it right.
Edit: Notice ##2(1-\frac{p^2}{p^2-K^2})=2 \frac{K^2}{K^2-p^2}=\frac{K}{K+p}+\frac{K}{K-p} ##.
I think the author got it right.