Recent content by rsq_a

Thank you for this note, and letting me know that I oversimplified things. I absolutely agree that in many cases, the use of complex numbers runs so deep that it's just not feasible (or of interest) to go back and see what whether one can formulate an analogous formulation without complex...

I'm sorry this is a few-month old thread, but I just came back today. You keep on saying "many calculations done in designing stem from complex numbers", but that's besides the point. Just because went from X to Y through the complex plane does not mean that it's the only way. A very simple...

It's not so much about age as it is about circumstance. Drive is definitely connected to age and age to circumstance. For example, when you're 20 years old, and your entire lift is devoted to schooling, then it's relatively easy to concentrate on your studies and pull an all-nighter. Try doing...

Yes, thank you for helping to get it back on topic. I did think of it this way: consider the series expansions around x = 0 of f(x) = \frac{1}{1+x^2} \quad \text{and} \quad g(x) =\frac{1}{1-x^2} Although these two functions have differing values, their rate of convergence is identical (via...

Oops. I think I understand. So for something like the circumference of a circle, pi would be expressed as the limit of some rational sequence, correct? I'm happy with that. You're right and it was a good point. Okay, so I'd like to understand the notion of radius of convergence sticking with...

Perfect. We're in agreement. So how do you explain the notion of radius of convergence as the distance from the point of expansion to the nearest 'singularity' without the notion of complex functions and the complex plane (e.g. for example, in terms of geometry in R^2)? For example, the...

Yes and no. Your example isn't valid, because we need our number system to be dense. It's important that pi is irrational, otherwise all our (real) results would be off. I'm not talking about numerical precision. If you could theoretically compute to arbitrary precision, then it's relevant...

Um. Thanks. I work with complex numbers every day (conformal mapping, contour integration, complex Fourier transforms, boundary integral methods, steepest descent approximations, etc.) ... so yes, I'm aware of their applicability, particularly as it relates to problems in mechanics. It doesn't...

Well, this is a mathematics forum, so presumably, we can pretend we're interested in answering the question for the question's sake... :smile: In any case, I think the question is fairly clear. The reason I ask is because I'm designing a lecture course and part of this requires me to explain...

Pretend that you are expaining the following to someone who knows nothing about complex numbers and within a universe where complex numbers have not been invented. In examining the function f(x) = \frac{1}{1 + x^2} we can derive the series expansion \sum_{n=0}^\infty (-1)^n x^{2n} We...

I can't say as an engineer, but simply as an applied mathematician: it's all mechanics, isn't it? Fluid mechanics and solid mechanics are intimately related in the sense that a lot of techniques and theories have analogues in one and the other. So at least from a theoretical standpoint, many...

I'm having a really hard time working with non-standard expressions for the curvature. This deals with two expressions for the curvature, one for a 2D plane curve, and the other for an axi-symmetric surface in 3D. The plane curve Suppose we have a plane curve given by x = h(y) for both x...

...I've no idea why I thought it was a few days ago. You're right, I was speaking nonsense.

I think in the case of a constant coefficient ODE, it's perhaps a bit easier to reason (??) Here is one argument just by counting unknowns. Maybe the first non-trivial example is with 3 dependent variables: \begin{align} x_1' &= p_{11} x_1 + p_{12} x_2 + p_{13} x_3 \\ x_2' &= p_{21} x_1 +...

Is there an easy way to show that the system: \begin{align} x_1' &= p_{11} x_1 + p_{12} x_2 + \ldots + p_{1n} x_n \\ x_2' &= p_{21} x_1 + p_{22} x_2 + \ldots + p_{2n} x_n \\ \ldots &= \ldots \\ x_n' &= p_{n1} x_1 + p_{n2} x_2 + \ldots + p_{nn} x_n \end{align} must be equivalent to...