# how to graph the following function ?

Dear guys

I would like to find the graph of the following function

let $$v(x)=\int_{-\infty}^{\infty}[\int_{0}^{\infty}t^{\alpha}e^{-t^{2}/2 +\kappa(b-y)t}dt(\int_{0}^{1}\frac{u^{\beta}}{(1-u^2)^{1/2}}e^{\frac{-\gamma[(x-b)u+(b-y)]^2}{(1-u^2)}}du)]dy$$

where $\alpha$ is any real number, $\kappa,b,\beta,\gamma\in (0,\infty),$

I know it is impossible to find the integral. What I did by far is :

1. Approximate the integral $$(\int_{0}^{1}\frac{u^{\beta}}{(1-u^2)^{1/2}}e^{\frac{-\gamma[(x-b)u+(b-y)]^2}{(1-u^2)}}du)$$ by Simpson method, or Newton-cotes method

2. Approximate the integral $$\int_{0}^{\infty}t^{\alpha}e^{-t^{2}/2 +\kappa(b-y)t}dt$$

3. Approximate the integral by plugging what I have in 1) and 2)

4. Graph the function

but using approximation methods I know are good enough.I meant the error is so big It would be great if some one could tell me a good method to find the graph of the above function

many thanks !

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Removed the "topological-graph-theory" tag, as plotting the graph of a function over a range is not at all what topological graph theory means. – sleepless in beantown Nov 28 '10 at 6:30
Removed the [ap.analysis-of-pde] tag as the question statement does not contain a partial differential equation – Willie Wong Nov 28 '10 at 7:54
I suspect your best bet here is to construct a specialized Gaussian quadrature method; you have quite a lot of leeway in choosing what your "weight function" ought to be, and what you ought to ultimately use would depend on the severity of the singularities. – J. M. Nov 30 '10 at 10:19