2
votes
1answer
81 views

Asymptotic expansion on 3 nonlinear ordinary differential equations

The 3 nonlinear differential equations are as follows \begin{equation} \epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber \end{equation} \begin{equation} \frac{ds}{dt}= ...
0
votes
1answer
14 views

First Order Approximation Taylor Series

I have the taylor series $f(z)=f(x_0)+(x-x_0)f'(z)+1/2(x-x_0)^2f''(z) ...$ and I am told that "As a first order approximation," $x-x_0$ ~ $\frac{f(x)-f(x_0)}{f'(x_0)}$ assuming $f'(x_0) \neq 0$ I ...
1
vote
3answers
113 views

A system of $n$ equations , how does it behave for growing $n$?

I read about the system of $n$ equations in the link below. I wonder how it behaves for growing $n$. Does it converge ? http://math.eretrandre.org/tetrationforum/showthread.php?tid=889 Here it is ...
3
votes
2answers
79 views

Integration by expansion

Consider the integral \begin{equation} I(x)= \frac{1}{\pi} \int^{\pi}_{0} \sin(x\sin t) \,dt \end{equation} show that \begin{equation} I(x)= \frac{2x}{\pi} +O(x^{3}) \end{equation} as ...
3
votes
0answers
64 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
0
votes
1answer
51 views

Expansion of Integration

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt \end{equation} show that \begin{equation} I(x)= 4+ \frac{8}{\pi}x +O(x^{2}) ...
1
vote
0answers
68 views

Big-O Notation for remainder terms in Taylor expansion

The Big-O notation is commonly used in Taylor expansions of the form $$f(x+\epsilon)=f(x)+\epsilon f'(x)+O(\epsilon^2)$$ to say that the remainder term grows at least quadratic around $\epsilon=0$. ...
3
votes
0answers
109 views

Question about Big O notation for asymptotic behavior in convergent power series

Examples of such use of Big O notation can be found for instance on Wolfram Alpha here. More details on the Wikipedia page. The idea, as I understand it, is that the term between parenthesis in Big O ...
0
votes
2answers
54 views

Expanding $\ln(1+f(x))$ around $f(x)=0$ when we do not know whether there is an $x$ such that $f(x)=0$.

I want to expand $\ln(1+f_T(x,\theta))$ about $1+f_T(x,\theta)=1$. What I have in mind is something like $$ \ln(1+f_T(x,\theta))=\ln(1)+f_T(x,\theta)-\frac{1}{2} \frac{1}{1+\tilde{f}} ...
3
votes
1answer
84 views

Finding an asymptotic expansion for a transcedental equation

I am new around here and was hoping you will be able to help me with the following. I have the equation: $x^3 - 3x^2 +(3-\epsilon ) x + \epsilon = sin(\frac{\pi}{2} x +\frac{\pi \epsilon}{2} ) $ and ...
1
vote
0answers
37 views

Question about asympotic expansions!! please help!

Question: Find the constants $$a_0, a_1, a_2$$ in the asympotic expansion $$\int_0^x t\sqrt{ln(t)} dt$$ = $a_0(x^2)(lnx)^\frac 12$ + $a_1\frac {x^2}{(lnx)^\frac 12}$ + $a_2\frac {x^2}{(lnx)^\frac ...
4
votes
0answers
91 views

Prove that a series is $O(t^a)$.

Consider the series $$ u(t,x) = \sum_{i \geq 1} {u_i(x) t_1^i } + \sum_{i+2j \geq k+2, j\geq 1} {\varphi_{i,j,k}(x) t_1^i t_2^j y^k} $$ where $t \in \tilde{\mathbb{C} \setminus \{ 0 \}}$, $x$ is ...
1
vote
1answer
42 views

Asymptotic Expansion in zero of $\frac{1}{\ln(1+x)}$

On wolfram the expansion is: $$\frac {1}{x} + \dfrac{1}{2} ...\,.$$ But I don't understand from where it outside comes the $\frac{1}{2}$ thanks
3
votes
2answers
143 views

The integral $\int\ln(x-\ln(x))~dx$

The integral $f(y)=\int_0^y\ln(x-\ln(x))~dx$ is on my mind. I'm not sure if this has a closed form? Maybe we need to use the lambert-W function to solve this one? If it cannot be done in closed ...
1
vote
0answers
28 views

Find the order of the following expression as x->0

Could someone help me find the order of the following expression without using the quotient rule? $\frac{1-\cos(x)}{1+\cos(x)}$ I expanded the denominator and the numerator but not sure how I get to ...
1
vote
1answer
59 views

Find “singular expansion” of a function

I have the function $(1-z)^{-z}$, analytic except on $\mathbb{R}_{\geq 1}$ Now in the text, it says the "singular expansion" at $z=1$ is $\displaystyle \frac{1}{1-z} + \log(1-z)+O((1-z)^{1/2})$ I'm ...
4
votes
2answers
104 views

how to find the asymptotic expansion of the following sum:

I need to determine an asymptotic expansion when $q \rightarrow 1$ of the sum $$S(q)=\sum_{n=0}^{\infty} \frac{q^n}{ (q^n + 1)^2 }.$$ Numerical computations suggest that $S(q)\sim\frac{c}{|q-1|}$ ...
3
votes
4answers
138 views

Taylor series of $\arctan(x+2)$ at $x=\infty$

The simple question is: what is the correct way to calculate the series expansion of $\arctan(x+2)$ at $x=\infty$ without strange (and maybe wrong) tricks? Read further only if you want more details. ...
2
votes
1answer
59 views

Asymptotic expansion of $\ln\left(\frac{x+a}{x-a}\right)$ in form of $\sum\limits_{n=0}^\infty a_n \left(\frac{1}{x}\right)^n$?

How can I find an expansion for $f(x)=\ln\left(\dfrac{x+a}{x-a}\right)$ in terms of powers of $x$, in the form of: $$f(x)=\sum_{n=0}^\infty a_n \left(\frac{1}{x}\right)^n$$ When I try a Taylor ...
1
vote
1answer
105 views

Asymptotic formula for complex gamma function at $+\infty+i \times y$

I am currently looking for the behaviour of the complex gamma function at real infinity: $\lim_{x \to \infty}\Gamma\left(x+i\times y\right)$ and more particularly for asymptotic formulas for the ...
2
votes
1answer
146 views

Numerical Analysis best estimate on polynomial order

I need to determine the best integer value of $k$ for the equation: \begin{equation} \arctan(x) = x + O(x^k) \text{ as $x\to 0$} \end{equation} Taylor's Theorem with Lagrange Remainder would ...
1
vote
3answers
69 views

Numerical Analysis and Big O

How can I show that $e^x -1$ is not $O(x^2)$ as $x\to0$ I'm not sure where to start. We can use Taylor's Theorem with remainder: \begin{equation} e^x = \sum\limits_{k=0}^n\dfrac{x^n}{n!} ...
0
votes
1answer
169 views

Confused about a limit proof and Big O.

I gave an incorrect proof here : How can evaluate $\lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x}$ I am confused as when considering the mistakes in my proof it seems the limit cannot be ...
1
vote
1answer
66 views

Estimate the scale of the power series with Poisson pdf-like terms

Sorry to bother you, but I guess that this question is not appropriate for MO, so I repost it here hoping that someone could give me a clue. I would like to have an estimate for the series $$P(t) = ...
1
vote
1answer
173 views

An issue with approximations of a recurrence sequence

By trying to give an approximation to a given recurrence sequence I encountered a problem. To be more precise I have a method but it fails if the right condition is not met and I wonder how I should ...
2
votes
1answer
49 views

Two-point Taylor expansion with one assymptotic point?

According to this paper, a two-point Taylor expansion can be definied like this: $$\text{Let }f\left(z\right)\text{ be an analytic function and }z_1 \text{and }z_2\in \mathbb{C}, z_1\neq ...
2
votes
0answers
84 views

Series expansion of a series

I would like to perform an asymptotic expansion of the function $$f(x) = \sum_{n=1}^{\infty}\frac{1}{(nx)^2}K_2(n x),$$ where $K_2(x)$ is the modified Bessel function of the second kind, around $x=0$. ...
3
votes
1answer
59 views

Exercise about MacLaurin's polynomial and small-o

In class the professor wrote the following limit: $\lim_{x\to 0} \frac{\sinh^2 (x) -x^2}{x^4}$ So he "expanded" (sorry for my English) the MacLaurin's formula for $\sinh x$ up to the 3rd power, and ...
0
votes
2answers
79 views

Asymptotic Expansion

I was trying to solve and ODE and while doing some asymptotics, I bumped into something like this $\left(1+\frac{\gamma}{z_{0}}+\epsilon \frac{z_{1}}{z_{0}}\right)^{-2}$ where $\gamma$ $\,$ is of ...