Binomial Theorem Application in Cauchy's and Sellmeier's Equations

In summary, the conversation discusses using the binomial theorem to expand Cauchy's Equation as an approximation of Sellmeier's Equation when \lambda >> \lambda_0_j. The hint suggests rewriting Sellmeier's Equation and then applying the binomial theorem, but it is also possible to first apply the binomial theorem and then take the square root. The conversation concludes with a helpful explanation of how to use the binomial theorem in this context.
  • #1
Luminous Blob
50
0
I am trying to do a question from Eugene Hecht's Optics book, which goes something like this:

Given the following equations:

Cauchy's Equation:

[tex]n = C_1 + \frac{C_2}{\lambda^2} + \frac{C_3}{\lambda^4} + ... [/tex]

Sellmeier's Equation:

[tex]n^2 = 1 + \sum_{j} \frac{A_j\lambda^2}{\lambda^2-\lambda_0_j^2}
[/tex]

where the [tex]A_j[/tex] terms are constants and each [tex]\lambda_0_j[/tex] is the vacuum wavelength associated with a natural frequency [tex]v_0_j[/tex], such that [tex]\lambda_0_jv_0_j = c [/tex].

Show that where [tex]\lambda >> \lambda_0_j [/tex], Cauchy's Equation is an approximation of Sellmeier's Equation.

Now it also gives a hint which is as follows:

Write the above expression with only the first term in the sum; expand it by the binomial theorem; take the square root of [tex]n^2[/tex] and expand again.

From the hint, I gather that it means to rewrite Sellmeier's Equation as:

[tex]n^2 = 1 + \frac{A\lambda^2}{\lambda^2 - \lambda_0^2}[/tex]

From there though, I have no idea how to apply the binomial theorem to expand it. I just don't see how anything in that equation has the form [tex](x+y)^n[/tex], except for where n = 1.

If anyone can explain to me how to apply the binomial theorem to the equation, or if I've misunderstood what the hint means, it would be much appreciated.
 
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  • #2
You can use the binomial theorem to expand [itex](1+x)^{1/2}[/itex] when x<<1.
 
  • #3
So you mean first take the square root of both sides, then expand it using the binomial theorem , letting [tex]x = \frac{A\lambda^2}{\lambda^2 - \lambda_0^2}[/tex], rather than first applying the binomial theorem, then taking the square root of both sides and then expanding again like the hint suggests?
 
  • #4
Rewrite [tex] \frac{A_j\lambda^2}{\lambda^2-\lambda_0_j^2} [/tex] as

[tex] \frac{A_j}{\lambda^2}\frac{1}{1-\frac{\lambda_0_j^2}{\lambda^2}} [/tex] and expand the second part as

[tex] \frac{1}{1-x^2} \approx 1 - x^2 + x^4 - x^6 \ldots [/tex] where [tex] x = \frac{\lambda}{\lambda_0_j} [/tex]
 
Last edited:
  • #5
Aah, I didn't think to do that. Thanks, that was a great help.
 

What is the binomial theorem?

The binomial theorem is a mathematical theorem that provides a way to expand the powers of a binomial expression. It states that for any real or complex numbers a and b, and any non-negative integer n, the expansion of (a + b)^n can be expressed as the sum of n+1 terms, each involving a power of a and b.

What is the formula for the binomial theorem?

The formula for the binomial theorem is (a + b)^n = ∑ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n, and (n choose k) = n! / (k! * (n-k)!). This formula allows us to easily find the coefficients of each term in the expansion.

How is the binomial theorem used in real life?

The binomial theorem has many applications in fields such as statistics, physics, and engineering. It can be used to calculate probabilities in statistics, to model the behavior of particles in physics, and to simplify complex equations in engineering problems.

What are some common misconceptions about the binomial theorem?

One common misconception is that the binomial theorem only applies to binomial expressions. In reality, it can be extended to any expression of the form (a + b)^n, where n is a non-negative integer. Another misconception is that the binomial coefficients (n choose k) represent multiplication. In fact, they are combinatorial numbers that represent the number of ways to choose k objects from a set of n objects.

How can the binomial theorem be proved?

The binomial theorem can be proved using mathematical induction or by using Pascal's triangle. It can also be proved using the concept of combinations and the binomial coefficient formula (n choose k). There are multiple ways to prove the binomial theorem, and the method used may vary depending on the level of mathematical knowledge and understanding of the person proving it.

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