Exploring Approximations by Expansion in Scientific Calculations

  • Thread starter eep
  • Start date
  • Tags
    Expansion
In summary, the conversation discusses the use of expansions in mathematics, specifically Taylor series and Fourier series, to approximate functions and make calculations easier. The example given is the expansion of p^t = m/(1-v^2)^1/2 for small values of v. The conversation also mentions the importance of calculating the radius of convergence for power series. There is a brief discussion on the signs in the expansion, with the conclusion that the correct sign is +.
  • #1
eep
227
0
Hi,
I've never really studied various ways of expanding expressions in order to obtain an approximation that can make calculations easier. For example,

[tex]
p^t = \frac{m}{\sqrt{1 - v^2}}
[/tex]

reduces to

[tex]
p^t = m + \frac{1}{2}mv^2 + ...
[/tex]

for v << 1.

How does one arrive at something like this? What other expansions are useful? I used to think that I'd just calculate everything exactly but I now realize these sorts of expansions are extremeley important.
 
Mathematics news on Phys.org
  • #2
Are you familiar with Taylor series? It says that if a function can be expanded in a power series, then it will be of the form

[tex]f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n[/tex]

Wheter this series converges towards the original function is another matter. (Actually, it is a matter of calculating the radius of convergence of a power series)

For exemple, the function [itex]f(x) = (1+x)^r[/itex] where r is any real number has as its Taylor expansion

[tex]\sum_{n=0}^{\infty} \frac{r(r-1)...(r-n+1)}{n!}x^n[/tex]

and the series converges to [itex]f(x) = (1+x)^r[/itex] for all |x|<1 but not for any other value of x.

This is exactly what has been done in your post. [itex]p(v)=m(1+v^2)^{-1/2}[/itex] is of the form [itex](1+x)^r[/itex] with x=-v² and r=-½, so it converges to the series expansion you wrote for all values of v such that |v²|<1.

N.B. in the context of special relativity, v is always lesser than 1 since it has been assumed that c=1, so the formula really is valid for all velocity.The only other expansion I know of is the expansion in a Fourier series.
 
Last edited:
  • #3
Btw, wouldn't it be
[tex]
p = m - \frac{1}{2}mv^2 + ...
[/tex]
instead? (with a "-" sign instead of a "+" sign on odd terms)
 
  • #4
The sign is +. It results from multiplying two - signs, -1/2 and -v2.
 

FAQ: Exploring Approximations by Expansion in Scientific Calculations

What is the concept of "Approximations by Expansion"?

"Approximations by Expansion" is a mathematical technique used to estimate a value or function by expanding it into simpler, more manageable terms. It involves using a known value or function to approximate an unknown value or function.

What are the benefits of using approximations by expansion?

The main benefit of using approximations by expansion is that it allows for the estimation of complex values or functions that would otherwise be difficult to calculate. It also allows for the simplification of complex equations and the identification of patterns and relationships between variables.

What are some common applications of approximations by expansion?

Approximations by expansion are commonly used in physics, engineering, and other sciences to solve problems that involve complex equations or values. They are also used in computer science and data analysis to approximate large datasets and make predictions.

What are the limitations of approximations by expansion?

One limitation of approximations by expansion is that they are only as accurate as the known value or function used for the approximation. They also become less accurate as the number of terms in the expansion increases. Additionally, they may not be applicable to all types of functions or equations.

How can I improve the accuracy of my approximations by expansion?

To improve the accuracy of approximations by expansion, it is important to use a known value or function that is as close as possible to the unknown value or function being approximated. It is also helpful to use more terms in the expansion and to check the accuracy of the approximation using multiple methods.

Similar threads

Replies
3
Views
3K
Replies
10
Views
389
Replies
4
Views
642
Replies
3
Views
3K
Replies
2
Views
2K
Replies
1
Views
1K
Back
Top