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I want to convert an integral from $(t_0, \infty)$ (or $(-\infty, t_0)$) range to $(-\infty, \infty)$ range by change of variable. What is the best transform function to do this - one that is simple, monotonic with $f(-\infty)=t_0$ and $f(\infty)=\infty$ (or $f(-\infty)=-\infty)$ and $f(\infty)=t_0$)?

I need this for an application of numerical quadrature with Gaussian weights.


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up vote 3 down vote accepted

$ f ( x ) = t_0 + e^x\ $ is an example, or $g(x)=t_0 - e^{-x}$.

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+1 Typical community wiki answer! :) – AD. Jun 19 '12 at 7:00

$$f(t)=t_0+\mathrm e^t$$ $ $ $ $ $ $

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