query: Prove that $\int_{-\infty}^{\infty} e^{-x^2} H_m(x)H_n(x)dx$
positive_passages.docid: 29230
positive_passages.text: Orthogonality of Hermite functions
negative_passages.docid: 8829
negative_passages.text: How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$
negative_passages.docid: 31792
negative_passages.text: show that $\int_{0}^{\infty}\frac{x\cos ax}{\sinh x}dx=\frac{\pi^2}{4} \operatorname{sech}^2 \left(\frac{a\pi}{2}\right) $
negative_passages.docid: 66852
negative_passages.text: Integral $ \int_{-\infty}^{\infty}\frac{x^{2}}{\cosh\left(x\right)}\,{\rm d}x $
negative_passages.docid: 129892
negative_passages.text: show $\int_0^{\infty} \frac{x\sin ax}{x^2+t^2}dx = \frac{\pi}{2}e^{-at}$
negative_passages.docid: 61133
negative_passages.text: Show that $\int|f(x)|dx=\int_0^\infty m(E_\alpha)d\alpha$
negative_passages.docid: 89602
negative_passages.text: $ \int_{0}^{ \infty} \int_{0}^{ \infty} \frac { e^{-(x+y)}}{x+y} dx dy $
negative_passages.docid: 157426
negative_passages.text: Proving that:$\int_X f_n g \, d\mu \to \int_X fg \, d\mu$ for all $g$ in $\mathscr{L}^q (X)$
negative_passages.docid: 142249
negative_passages.text: Show that $\lim_{n\to \infty} \int_E \cos^2(nx + a_n) dx = \frac{1}{2}m(E)$
negative_passages.docid: 10093
negative_passages.text: $\lim_{x\to \infty}e^{-x^2}\int_x^{x+1/x}e^{t^2}dt$
negative_passages.docid: 84413
negative_passages.text: Why is $\int_{-\infty}^\infty |g(x)|\,dx = \int_0^\infty \mu(\{x : |g(x)| \ge t\})\,dt$ true?
negative_passages.docid: 89956
negative_passages.text: Is the integral $\int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$ equal for all $a \neq 0$?
negative_passages.docid: 12865
negative_passages.text: Prove that $\lim_{n\to\infty} H_n/n = 0$ ($H_n$ is the $n$-th harmonic number) using certain techniques
negative_passages.docid: 72748
negative_passages.text: Proof that $\sum\limits^{\infty}_{i=0}{iP(X > i)} = \frac12(E(X²)-E(X))$ for $X$ nonnegative and integer valued
negative_passages.docid: 100642
negative_passages.text: Proof of the L'Hôpital Rule for $\frac{\infty}{\infty}$
negative_passages.docid: 102636
negative_passages.text: Contour Integral: $\int^{\infty}_{0}(1+z^n)^{-1}dz$
negative_passages.docid: 119133
negative_passages.text: there is $M<\infty$ such that $\sum_{n} |\hat{f}(n)|\le M\int_{0}^{2\pi}|f(t)|dt$ for each $f\in X$