Fyneman exercise I ch28

1.

Interpret the following two problems in complex numbers geometrically, and thus find absolute value of A in each case
a) $A=re^{i\frac{\theta}{2}}+re^{-i\frac{\theta}{2}}$
b) $A=\sum\limits_{n=0}^{N}{re^{in\theta}}$

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Ans

a)$|A|=2r\cos{\frac{\theta}{2}}$
b)

Let radius of circle R(r in fig., but we have already r in problem. $\theta = \phi$),
$r= 2R\sin{\frac{\theta}{2}}$ —-(1) $A=2R\sin{\frac{(N+1)\theta}{2}}$ —-(2) substitute (1) to (2)
$|A| = r \left| \frac{\sin{\frac{(N+1)\theta}{2}}}{\sin{\frac{\theta}{2}}} \right|$