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The constant is $2$. Let $n=\lfloor d_2/d_1 \rfloor \geq 1$, and let $t_k$ be the time which the long distance runner takes to arrive at the distance $kd_1$ from the origin, $1\leq k\leq n$. Proving …

There are two outstanding books which I found very readable (I belong to the class of mathematicians who have great difficulties reading physics books and papers): Landau and Lifshitz, Mechanics, and …

Carlo Beenakker's answer is right. When t>0, you cannot close the contour in the lower half-plane, because the exp in the numerator is large in the lower half-plane. You must close the contour in the …

Most explanations miss a very simple point. Mercator projection becomes simple if we use complex numbers. It is the stereographic projection which sends the North pole to $\infty$ and the South pole t …

Lifshitz, Course of theoretical physics, vol. I, and, of course, V. Arnold, Mathematical methods of classical mechanics. …

The earliest application is the Mercator projection which was introduced long before the complex exponential was defined in the way we define it nowadays. $z\mapsto e^z$ is considered as a map from th …

Besides the important physical motivation pointed out by Carlo Beenakker, there is another one, purely mathematical. Laplace transform is a generalization of a power series (and Dirichlet series). In …

Now, this problem arises from problems of mathematical physics: to solve the Laplace equation in $R^3$, or to solve the eigenvalue problem for Laplace's equation. …