- #1

thomas49th

- 655

- 0

E_{z} = E_{x} + E_{y}:

Proof:

[tex] E_{z} => \int^{\infty}_{-\infty} {(x(t) + y(t))^{2}} dt

=> \int {(x(t) + y(t))^{2}}^{2} dt

=> \int (x^{2}(t)) + \int(y^{2}(t))dt + \int x(t)y(t)dt

=> E_{x} + E_{y}

[/tex]

because [tex]\int x(t)y(t)dt[/tex] = 0 because of integration by parts:

u = x(t) dv/dt = y(t)

u' = dx/dt, v = [tex]frac{y^{2}(t)}{2}[/tex]

so [tex]x(t)\frac{y^{2}(t)}{2} - \int {\frac{y^{2}(t)}{2}\frac{dx}{dt}}dt[/tex]

[tex]x(t)\frac{y^{2}(t)}{2} - \int {\frac{y^{2}(t)}{2}}dx[/tex]

we can treat y^2(t) as a constant so:

[tex]x(t)\frac{y^{2}(t)}{2} - \int^{\infty}_{-\infty} {\frac{y^{2}(t)}{2}}dx[/tex]

[tex]x(t)\frac{y^{2}(t)}{2} - } [{\frac{y^{2}(t)x}{2}}]^{\infty t}_{-\infty t}[/tex]

but the problem is that the limits were destined for integrating with respect to time. I'm not integrating with respect to x.

Any suggestions?

Thanks

Thomas