updates
Find the solution of the following initial value problem.
I have the solve the following exercise: Exercise: Describe the qualitative nature of the solution of the initial value problem $$ y'' + 4y = g(...
Apr 15, 2026
Archive: Updates
updates
Understanding Lipschitz Continuity
I have heard of functions being Lipschitz Continuous several times in my classes yet I have never really seemed to understand exactly what this concept re...
Apr 15, 2026
updates
What does a circle with a dot in it represent?
Consider the following formula for finding the area of an arbitrary planar polygon: $$ A=N\odot\sum_{i=0}^{n-1}\left(\frac{P_i\times P_{i-1}}{2}\right)$$ ...
Apr 15, 2026
updates
How to find Laurent series Expansion
$f(z)$ is defined like this: $$ f(z) = \frac{z}{(z-1)(z-3)} $$ I need to find a series for $f(z)$ that involves positive and negative powers of $(z-1)$, w...
Apr 15, 2026
updates
positive definite matrix plus positive semi matrix equals positive definite?
I have a questions related to the positive definite[PD] matrix and positive semi definite[PSD] matrix I see and get the property about PD and PSD 1) PD + ...
Apr 15, 2026
updates
Is there a proof for the fact that, if you perform the same operation on both sides of an equality, then the equality holds?
Is there a proof for this, or is it just taken for granted? Does one need to prove it for every separate case (multiplication, addition, etc.), or only wh...
Apr 15, 2026
updates
Calculating half-angle of cone of rays entering pinhole camera
I have the following pinhole camera: The cone of rays that would enter the pinhole from the object would resemble the following: (Image from https://www.o...
Apr 15, 2026
updates
Simplify $\frac{5}{6}\log\left(\frac{5}{4}\right) - \frac{1}{6}\log(2)$ to $\log\left(\frac{5}{4}\right) - \frac{1}{6}\log\left(\frac{5}{2}\right)$
I'm trying to bring this expression: $$\frac{5}{6}\log\left(\frac{5}{4}\right) - \frac{1}{6}\log(2)$$ To this one: $$\log\left(\frac{5}{4}\right) - \...
Apr 15, 2026
updates
Are functions still continuous even with removable discontinuities?
If we define a continuous function as a function that is continuous at all points in its domain, then would a removable discontinuity not impact this? For...
Apr 15, 2026
updates
Comparing $2^n$ to $n!$
Comparing $\;2^n\;$ and $\;n!\;$ I need to rewrite one or the other so they can be more easily compared (i.e. have similar form). Because of the factorial...
Apr 15, 2026
