Continuous Time Function

**sooyewguan** · Jun18-09, 10:22 PM

Lets say I have an equation,

$LaTeX Code: y=\\alpha e^{\\beta W}$

where,
$LaTeX Code: \\alpha = a e^{b f}$ and $LaTeX Code: \\beta = c f + d$

$LaTeX Code: W = \\int^{T}_{0}f dt$

My problem now is, what happen if

is changing with time

,

How do I modify my main equation,

, so that it become an continuous-time function,

.

Thank you.

**HallsofIvy** · Jun19-09, 07:39 AM

I'm not sure what you mean: what you give
$LaTeX Code: y= \\alpha e^\\beta W(t)$
is a "continuous-time function"- or at least a continuous function of t.

If you want to you can replace each of $LaTeX Code: \\alpha$ , $LaTeX Code: \\beta$ , and W with their explicit dependence on t:
$LaTeX Code: y(t)= ae^{bf(t)} e^{(cf+d)\\int_0^t f(u)du}$
(I've changed the dummy variable in the integral to u so as not to confuse it with the variable t.)

But I don't think that really adds anything as long as you don't know the explicit form of f.