Applications of Taylor polynomials to Planck's Law

In summary, the conversation was about finding the Taylor Polynomial for g(x) = \frac{1}{x^5 ( e^{b/x} -1)} and using it to show the similarity between Planck's Law and the Raleigh-Jeans Law for large wavelengths. The series representation of e^{-bx^{-1}} was used to simplify the equation and obtain the Taylor Polynomial for g(x).
  • #1
coneyaw
10
0
Due to too much wrong information being posted on my behalf, I am resubmitting a cleaned up version of my last post. I have 2 hours to get this problem done :(. Essentially, I don't know at all how to find the Taylor Polynomial for [tex]
g(x) = \frac{1}{x^5 ( e^{b/x} -1)}
[/tex]

coneyaw said:

Homework Statement


f([tex]\lambda[/tex]) = [tex]\frac{8\pi hc\lambda ^{-5}}{e^{hc/\lambda kT}-1}[/tex]
Is Planck's Law
where
[tex]h\ =\ Planck's\ constant\ =\ 6.62606876(52)\ \times\ 10^{-34}\ J\ s;[/tex]
[tex]c\ =\ speed\ of\ light\ =\ 2.99792458\ \times\ 10^{8}\ m\ s^{-1};[/tex]
[tex]and\ Boltzmann's\ constant\ =\ k\ =\ 1.3806503(24)\ \times\ 10^{-23}\ J\ K^{-1}[/tex]

For my calculus class, I am asked to use a Taylor polynomial to show that the values for Planck's Law gives approximately the same values as the Raleigh-Jeans Law for large wavelengths [tex]\lambda[/tex].

Homework Equations


Basically I need some help regarding leading me in the right direction. I need to know how to pursue the correct center and basically someone to give me starting conditions, then I can figure the inequality and error on my own

The Attempt at a Solution


coneyaw said:
Here is an example of one that I've done before
Isn't this similar to how I find the taylor polynomial for [tex]
g(x) = \frac{1}{x^5 ( e^{b/x} -1)}
[/tex]
http://img509.imageshack.us/img509/1417/fasdfci8.jpg [Broken]
[/URL]
dynamicsolo said:
You will want to replace the exponential function in the denominator with its Taylor approximation. What is the series for e^Kx? (Keep in mind that [tex]\lambda[/tex] is the variable of interest.) IIRC, to end up with the R-J result, you will only need to use the first two terms (for large wavelengths, the exponent will be small compared to 1). The "1"s cancel, after which you simplify the resulting algebraic expression...

Gib Z said:
Your misunderstanding for a lot of this may stem from your definition of the Taylor series. Check your notes again =] Once you get that right, you will see the the MacLaurin series is just the Taylor series when a=0, and the one you want to use here. You will also see what dynamicsolo meant by "first two terms".

I would personally not take dynamicsolo's route though (no offense intended) as by only taking two terms, canceling 1s and saying only the leading term will matter is not as rigorous as taking the Taylor series of [tex]f(x) = \frac{1}{x^5 ( e^{kx}-1) }[/tex] and then manipulating constants.

coneyaw said:
[tex]e^{b/x}\ =\ e^{bx^{-1}}[/tex] which is much like the form [tex]e^{x^{2}}[/tex]

So if my assumptions are right, [tex]
g(x) = \frac{1}{x^5 ( e^{b/x} -1)}
[/tex]

Is going to be equal to the series representation of [tex]x^{-5}[/tex] times the series representation of [tex]\frac{1}{e^{bx^{-1}}}\ =\ e^{-bx^{-1}}[/tex]

that being said, I could find the representation of [tex]e^{-bx^{-1}}[/tex] similarly to the above problem about [tex]e^{x^{2}}[/tex]
 
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  • #2
e^{-bx^{-1}}\ =\ 1-bx^{-1}+\frac{b^2x^{-2}}{2!}-\frac{b^3 x^{-3}}{3!}+...plugging this back into the equation for g(x)g(x) \ = \ \frac{1}{x^5} - \frac{b}{x^4} + \frac{b^2}{2x^3} - \frac{b^3}{6x^2} + ...And that is the Taylor Polynomial of g(x).
 

1. What is Planck's Law and how is it related to Taylor polynomials?

Planck's Law is a fundamental principle in physics that describes the relationship between the energy of a photon and its frequency. This law is derived using Taylor polynomials, which are mathematical expressions used to approximate a function. In the case of Planck's Law, Taylor polynomials are used to approximate the energy-frequency relationship of photons.

2. How are Taylor polynomials used to improve the accuracy of Planck's Law?

Taylor polynomials allow us to approximate a function to any degree of accuracy by adding more terms to the polynomial. This means that by using higher degree Taylor polynomials, we can improve the accuracy of Planck's Law and make more precise predictions about the energy of photons at different frequencies.

3. Can Taylor polynomials be used to model other physical phenomena?

Yes, Taylor polynomials are a versatile tool in mathematics and can be used to model a wide range of physical phenomena. In addition to Planck's Law, they have applications in areas such as fluid dynamics, quantum mechanics, and statistical mechanics.

4. How does the order of a Taylor polynomial affect its accuracy?

The order of a Taylor polynomial refers to the highest degree of the polynomial. As the order increases, the accuracy of the polynomial also increases. This is because higher order polynomials have more terms, allowing them to better approximate the behavior of a function.

5. Are there any limitations to using Taylor polynomials to approximate Planck's Law?

While Taylor polynomials can greatly improve the accuracy of Planck's Law, they are still approximations and may not perfectly match the actual behavior of photons. Additionally, Taylor polynomials can only be used to approximate smooth functions, so they may not be suitable for modeling phenomena with sudden changes or discontinuities.

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