[Q] square of 'sinc function' integral

In summary, the conversation revolves around a problem of finding the probability density of an eigenstate of momentum chosen after a momentum measurement. The speaker encountered difficulties in solving the integral \int_{k_o}^0\frac{sin^{2}(2x)}{2x^{2}}dx and attempted to transform it into \int_{k_o}^0\frac{1-cos(2x)}{2x^{2}}dx. However, they faced a singularity issue and were unable to solve it. They sought help in finding a solution and suggested trying to write cos as a series form.
  • #1
good_phy
45
0
Hi.

I tried to solve some problem that i should get probability density with which eigenstate of

momentum is chosen after momentum measurement by using [tex]<\varphi_{k}|\Psi>[/tex]

I faced some stuck integral problem such as [itex]\int_{k_o}^0\frac{sin^{2}(2x)}{2x^{2}}dx[/itex]

I transformed [itex]sin^{2}(2x) = \frac{1 - cos(2x)}{2}[/itex] so i obtained [itex]\int_{k_o}^0\frac{1-cos(2x)}{2x^{2}}dx[/itex] but i don't know next step because, [itex]\int_{k_o}^0\frac{1}{x^2}dx[/itex] go up to infinity,diverse.

i tried to do partial integral such as [itex]\int udv = uv - \int vdu[/itex] but encountered same problem.

How can i overcome this singular point problem? i convinced that [itex]\int_{k_o}^0\frac{sin^{2}(2x)}{2x^{2}}dx[/itex]

should be solved to convegence because graphic of [itex]\frac{sin^{2}(2x)}{2x^{2}}[/itex].

Please help me and give me an answer.
 
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  • #2
i think you can try to write the cos as a series form.
P.S. the third formula in your statements seems...
 
  • #3


Firstly, it is important to note that the question is not very clear and lacks specific context. However, based on the information provided, it seems that the problem at hand involves calculating the probability density of an eigenstate of momentum after a momentum measurement. This involves solving an integral with a sinc function, and the author has attempted to transform it into a simpler form but has encountered a problem with the singularity at the lower limit of integration.

In this case, it might be helpful to use a different approach to solve the integral. One possible method is to use the substitution u = 2x, which would transform the integral into \int_{2k_o}^0\frac{sin^{2}(u)}{u^2}du. This new integral can then be solved using techniques such as integration by parts or by recognizing it as a well-known integral.

Another approach could be to use the Taylor series expansion of the sinc function, which is given by sinc(x) = \frac{sin(x)}{x} = 1 - \frac{x^2}{6} + \frac{x^4}{120} - \frac{x^6}{5040} + .... This expansion can be used to approximate the integral and then take the limit as the lower limit of integration approaches 0. This would give a finite result and help overcome the singularity problem.

Furthermore, it is important to carefully consider the limits of integration as they can greatly affect the convergence of the integral. In this case, the limit of integration should be chosen such that the integral converges. This may require some further analysis or manipulation of the original integral.

In conclusion, there are various approaches that can be used to solve the integral involving the sinc function. It is important to carefully consider the limits of integration and use appropriate techniques to overcome any singularity problems. Additionally, providing more specific context and information about the problem would make it easier to provide a more tailored and accurate solution.
 

What is the formula for the square of the sinc function integral?

The formula for the square of the sinc function integral is ∫(sinc(x))^2 dx = πx - sin(2x) + C.

What is the purpose of taking the square of the sinc function integral?

Taking the square of the sinc function integral allows for the calculation of the energy spectrum of a signal. It is also used in signal processing and digital communications.

How can the square of the sinc function integral be interpreted graphically?

The square of the sinc function integral can be interpreted as the area under the curve of the squared sinc function graph. It represents the energy contained in the signal at different frequencies.

What is the relationship between the square of the sinc function integral and the Fourier transform?

The square of the sinc function integral is closely related to the Fourier transform. It is the Fourier transform of the autocorrelation function of a signal.

How is the square of the sinc function integral used in practical applications?

The square of the sinc function integral is used in a variety of practical applications, including signal processing, digital communications, and spectral analysis. It is also used in the design of filters and in the calculation of signal-to-noise ratios.

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