User:Egm4313.s12.team8.colocar/R7.1

Problem Statement
$$ \displaystyle \text{Verify (1)-(2) }\ $$ ((4) and (5) p19.9, respectively)

$$ \displaystyle \text{Given:} $$

Verifying (1)
$$ \displaystyle \text{From equation } \, $$($$) $$ \displaystyle \, \text{ we have } $$ $$ \displaystyle \langle \bar f, \bar g \rangle := \int_0^L \bar f(x) \, \bar g(x) \, dx, \text{ where we can plug in } \phi \, \text{ from } \, $$ ($$) $$ \displaystyle \, \text{ to get} $$

$$ \displaystyle \text{From } $$ ($$), $$ \displaystyle \text{ and using equation } \, $$ ($$), $$ \displaystyle \, \text{ we get}$$

$$ \displaystyle \text{or} $$

$$ \displaystyle \text{From equation } \, $$ ($$), $$ \displaystyle \text{ for all constants of } \omega_i \text{ and } \omega_j \text{ yield } 0 \text{ as all values of } \sin (\pi) \text{ equal zero. Thus,} $$ $$ \displaystyle \langle \phi_i, \phi_j \rangle = 0 \ \text{ for } \ i \ne j $$

Verifying (2)
$$ \displaystyle \text{Given equation } \, $$($$) $$ \displaystyle \, \text{ and from } $$ ($$), $$ \displaystyle \, \text{ we have } $$

$$ \displaystyle \text{Plugging in the identity, and substituting } L=\pi \text{ from } [0,\pi] \text{ gives } $$

$$ \displaystyle \text{or} $$ $$ \displaystyle \langle \phi_j, \phi_j \rangle := \frac{1}{2} \left [ x - \frac{1}{2 \omega_j} \sin(2 \omega_j x) \right ]_0^\pi $$ $$ \displaystyle \langle \phi_j, \phi_j \rangle := \frac{1}{2} \left [ \pi - \frac{1}{2 \omega_j} \sin(2 \omega_j \pi) \right ] $$ $$ \displaystyle \text{Thus} $$ $$ \displaystyle \langle \phi_j, \phi_j \rangle := \frac{\pi}{2} = \frac{L}{2} \text{ for } i=j. $$