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:

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

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

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 $$y$$:

$$\omega = -\alpha \mbox{log}(1 - y), \alpha > 0$$

$$\omega = \frac{y}{1-y}$$

$$\omega = \left ( \frac{y}{1 - y} \right) ^ 2$$


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.