Recent content by pasmith

It may help to note that $$\frac{n^2}{n + \frac15} = n - \frac{1}{5 + \frac 1n}.$$

Yes. Expanding ##(z - a)^{-k}## in binomial series can be done in two ways; one converges for ##|z| < |a|## and the other for ##|z| > |a|##.

A lot o numerical analysis involves dealing with spaces of polynomials of degree at most $$N$$ on $$[-1,1]$$. There are various families of polynomials which can be taken as basis functions, and these will be orthogonal with respect to a particular inner product. Quite often, one wishes to...

The equation of a line in ##\mathbb{R}^2## can be written as $$\mathbf{n} \cdot (\mathbf{x} - \mathbf{x}_0) = 0$$ where ##\mathbf{x}_0## is a point on the line and ##\mathbf{n}## is a vector normal to the line. Here we have ##\mathbf{n} = (\cos \omega, \sin \omega)## and ##\mathbf{x}_0 = (5\cos...

You can use linear interpolation between the data points. Between z_i and z_{i+1} that gives you \int_{z_i}^z \frac{1}{\lambda(z)}\,dz = \int_{z_i}^z \frac{1}{A_i + B_iz}\,dz which you can do analytically.

Yes, where D is a Jordan normal form, ie. the expression of the map with respect to a basis of (generalized) eigenvectors \{v_1, \dots, v_n\}. Conjugation by P then gives the expression of the map with respect to the standard basis.

For the purpose of this exercise, does it make a difference to the mathematical analysis if the boundary condition is the dimensionally consistent kUt^p or U(t/t_0)^p rather than sloppy Ut^p? It is common to use scaled units in order sweep such constants of proportionality under the carpet. The...

Can you find a Liapunov function for the system?

I would assume the first, since the second would have been written as y \ln x. But the first leads to a non-linear, non-separable equation and the second leads to a linear equation.

A graphical description is not really a rigorous proof, although it might help you to find one. Ultimately, showing that X/\sim is homeomorphic to Y requires finding a continuous function f: X/\sim \to Y and showing that it has a continuous inverse. For example, showing that [0,1]/\sim where 0...

It should not be difficult to get from a graphical description to a parametrisation. For example, for the Mobius strip one can take a line segment with centre on a circle of radius R such that the angle to the horizontal plane goes through a half rotation as the centre moves through a full...

\epsilon < 8 renders as \epsilon < 8. Originally discovered in my post at https://www.physicsforums.com/threads/limit-of-piecewise-function-using-epsilon-delta.1081023/post-7267512. Looking at the preview of this post (with LaTeX not being rendered due to a known bug) suggests that < is being...

You haven't told us the definition of f. You can assume \epsilon < 8 (a \delta which works for such an \epsilon will also work for any larger \epsilon) so that 0 < 8 - \epsilon < 8 + \epsilon.

Most calculators will label it as "tan-1" rather than "arctan". In mathematics we generally measure angles in radians rather than degrees, so make sure that the calculator is set to the right unit.

How do you define [z] for z \in \mathbb{C}?