Solving Integrals with "Sommerfeld substitution

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In summary, the conversation discusses a method for solving integrals involving the function 1+\epsilon \cdot cos \theta, which can be challenging. The solution involves using a substitution and rearranging the integrals to make them easier to solve. This substitution is known as the "Sommerfeld substitution" and is considered a useful and elegant method for solving such integrals. The conversation ends with appreciation and thanks for the information shared.
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coomast
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Some time ago I had to do a calculation concerning a physical problem. In this calculation some integrals were needed to be solved and the method on how to do this was briefly described. I was able to solve the problem completely and found the method so beautiful that I would like to share it with all of you. Consider the following integrals:

[tex]\int \frac{1}{1+\epsilon \cdot cos \theta}d\theta[/tex]

[tex]\int \frac{1}{\left(1+\epsilon \cdot cos \theta\right)^2}d\theta[/tex]

[tex]\int \frac{1}{\left(1+\epsilon \cdot cos \theta\right)^3}d\theta[/tex]

[tex]\int \frac{cos \theta}{1+\epsilon \cdot cos \theta}d\theta[/tex]

[tex]\int \frac{cos \theta}{\left(1+\epsilon \cdot cos \theta\right)^2}d\theta[/tex]

[tex]\int \frac{cos \theta}{\left(1+\epsilon \cdot cos \theta\right)^3}d\theta[/tex]

[tex]\int \frac{sin \theta}{1+\epsilon \cdot cos \theta}d\theta[/tex]

[tex]\int \frac{sin \theta}{\left(1+\epsilon \cdot cos \theta\right)^2}d\theta[/tex]

[tex]\int \frac{sin \theta}{\left(1+\epsilon \cdot cos \theta\right)^3}d\theta[/tex]

Some of these are straightforward, but some are not. In order to solve the non trivial ones in a systematic way, one can use the following substitution:

[tex]1+\epsilon \cdot cos \theta = \frac{1-\epsilon^2}{1-\epsilon \cdot cos \gamma}[/tex]

With:

[tex]0\leq \theta \leq 2\pi[/tex]

[tex]0\leq \gamma \leq 2\pi[/tex]

The following relations can be obtained:

[tex]cos \theta = \frac{cos \gamma -\epsilon}{1-\epsilon \cdot cos \gamma}[/tex]

[tex]sin \theta = \frac{\sqrt{1-\epsilon^2} sin \gamma}{1-\epsilon \cdot cos \gamma}[/tex]

[tex]cos \gamma = \frac{\epsilon +cos \theta}{1+\epsilon \cdot cos \theta}[/tex]

[tex]sin \gamma = \frac{\sqrt{1-\epsilon^2} sin \theta}{1+\epsilon \cdot cos \theta}[/tex]

[tex]d \theta = \frac{\sqrt{1-\epsilon^2}}{1-\epsilon \cdot cos \gamma}d \gamma[/tex]

[tex]d \gamma = \frac{\sqrt{1-\epsilon^2}}{1+\epsilon \cdot cos \theta}d \theta[/tex]

After rewriting some of the integrals into smaller ones, substituting this and rearranging it is possible to solve them in a fairly easy way. As an example, let's take the one before the last, it was:

[tex]\int \frac{sin \theta}{\left(1+\epsilon \cdot cos \theta\right)^2}d\theta[/tex]

Using the substitution we get:

[tex]\int \frac{sin \gamma}{1-\epsilon^2}d\gamma=\frac{-cos \gamma}{1-\epsilon^2}+C[/tex]

The solution is now obtained by substituting either

[tex]\gamma = arccos \left( \frac{\epsilon +cos \theta}{1+\epsilon \cdot cos \theta} \right)[/tex]

or

[tex]\gamma = arcsin \left( \frac{\sqrt{1-\epsilon^2} sin \theta}{1+\epsilon \cdot cos \theta} \right)[/tex]

This substitution is called the "Sommerfeld substitution" after the "inventor" and is one of the nicest substitutions I've ever encountered for solving integrals in the real with "classical" functions. I hope this is helpful in solving other types of integrals you might be working on.

best regards,

Coomast
 
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  • #2
your information was very useful thanks
 
  • #3
very nice
and very useful
thanx
 

Related to Solving Integrals with "Sommerfeld substitution

What is the "Sommerfeld substitution" method for solving integrals?

The Sommerfeld substitution method is a technique used to solve integrals involving trigonometric functions. It involves making a substitution of variables that simplifies the integral, making it easier to solve.

How do you perform the Sommerfeld substitution?

To perform the Sommerfeld substitution, you first identify the integral as one that involves trigonometric functions. Then, you make a substitution of variables, typically using the tangent function, to simplify the integral. Finally, you solve the integral and substitute back in the original variable to find the final solution.

When should I use the Sommerfeld substitution method?

The Sommerfeld substitution method is most useful when you encounter integrals that involve trigonometric functions, particularly those with a mix of sine and cosine terms. It can also be used when other methods, such as u-substitution or integration by parts, are not applicable.

What are the benefits of using the Sommerfeld substitution method?

The Sommerfeld substitution method can greatly simplify integrals involving trigonometric functions, making them easier to solve. It also allows for the use of trigonometric identities to further simplify the integral. Additionally, it can be a useful tool for solving integrals that would be difficult or impossible to solve with other methods.

Are there any limitations to the Sommerfeld substitution method?

While the Sommerfeld substitution method is a useful tool for solving integrals involving trigonometric functions, it may not always be the most efficient method. In some cases, other techniques such as u-substitution or integration by parts may be faster or easier to use. Additionally, the Sommerfeld substitution method may not work for all types of integrals and may require multiple substitutions to fully solve the integral.

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