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Reading a book on fractional calculus reminded me that I'd like to know more on the following method/idea.

Given an integral: $\displaystyle \int_{-\pi}^\pi{\frac{\displaystyle\sum_{j=a}^b{e^{i\cdot c_j \cdot t}}}{\displaystyle\sum_{k=a}^b{e^{i \cdot d_k \cdot t}}} dt}$

I can use recurrences and generating functions to find all of the derivatives. Using all of the derivatives (evaluated at a point), I can essentially create another function (for example, the function at point $n$ is the $n$th derivative). The function brings up a curiousity: how accurate could it be at predicting the integral itself?

EDIT An additional thought: If this method fails, perhaps the derivatives could still be used to aid in a very good spline extrapolation using only a few points.

So I'm wondering what problems arise when trying to extrapolate/interpolate an integral of the form above using derivatives. I'd like to know any studies of this problem if they exist, and what I can expect. I realize I'm being a bit vague, but I'm not exactly sure what the good questions are regarding this issue. I would like to explore this possibility as much as possible, and hope that answers arrive with hope and how to evaluate the integral this way.

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A ratio of sums of complex exponentials? That's going to be rather oscillatory on the real line, depending on the values of the $c_j$ and the $d_k$ . Maybe constructing a nice contour (avoiding any poles that might occur, of course) might help here? –  J. M. Nov 10 '10 at 15:28

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If you can calculate all the derivatives at a point, say t=0, you can create a Laurent series around zero. As long as all the poles of the function are outside a circle of radius $\pi$, this will converge to the function. You can then integrate term by term to get the integral.

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