Transforming limits of integration to a bounded region

[nkinar]

Hello---

I've been working on a problem which requires the numerical evaluation of an improper integral. I would like to transform the limits of integration on [0,\infty) to the bounded region [a,b] by replacing the variable \omega with another variable. Here is the integral:

<br /> u(t,\tau)=\frac{1}{\pi}\int_{0}^{\infty}\! G(\omega)\, d\omega<br />

<br /> G(\omega)=4\sqrt{\pi}\frac{\omega^{2}}{\omega_{0}^ {3}}\mbox{exp}\left(-\frac{\omega^{2}}{\omega_{0}^{2}}\right)\mbox{cos \left(\omega t-\left(\frac{\omega}{\omega_{0}}\right)^{-\gamma}\omega\tau\right)\mbox{exp}\left(-\frac{1}{2Q}\left(\frac{\omega}{\omega_{0}}\right) ^{-\gamma}\omega t\right)}<br />

How should I proceed? Is there a relevant reference which could point me in the proper direction?

 

[nkinar]

Perhaps contour integration could provide a technique to evaluate real definite integrals? Is there a good textbook/monograph available on contour integration?

 

[nkinar]

I've found a monograph with some recommended substitutions [1], but I can't immediately see how this would transform the limits of integration.

Here are three recommended substitutions found in the monograph to transform the limits of integration from [0,\infty) to [0,1), using the variable y:

<br /> \omega = -\alpha \mbox{log}(1 - y), \alpha &gt; 0<br />

<br /> \omega = \frac{y}{1-y}<br />

<br /> \omega = \left ( \frac{y}{1 - y} \right) ^ 2<br />


But what is particularly confusing is that as y \rightarrow 1, \omega \rightarrow |\infty|.

Is there anything that can be done to transform [0,\infty) to [0,1] or similar?


[1] A.R. Krommer and C.W. Ueberhuber, Computational Integration, Philadelphia: Society for Industrial and Applied Mathematics, 1998.

 

[nkinar]

Is there a way to numerically integrate this integral? I've tried to perform the integration on a truncated interval such as [0,1000] or [0,10000] instead of [0,\infty) but I've found that the truncated integral cannot adequately approximate the improper integral.

There's got to be a way to properly do this.

 

[nkinar]

Oh, and I've also learned that as \omega \rightarrow 0, G(\omega) \rightarrow \infty, so the interval over which the integration is performed would have to be (0,\infty), or the transformed interval would have to be (0,1).

 

[nkinar]

So perhaps the best way to proceed would be to use a numerical integration procedure which does not use the endpoints of the interval. One of these methods is the "open" Newton-Coates integration method applied to the transformed interval (0, 1).

 

[nkinar]

Okay, well - the following method worked for me. An algorithm to perform adaptive quadrature can be found in the paper by Shampine [1]. This method has been implemented in Matlab as "quadgk." Running the quadgk function on the interval [0,\infty) worked well for u(t, \tau) above.[1] L. Shampine, “Vectorized adaptive quadrature in MATLAB,” Journal of Computational and Applied Mathematics, vol. 211, Feb. 2008, pp. 131-140.