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I'm interested in showing that:

$ \frac{d}{dt}P \; \int_{-\infty}^{\infty} \frac{\phi(x)}{x-t}dt = P \int_{-\infty}^{\infty}\frac{\phi(x)-\phi(t)}{(x-t)^2}dt $

where $\phi(x)$ is a test function (goes to zero at $-\infty$ and $\infty$ fast enough that we don't have to worry about x not going to zero fast enough)

The problem is that when I attempt this:

$ \frac{d}{dt}P \; \int_{-\infty}^{\infty} \frac{\phi(x)}{x-t}dt = \lim_{\epsilon \rightarrow 0} \left\{ -\frac{\phi(t+\epsilon)}{\epsilon}-\frac{\phi(t-\epsilon)}{\epsilon} +\int_{-\infty}^{t-\epsilon}\frac{\phi(x)}{(x-t)^2} + \int_{t+\epsilon}^{\infty}\frac{\phi(x)}{(x-t)^2} \right\} $

But I don't know where to go from here... and I'm not sure I'm on the right track. The $\phi(t)$ term doesn't seem to want to pop out.

Could this be some sort of dirac delta identity I'm missing?

Thanks!

  • 0
    how do you end up with P∫∞−∞dx 1/(x−t)^2=2/ϵ? I got 1/(ϵ+t)+1/(ϵ-t)2013-11-14

2 Answers 2

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I will assume you are trying to show $\frac{d}{d t} \mathrm{P} \int_{-\infty}^\infty dx \ \frac{\phi(x)}{x-t} = \mathrm{P} \int_{-\infty}^\infty dx \ \frac{\phi(x)-\phi(t)}{(x-t)^2}$ Using the definition of principal value and the Leibniz integral rule you will find $\frac{d}{d t} \mathrm{P} \int_{-\infty}^\infty dx \ \frac{\phi(x)}{x-t} = -\lim_{\epsilon\rightarrow 0}\frac{2}{\epsilon} \phi(t) + \mathrm{P} \int_{-\infty}^\infty dx \ \frac{\phi(x)}{(x-t)^2}$ Then notice that $\mathrm{P} \int_{-\infty}^\infty dx \ \frac{1}{(x-t)^2} = \frac{2}{\epsilon}$ The result follows immediately.

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One can begin with $\frac{d}{dt}P \; \int_{-\infty}^\infty \frac{\phi(x)-\phi(t)}{x-t}dx.$