Solving Trig Integrals with Residue Theorem

In summary, the conversation is about a person's attempt to understand an example of using the residue theorem to solve a trigonometric integral. They mention finding a singularity and questioning whether it is obvious that it lies inside the unit circle. The other person suggests using the hint of letting a=sin(x) and simplifying, but the first person is still unsure and wondering about the range of x.
  • #1

dyn

770
61
Hi.
I have looked through an example of working out a trig integral using the residue theorem. The integral is converted into an integral over the unit circle centred at the origin. The singularities are found.
One of them is z1 = (-1+(1-a2)1/2)/a
It then states that for |a| < 1 , z1 lies inside the unit circle.
Should this be obvious just by looking at z1 ? Because I can't see it. I have tried a few values and it seems to be true but that doesn't prove it. If its not obvious how do I go about trying to show it ?
Thanks
 
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  • #2
Hint: Let a=sin(x) for real |x|<pi/2 and simplify.
 
  • #3
I must be missing something as the answer I get doesn't seem to help : -cosec x + cot x
Also I'm not sure why I can specify | x | < π/2
 
  • #4
Hmm... I was assuming a is real but it is probably complex? It might still work but the range of x will need more thought.
 

What is the Residue Theorem?

The Residue Theorem is a powerful tool in complex analysis that allows us to evaluate certain types of integrals using the residues of a function. It states that the integral of a function around a closed curve is equal to the sum of the residues of the function inside the curve.

How does the Residue Theorem apply to trigonometric integrals?

The Residue Theorem can be used to solve trigonometric integrals by converting them into complex valued functions. This allows us to use the properties of complex numbers to simplify the integration process and evaluate the integral using the residues of the function.

What are the steps for solving trigonometric integrals using the Residue Theorem?

The first step is to convert the trigonometric integral into a complex valued function by using Euler's formula. Then, we need to find the poles and residues of the function. After that, we need to determine the contour that encloses all the poles and use the Residue Theorem to evaluate the integral. Finally, we convert the solution back into a real valued function to get the final answer.

What are the advantages of using the Residue Theorem for solving trigonometric integrals?

The Residue Theorem provides a systematic and efficient method for solving trigonometric integrals. It eliminates the need for complicated trigonometric identities and reduces the integration process to simple algebraic manipulations. Additionally, it can be used to solve a wide range of integrals that may not be solvable using other methods.

Are there any limitations to using the Residue Theorem for solving trigonometric integrals?

The Residue Theorem can only be used for integrals that can be expressed as a rational function, which means that it may not be applicable to all trigonometric integrals. Additionally, finding the poles and residues of a function can be a challenging task in some cases, which may make the method more time-consuming compared to other integration techniques.

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