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Continuous Time Function

  1. Jun 18, 2009 #1
    Lets say I have an equation,

    [tex]y=\alpha e^{\beta W}[/tex]

    where,
    [tex]\alpha = a e^{b f}[/tex] and [tex]\beta = c f + d[/tex]

    [tex]W = \int^{T}_{0}f dt[/tex]

    My problem now is, what happen if [tex]f[/tex] is changing with time [tex]t[/tex], [tex]f(t)[/tex]

    How do I modify my main equation, [tex]y[/tex], so that it become an continuous-time function, [tex]y(t)[/tex].

    Thank you.
     
  2. jcsd
  3. Jun 19, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

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

    If you want to you can replace each of [itex]\alpha[/itex], [itex]\beta[/itex], and W with their explicit dependence on t:
    [tex]y(t)= ae^{bf(t)} e^{(cf+d)\int_0^t f(u)du}[/tex]
    (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.
     
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