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First of all I strongly recommend a great two-volume book by R.Courant and D.Hilbert Methods of Mathematical Physics, vol.1 (linear algebra, series expansions, integral equations, calculus of ...

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There are a few books written in Mathematica and dealing with mathematical research (to some extent), they are really interesting, e.g. : "Mathematica in Action" by Stan Wagon (a ...

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If you are looking for practical solving of differential equations I recommend : Advanced Mathematical Methods for Scientists and Engineers by Carl M. Bender and Steven A.Orszag. Methods of ...

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Given theorem on existence and uniquenes of solutions to ordinary differential equations (Picard's theorem) there is only one local solution if the Lipschitz condition is satisfied, in fact it is when ...

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This integral can be expressed with help of the residue theorem: $$I=\oint \frac{\cos(z)}{z(e^{z}-1)}dz=2\pi i \sum_{k}\; \mathrm{Res}(f,a_k)$$ There is only one removable singularity at $z=0$ and you ...

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I strongly recommend to read a review paper with many interesting references therein : PDE as a Unified Subject by Sergiu Klainerman. An essay on partial differential equations written by a leading ...

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It is more convenient to calculate the integral with respect to $z$ first and then with respect to $x$ and $y$. We have $$\int_{0}^{\infty}\frac{1}{(1+x^2 z^2)(1+y^2 z^2)} dz=\frac{x \operatorname{... View answer 3 answers 3 votes 3k views 2 votes Method I Since we have a fraction going to a non-vanishing value given its denominator is going to 0 we have to assume that its numerator also tends to 0, therefore we should solve : Reduce[Limit[... View answer 2 answers 0 votes 86 views 1 votes Let's write$$\int_0^\infty \frac{x \ln(x)}{x^4+4x^2+1}dx=\int_0^1\frac{x \ln(x)}{x^4+4x^2+1}dx+\int_1^\infty \frac{x \ln(x)}{x^4+4x^2+1}dx$$For the first integral we change the variable x\to z=1/x ... View answer 3 answers 0 votes 69 views 1 votes$$\lim_{n\to \infty}(\sqrt{n^2+6b}-\sqrt{n^2-n})=\lim_{n\to \infty}\frac{n^2+6b-(n^2-n)}{\sqrt{n^2+6b}+\sqrt{n^2-n}}=\lim_{n\to \infty}\frac{6b+n}{\sqrt{n^2+6b}+\sqrt{n^2-n}}=\\ \lim_{n\to \infty}\...

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Since $34 max(p,q,r)$ is even, it means that $p q + q r + r p$ must be even since $2018$ is even. Now, all three numbers of $p q, q r, r p$ cannot be odd, thus at least one is even, i.e. say $p=2$, ...