# Incomplete solution of integral

• freddyfish
In summary, the task is to integrate dθ/(k-cosθ) from -pi to pi. The method of residues was used and the poles were evaluated for |k| greater than 1. However, for values of k that are not greater than 1, the residue method cannot be used. An alternative method, such as Cauchy's principal value of the integral, may be necessary to evaluate the integral.
freddyfish
The task is to integrate dθ/(k-cosθ) from -pi to pi.

This is the case - I have solved this and also seen the teacher solve this on the board but I am not satisfied with the method and answer. We used the method of resdues.

This is how my instructor would solve it:

F(cosθ,sinθ)=1/(k-cosθ)

F(x,y)=1/(k-x)=P(x,y)/Q(x,y)

Q(x,y)=k-x

x^2+y^2=1 implies Q(x,y)≠0 iff |k| is greater than 1.

-1≤k≤1 implies that Q(x,y)=0 iff x=k.

Indeed this seems legitimate.

He then evaluates the residues for |k| greater than 1 only.

However, when k is not greater than one, the poles lie on the unit circle in the finite/complex plane. In this case the residue method cannot be used and the answer is therefore not complete. What is the solution of this integral without the assumption that |k| is greater than 1?

Thanks

Assuming real k: with ##|k| \leq 1##, your function is not well-defined everywhere in the integration range, and you need a different way to write the problem.

Yeah, your assumtion is correct. Well, that is how the problem was stated. All you are given is the integral, which I have presented in words instead of using the integral sign. Thus, how would I write it in a different way to make it possible to solve?

It depends on the integral you want to calculate, and the reason why you want to do that.

Something like
$$\lim_{\epsilon \to 0} \left( \int_{-\pi}^{\arccos(k)-\epsilon} \frac{d\theta}{k-cos(\theta)} + \int_{\arccos(k)+\epsilon}^{\pi} \frac{d\theta}{k-cos(\theta)} \right)$$
might be possible to evaluate.

Ah, of course! That's Cauchy's principal value of the integral. :)

## 1. What is an incomplete solution of an integral?

An incomplete solution of an integral refers to a solution that is not fully evaluated or does not include all possible solutions. This can happen when there are multiple ways to solve an integral or when the integral is very complex.

## 2. How do I know if my solution to an integral is incomplete?

If your solution to an integral only includes certain terms or does not have a definite value, it is likely that it is an incomplete solution. Additionally, if you are unsure if your solution is complete, you can always check it by using integration techniques or plugging it back into the original equation to see if it satisfies all conditions.

## 3. Can an incomplete solution of an integral still be useful?

Yes, an incomplete solution of an integral can still be useful in certain cases. For example, it can help in approximating the value of the integral or in finding bounds for the integral. However, it is important to note that an incomplete solution is not considered a complete and accurate solution.

## 4. What are some common reasons for an incomplete solution of an integral?

There are various reasons why an integral may have an incomplete solution. Some common reasons include the complexity of the integral, the use of integration techniques that may not give the complete solution, or the presence of multiple solutions for the integral.

## 5. How can I improve my chances of getting a complete solution for an integral?

To increase the likelihood of obtaining a complete solution for an integral, it is important to have a good understanding of integration techniques and how to apply them correctly. Additionally, checking your solution and using multiple approaches to solve the integral can also help in getting a complete solution.

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