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Math Help - expansion of a function

  1. #1
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    expansion of a function

    when reading a paper, the author said that
    y(t) = Ur(t)\int_{-\infty}^t dt^\prime \exp\left(-\frac{t-t^\prime}{\tau} - U\int_{t^\prime}^t d t^{\prime\prime} r(t^{\prime\prime})\right)
    can be expanded as expression involving r(t), r^\prime(t) ...
    (if r^\prime / r \ll r )
    y(t) \approx \frac{r}{1+rU\tau} +r^\prime\frac{Ur\tau^2}{(1+rU\tau)^3} + \cdots ,
    I don't know how he arrived at this conclusion. Can anyone enlighten me with some suggestions?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by wclayman View Post
    when reading a paper, the author said that
    y(t) = Ur(t)\int_{-\infty}^t dt^\prime \exp\left(-\frac{t-t^\prime}{\tau} - U\int_{t^\prime}^t d t^{\prime\prime} r(t^{\prime\prime})\right)
    can be expanded as expression involving r(t), r^\prime(t) ...
    (if r^\prime / r \ll r )
    y(t) \approx \frac{r}{1+rU\tau} +r^\prime\frac{Ur\tau^2}{(1+rU\tau)^3} + \cdots ,
    I don't know how he arrived at this conclusion. Can anyone enlighten me with some suggestions?
    That is awfully hard to read. Could you maybe explain the variables? Is U just a constant? When you write U\int_t^t' dt'' r(t'') is that just then U(t-t')?
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  3. #3
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    Sorry for the inconvenience, the only functions these formulas is r(x) and y(t), both U and \tau are constants.
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