Having trouble evaluating an integral

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In summary, the conversation discusses the problem of finding a series expression for the triangle waveform and solving the integral for it. The person has reduced the problem and tried different methods such as using InverseFourierTransform and residues, but is still struggling to find a solution. They are now considering expanding the expression and using trigonometric identities.
  • #1
mcah5
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Hello,

I'm trying to find a series expression for the triangle waveform through some messy math.

I've reduced the problem down to solving the integral:

Integrate[ 1/w^2 * (1-Cos[T*w/2]) * Exp[(1-n)*I*t*w ] with respect to w from -Infinity to Infinity

T is a constant, I is Sqrt[-1], and n is an integer.

Mathematica cannot evaluate this integral, but if I use the function InverseFourierTransform and substitute a specific value of n, mathematica works. I plotted for a couple of n's and I know this is the integral I want.

I tried doing the integral through residues, but the only singularity I can see is w = 0 and the residue there is zero.

The problem is Ex. #5 part d of http://www.hep.caltech.edu/~fcp/math/distributions/distributions.pdf (scroll all the way down to the bottom)

Is there some residue I'm not seeing?

Thanks!
 
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  • #2
Why not simply expand the exp(i[...]) in sin and cos and separate the integral in real and imaginary part and integrate them separately.

Pehaps use the cosAcosB and CosASinB identities on the resulting integrals.
 
  • #3
Actually, I just re-read the question. It asks that I find the Fourier transform of f, not f itself so I don't need to do this integral :eek:

Now I feel stupid. I'll try that method though, thanks :)
 

Related to Having trouble evaluating an integral

1. How can I determine which method to use when evaluating an integral?

The method used to evaluate an integral depends on the complexity of the function and the limits of integration. Common methods include substitution, integration by parts, and trigonometric substitution. It is important to carefully analyze the integral and choose the most appropriate method for solving it.

2. What are some common mistakes to avoid when evaluating an integral?

Some common mistakes to avoid when evaluating an integral include forgetting to apply the chain rule, not considering the limits of integration, and making incorrect substitutions. It is also important to check for any potential discontinuities or singularities in the function that may affect the final result.

3. Can I use a calculator to evaluate integrals?

While some calculators have built-in functions for evaluating simple integrals, it is not recommended to solely rely on a calculator when solving more complex integrals. It is important to understand the underlying principles and methods used to evaluate integrals in order to accurately solve them.

4. How do I know if I have correctly evaluated an integral?

To verify the accuracy of your evaluation, you can take the derivative of the resulting function and check if it matches the original integrand. Additionally, you can use online tools or software to graph both the integral and its derivative and compare the two graphs to confirm the accuracy of your solution.

5. Are there any tips for simplifying integrals and making the evaluation process easier?

One tip for simplifying integrals is to break them down into smaller, more manageable parts. This can be achieved through techniques such as partial fraction decomposition, splitting the integral into multiple integrals, or using trigonometric identities. It is also helpful to practice regularly and familiarize yourself with common integrals and their solutions.

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