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Use rescaling or translation to normalize parameters

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Quick description

One can often use rescaling transformations such as x \mapsto \lambda x or translation transformations such as x \mapsto x-x_0 to normalize one or more parameters in the integral to equal a particularly simple value, such as 0 or 1. These changes of variables can create additional factors, but usually such factors can be moved outside of the integral, thus simplifying the remaining integrand.

Prerequisites

Undergraduate calculus; complex analysis

Example 1

In the integral \int_0^\infty \frac{1}{a^7+x^7}\ dx discussed on this page, one can use the substitution x = ay to change this integral to a^{-6} \int_0^\infty \frac{dy}{1+y^7}, which in fact already solves the problem except for the task of computing the numerical quantity \int_0^\infty \frac{dy}{1+y^7} (but if one is willing to lose multiplicative constants, one does not even need to do that, once one checks of course that this quantity is finite).

Example 2

Problem: (Fourier transform of Gaussians) Compute the integral \int_\R e^{-\pi (x-x_0)^2 / r^2} e^{- 2\pi i x \xi}\ dx, where r > 0 and x_0, \xi \in \R are parameters.

Solution: We begin by using the translation substitution = x-x_0 to eliminate the x_0 factor:

 \int_\R e^{-\pi (x')^2 / r^2} e^{- 2\pi i x' \xi} e^{-2\pi i x_0 \xi}\ dx'.

The factor e^{-2\pi i x_0 \xi} is independent of x' and can be moved out of the integral. Next, we use the scaling substitution  ry to rewrite the integral as

 r e^{-2\pi i x_0 \xi} \int_\R e^{-\pi y^2} e^{-2\pi i r y \xi}\ dy.

(One could also have used a similar rescaling to eliminate the \pi factors, but they will turn out to be rather convenient for us, actually, so we will keep them.) The next trick is to complete the square, writing the above integral as

 r e^{-2\pi i x_0 \xi} \int_\R e^{-\pi (y+ir\xi)^2} e^{-\pi r^2 \xi^2}\ dy.

One of the reasons we complete the square is because we can move the y-independent factor out of the integral:

 r e^{-2\pi i x_0 \xi} e^{-\pi r^2 \xi^2} \int_\R e^{-\pi (y+ir\xi)^2} \ dy.

Now we would like to normalize away the ir\xi factor by making the substitution z = y+ir\xi. This turns the real integral into a contour integral, but one should not be afraid of this, and obtain

 r e^{-2\pi i x_0 \xi} e^{-\pi r^2 \xi^2} \int_{-\infty+ir\xi}^{\infty+ir\xi} e^{-\pi z^2} \ dz.

But now we can apply the contour shifting method and get rid of ir\xi altogether:

 r e^{-2\pi i x_0 \xi} e^{-\pi r^2 \xi^2} \int_{-\infty}^{\infty} e^{-\pi z^2} \ dz.

Now we use the standard integral \int_\R e^{-\pi x^2}\ dx = 1 (computed at the square and rearrange page) and arrive at our final answer of r e^{-2\pi i x_0 \xi} e^{-\pi r^2 \xi^2}.

General discussion

It is often a good tactic to perform as many normalizations as one can first, as this can eliminate many parameters which would otherwise clutter up the arguments later. However, there are exceptions to this rule:

  • Sometimes, one wants to cut up an integral into many pieces, and perform a different normalization for each piece. In such cases, a premature normalization may be counter-productive.

  • In other cases, one wants to do something sneaky with the parameter, e.g. differentiate with respect to it. Again, in that case it may be counter-productive to normalize the parameter prematurely.

One can also use dimensional analysis as a substitute for the rescaling normalisation (examples needed).