Articles of analysis

Proof to show function f satisfies Lipschitz condition when derivatives f' exist and are continuous

The question is as follows: Given a function $f$, 2 known information: (1) $f'(x)$ exist (2) $f'(x)$ are continuous Goal: function $f$ satisfies Lipschitz condition on any bounded interval $[a,b]$ Here is my attempt: 1/ Recall Lipschitz condition: a function $f$ satisfies Lipschitz if there is a real number $N$ such that $|f(x) – f(y)| […]

Integration of $\ln $ around a keyhole contour

I want to evaluate the following integral: $$\int_{0}^{\infty}\frac{\ln^2 x}{x^2-x+1}{\rm d}x$$ I use the following contour in order to integrate. I considered the function $\displaystyle f(z)=\frac{\ln^3 z}{z^2-z+1}$. The poles of the function are $\displaystyle z_1=\frac{1+i\sqrt{3}}{2}, \; z_2=\frac{1-i\sqrt{3}}{2}$ and these are simple poles. I evaluated the residues $\displaystyle \mathfrak{Res}(z_1)=\mathfrak{Res}(z_2)=-\frac{i\pi^2}{9\sqrt{3}}$. If we declare $\gamma$ the entire contour , […]

What kind of “mathematical object” are limits?

When learning mathematics I tend to try to reduce all the concepts I come across to some matter of interaction between sets and functions (or if necessary the more general Relation) on them. Possibly with some extra axioms thrown in here and there if needed, but the fundamental idea is that of adding additional structure […]

A difficult integral evaluation problem

How do I compute the integration for $a>0$, $$ \int_0^\pi \frac{x\sin x}{1-2a\cos x+a^2}dx? $$ I want to find a complex function and integrate by the residue theorem.

Examples of function sequences in C that are Cauchy but not convergent

To better train my intuition, what are some illustrative examples of function sequences in C[0,1] that are Cauchy but do not converge under the integral norm?

Why is the empty set considered an interval?

What is the definition of an interval and why is the empty set an interval by that definition?

How to construct this Laurent series?

How do I construct the following Laurent series (clipped off Wolfram Alpha)? I know that the numerator can be written as $-1+\frac{\pi}2 z-…$ Alternatively (without the Laurent series), how can I see that the pole at $1$ is of order $2$?

$f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$

Trying to solve $f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx $ converges. I need to prove that: $$\lim \limits_{x \to \infty} f(x) = 0$$ Would appreciate your help!

Pointwise vs. Uniform Convergence

This is a pretty basic question. I just don’t understand the definition of uniform convergence. Here are my given definitions for pointwise and uniform convergence: Pointwise limit: Let $X$ be a set, and let $F$ be the real or complex numbers. Consider a sequence of functions $f_n$ where $f_n:X\to F$ is a bounded function for […]

recurrence relation $f(n)=5f(n/2)-6f(n/4) + n$

I’ve been trying to solve this recurrence relation for a week, but I haven’t come up with a solution. $f(n)=5f(n/2)-6f(n/4) + n$ Solve this recurrence relation for $f(1)=2$ and $f(2)=1$ At first seen it looks like a divide and conquer equation, but the $6f(n/4)$ confuses me. Please help me to find a solution. Kind regards,