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 Jan 6 comment $2\cdot\int_0^\infty \frac{a-u^2}{\left( u^2+\frac{a^2}{b-a}\right) \left(u^2+\frac{b^2}{b-a} \right) \sqrt{\cdots } }\mathrm {d}u$ @Moron: Thank you for the push :) I hope it is much more convenient now. Greets Jan 6 awarded Editor Jan 6 revised $2\cdot\int_0^\infty \frac{a-u^2}{\left( u^2+\frac{a^2}{b-a}\right) \left(u^2+\frac{b^2}{b-a} \right) \sqrt{\cdots } }\mathrm {d}u$ Clearified question Jan 6 comment $2\cdot\int_0^\infty \frac{a-u^2}{\left( u^2+\frac{a^2}{b-a}\right) \left(u^2+\frac{b^2}{b-a} \right) \sqrt{\cdots } }\mathrm {d}u$ @Moron: Thank you to point out that my question is not really clear. I want to calculate the given integral with respect to the constant $a$ and $b$. Today, I tried to calculate the contribution at the pole via Cauchy principal value integration but my vanishing result seams erroneous. I think I am just missing some basic "singular" integral skills to calculate the integral efficiently and I thought this might be just the right place to ask :) PS.: I will update the question tomorrow due to my further calculations. Sincerely Jan 6 asked $2\cdot\int_0^\infty \frac{a-u^2}{\left( u^2+\frac{a^2}{b-a}\right) \left(u^2+\frac{b^2}{b-a} \right) \sqrt{\cdots } }\mathrm {d}u$ Dec 22 answered How to check if transformation is affine? Dec 12 comment Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense? @Raskolnikov: Thank you for the comment which pointed directly in the right direction. Nevertheless my experience in functional analysis on manifolds is very limited so I could not really grasp the idea. I think Willie Wong pointed out these lines of thought in a very explicit and decent way. Dec 11 awarded Scholar Dec 11 accepted Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense? Dec 11 comment Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense? Thank you very much for your thoughtful and detailed answer. If I got you correctly, the notion only makes sense for the existence of a timelike killing vector, say $\partial_t$. This should imply that a time <> frequency FT only makes sense for stationary solutions in which the quotient manifold $M/\psi_t$ ($\psi_t$ being the local flux to $\partial_t$) exists. Hence, the FT is only feasible for dynamical field theories if the dynamics are independent from the (background) metric. Dec 11 awarded Supporter Dec 11 awarded Student Dec 11 asked Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense? Dec 7 awarded Teacher Dec 6 answered Why study “curves” instead of 1-manifolds?