# Tagged Questions

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### Asymptotic expansion on 3 nonlinear ordinary differential equations

The 3 nonlinear differential equations are as follows $$\epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber$$ \frac{ds}{dt}= ...
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### 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 ...
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### 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 ...
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### Integration by expansion

Consider the integral $$I(x)= \frac{1}{\pi} \int^{\pi}_{0} \sin(x\sin t) \,dt$$ show that $$I(x)= \frac{2x}{\pi} +O(x^{3})$$ as ...
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### 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 ...
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### Expansion of Integration

Consider the integral $$I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt$$ show that I(x)= 4+ \frac{8}{\pi}x +O(x^{2}) ...
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### 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 $\epsilon=0$. ...
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### 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 ...
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### 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 ...
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According to this paper, a two-point Taylor expansion can be definied like this: $$\text{Let }f\left(z\right)\text{ be an analytic function and }z_1 \text{and }z_2\in \mathbb{C}, z_1\neq ... 0answers 84 views ### Series expansion of a series I would like to perform an asymptotic expansion of the function$$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 $x=0$. ...
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 ...
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 ...