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1. ### Fourier transform

I made a mistake. The Fourier transform of 1/x actually has a sign function in it: http://www.wolframalpha.com/input/?i=fourier+1%2Fx because depending on whether k in the exponential is positive or negative, you have to complete your contour in the upper or lower half respectively of the...
2. ### Fourier transform

Well if the Fourier transform of 1/x exists, it would make the most sense that it's the principal part. However, can you really ignore what's going on at x=0 like that? For example, can you reconstruct the function from the transform? f(x)=\int \frac{dk}{2 \pi} (i \pi) e^{-ikx}=(i...
3. ### Fourier transform

Does it make sense to take the Fourier transform of a function that blows up at some point? For example the Fourier transform of f(x)=1/x, which blows up at zero? Doesn't the integral: \int^{\infty}_{-\infty} \frac{dx}{x} e^{-ikx} not converge because of x=0? Yet for some reason analytical...
4. ### Fourier transform

Suppose a function f(k) has a power series expansion: f(k)=\Sigma a_i k^i Is it possible to inverse Fourier transform any such function? For example: f(k)=\Sigma a_i k^{i+2}\frac{1}{k^2} Since g(k)=1/k^2 should have a well-defined inverse Fourier transform, and the inverse Fourier...
5. ### Volume of a simplex using delta function

I figured it out. The method I outlined is correct. So we had this: \frac{i^{d+2}}{2\pi} \int dk \mbox{ } e^{ik} \left(\frac{1}{k}\right)^{d+2}\left(e^{-ik}-1 \right)^{d+2} Manipulating: \frac{2^{d+1}}{\pi} \int dk...
6. ### Volume of a simplex using delta function

I tried to calculate the volume of a simplex, but got an integral I couldn't do. For simplicity take a 2-simplex (the volume of a 2-simplex is 1/6) V=\int da \int dx \int dy \int dz \mbox{ } \delta(1-a-x-y-z) where the integration limits are over the 4-cube. My reasoning for this formula...
7. ### Logarithms and online integrators

The integral of ln(1-x) is -(1-x)ln(1-x)-x, when 0<=x<=1. So for example: \int_{0}^{1} ln (1-x)dx= (-(1-x)ln(1-x)+x)_{0}^{1}=-1 However, going to an online integrating site: http://integrals.wolfram.com/index.jsp?expr=Log[1-x]&random=false they give the integral of ln(1-x) as...
8. ### L^1 convergence and uniform convergence

What is the L^1 norm?
9. ### L^1 convergence and uniform convergence

Doesn't L^1 mean the space of integrable functions? So why isn't f=0 in this space?
10. ### L^1 convergence and uniform convergence

If your measure space is Borel then I don't think this statement is true. A sequence of Borel measurable functions that has a limit converges to a Borel function.