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Natural Science Forum / Physics / General Physics / July 2008Convergent series for Exponential integral
Hi I need a little help here. I have a function, f = exp (i*k*sqrt(x^2+a^2)) / sqrt(x^2+a^2). I need to integrate this function with respect to x. As this is an exponential integral, exact integral is not possible. what could be the convergent series for this? i = sqrt (-1) and a, k are constants. > Hi > I need a little help here. [quoted text clipped - 5 lines] > As this is an exponential integral, exact integral is not possible. > what could be the convergent series for this? Asking sci.math would be better for this type of thing. My suggestion is expand f in a power series, then integrate term by term. I cannot promise convergence though. However if you want to integrate along (-\infty,\infty) or [0,\infty) I believe this can be solved exactly using contour integration. > Hi > I need a little help here. > I have a function, > f = exp (i*k*sqrt(x^2+a^2)) / sqrt(x^2+a^2). > I need to integrate this function with respect to x. > As this is an exponential integral, exact integral is not possible. > what could be the convergent series for this? Do you need a definite or an indefinite integral? If you write x = a sinh u, you'll simplify it; the definite integral from zero to infinity will be a comnination of Bessel functions. Steve Carlip > Do you need a definite or an indefinite integral? I need a definite integral int(f,x,0,1) > If you write > x = a sinh u, you'll simplify it; the definite integral from zero > to infinity will be a comnination of Bessel functions. could you elaborate? Leo > > Do you need a definite or an indefinite integral? > I need a definite integral > int(f,x,0,1) That's harder... > > If you write > > x = a sinh u, you'll simplify it; the definite integral from zero > > to infinity will be a comnination of Bessel functions. > could you elaborate? Do the change of variables, and you'll get something proportional to an integral of exp{ika cosh u} du. The definite integral from u=0 to infinity is a linear combination of J_0(ka) and Y_0(ka) -- look up the integral representations of the Bessel functions. Steve Carlip |
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