"Mathematical Physics by Carl Bender"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 4번째 줄: | 4번째 줄: | ||
==lecture 1 perturbation method== | ==lecture 1 perturbation method== | ||
| − | * solve $x^5+x=1$ | + | * solve $x^5+x=1$=== |
===method 1=== | ===method 1=== | ||
* try $x^5+\epsilon x=1$ | * try $x^5+\epsilon x=1$ | ||
| 22번째 줄: | 22번째 줄: | ||
* can we get a meaningful number out of this? | * can we get a meaningful number out of this? | ||
* yes, for example, Pade summation can be used | * yes, for example, Pade summation can be used | ||
| + | |||
| + | |||
| + | ===asymptotics=== | ||
| + | * $f\sim g\, \quad (x\to x_0)$ iff $$\lim_{x\to x_0}\frac{f(x)}{g(x)}=1$$ | ||
| + | * method of dominant balance | ||
| + | |||
==books== | ==books== | ||
2014년 2월 27일 (목) 11:01 판
list
lecture 1 perturbation method
- solve $x^5+x=1$===
method 1
- try $x^5+\epsilon x=1$
- find $x(\epsilon)$ satisfying $x(\epsilon)^5+\epsilon x(\epsilon)=1$
- answer
$$x(\epsilon)=1-\frac{\epsilon }{5}-\frac{\epsilon ^2}{25}-\frac{\epsilon ^3}{125}+\frac{21 \epsilon ^5}{15625}+\frac{78 \epsilon ^6}{78125}+\cdots$$
- Setting $\epsilon=1$ gives numerical value $0.75\cdots$
weak coupling approach
- use the similar idea to Feynman diagrams
- try $\epsilon x^5+ x=1$
- we get
$$ x(\epsilon)=1-\epsilon +5 \epsilon ^2-35 \epsilon ^3+285 \epsilon ^4-2530 \epsilon ^5+23751 \epsilon ^6+\cdots $$
- can we get a meaningful number out of this?
- yes, for example, Pade summation can be used
asymptotics
- $f\sim g\, \quad (x\to x_0)$ iff $$\lim_{x\to x_0}\frac{f(x)}{g(x)}=1$$
- method of dominant balance
books
- Bender, Carl M., and Steven A. Orszag. 1999. Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. Springer.