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drorspei.com | ||
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ckrao.wordpress.com
In this post I would like to prove the following identity, motivated by this tweet. $latex \displaystyle n! \prod_{k=0}^n \frac{1}{x+k} = \frac{1}{x\binom{x+n}{n}} = \sum_{k=0}^n \frac{(-1)^k \binom{n}{k}}{x+k}$ The first of these equalities is straightforward by the definition of binomial coefficients. To prove the second, we make use of partial fractions. We write the expansion $latex \displaystyle... |
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mathematicaloddsandends.wordpress.com
I recently learned of Craig's formula for the Gaussian Q-function from this blog post from John Cook. Here is the formula: Proposition (Craig's formula). Let $latex Z$ be a standard normal random variable. Then for any $latex z \geq 0$, defining $latex \begin{aligned} \mathbb{P}\{ Z \geq z\} = Q(z) = \dfrac{1}{\sqrt{2\pi}} \int_z^\infty \exp \left( -... |
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nhigham.com
The spectral radius $latex \rho(A)$ of a square matrix $latex A\in\mathbb{C}^{n\times n}$ is the largest absolute value of any eigenvalue of $LATEX A$: $latex \notag \rho(A) = \max\{\, |\lambda|: \lambda~ \mbox{is an eigenvalue of}~ A\,\}. $ For Hermitian matrices (or more generally normal matrices, those satisfying $LATEX AA^* = A^*A$) the spectral radius is just... |
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jackkelly.name
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