What type of expansion is this?

In summary, the conversation discusses the Binomial series expansion of ##(1+x)^{1/2}## and how the first order derivative can be taken to get the Taylor Expansion. The conversation also clarifies that x represents a trigonometric function and n is equal to 1/2.
  • #1
Chuck88
37
0
When I am reading the paper about Rayleigh instability, I found this type of expanding method.

[tex]
\sqrt{1+(\frac{2\pi\delta}{\lambda})^2 \cos^2(\frac{2\pi x}{\lambda})} = 1 + \frac{1}{2}(\frac{2\pi\delta}{\lambda})^2\cos^2 (\frac{2\pi x}{\lambda}) + \cdots
[/tex]

Can someone tell me what type of expansion is this?
 
Physics news on Phys.org
  • #2
It's the Binomial series expansion of ##(1+x)^{1/2}##
 
  • #3
AlephZero said:
It's the Binomial series expansion of ##(1+x)^{1/2}##

If we suppose that ##f(x) = (\frac{2\pi\delta}{\lambda})^2 \cos^2(\frac{2\pi x}{\lambda})##, is it right that we could take derivative with respect to ##f(x)## to get the Taylor Expansion? The first order derivative I mean is presented below:

[tex]
\frac{d(1 + f(x))^{\frac{1}{2}}}{df(x)} = \frac{1}{2} \frac{1}{\sqrt{1 + f(x)}}
[/tex]
 
  • #4
I'm not sure where that is leading to. I meant
##(1+x)^n = 1 + n x + n(n-1)x^2 / 2! + n(n-1)(n-2)x^3/3! + \dots##
where x is the trig function and n = 1/2.
 
  • #5
AlephZero said:
I'm not sure where that is leading to. I meant
##(1+x)^n = 1 + n x + n(n-1)x^2 / 2! + n(n-1)(n-2)x^3/3! + \dots##
where x is the trig function and n = 1/2.

OK. Now I comprehend. Thanks a lot.
 

What type of expansion is this?

This question is asking about the type of expansion in a scientific context. There are several types of expansion that can occur in scientific fields, such as thermal expansion, chemical expansion, and biological expansion.

What is thermal expansion?

Thermal expansion is the increase in size or volume of a substance due to an increase in temperature. This expansion occurs because the molecules in the substance start moving faster and require more space, causing the substance to expand.

What is chemical expansion?

Chemical expansion is the increase in volume or size of a substance due to a chemical reaction. This type of expansion can occur when a substance reacts with another substance, resulting in a change in the number of molecules and therefore, an increase in volume.

What is biological expansion?

Biological expansion is the increase in size or volume of living organisms. This can occur through processes such as cell division, growth, or the absorption of water and nutrients. Biological expansion is essential for the development and survival of organisms.

What are some examples of expansion in everyday life?

Some examples of expansion in everyday life include the expansion of metal objects when heated, the expansion of bread dough during baking, and the expansion of a balloon when filled with air. These are all examples of thermal, chemical, and biological expansion, respectively.

Similar threads

Replies
12
Views
1K
Replies
4
Views
189
Replies
16
Views
1K
  • Calculus
Replies
3
Views
756
  • Calculus
Replies
29
Views
516
Replies
2
Views
180
  • Calculus
Replies
4
Views
1K
Replies
1
Views
830
Replies
3
Views
936
Replies
2
Views
978
Back
Top