User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/existence

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/integrand

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/evaluation

User:GFauxPas/Sandbox/Zeta2/FourierSeries/

User:GFauxPas/Sandbox/Zeta2/Informal Proof

$\mathcal L \left\{{}\right\}$


 * $\mathcal L \left\{{e^{at}f\left({t}\right)}\right\} = F\left({s-a}\right)$

Theorem
Let $P_n$ be a real polynomial, of degree $n$.

Let $e^z$ be the complex exponential, where $z = x + iy$.

Then for every $n \in \N_{\ge 0}$:


 * $\displaystyle \lim_{x \mathop \to +\infty} \frac {P_n}{e^z} = 0$

Proof
The proof proceeds by induction on $n$, the degree of $P_n$.

Basis for the Induction
The case $n = 0$ is verified as follows:

This is the basis for the induction.

Induction Hypothesis
Fix $n \in \N$ with $n \ge 0$.

Assume:

holds for $n$.

This is our induction hypothesis.

Induction Step
This is our induction step:

The result follows by the Principle of Mathematical Induction.

Am I breaking any rules here? I feel uncomfortable about this for some reason. --GFauxPas (talk) 10:17, 12 May 2014 (UTC)

Limit at Infinity of Polynomial

 * $P\left({1}\right)$: Limit at Infinity of Identity Function

Induction Hypothesis: $P\left({n}\right)$:

Induction Step:

But $P_n - P_{n-1}$ is a polynomial of degree $n$.

This approaches $+\infty$ or $-\infty$ from the induction hypothesis.

The result follows by the Principle of Mathematical Induction.