- #1
Forny
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Somebody could please tell me how to evaluate the integral:
integral(sin(x^2)) from o to infinity
integral(sin(x^2)) from o to infinity
Forny said:well, I had already checked the link, but I don't undestand the strategy, could somebody please explain it to me?, or if you have a text where I can find some explanation about that kind of process, I would really apreciate if you help me.
Forny said:Somebody could please tell me how to evaluate the integral:
integral(sin(x^2)) from o to infinity
ZZappaZZappa said:This is why:
[tex]
\left\arrowvert\int_0^{\pi/4}e^{iR^2e^{i2\theta}}iRe^{i\theta}d\theta \right\arrowvert = \left\arrowvert\int_0^{\pi/4}exp\{iR^2(\cos(2\theta)+i \sin(2\theta))\}iRe^{i\theta}d\theta \right\arrowvert
[/tex]
Evaluating the absolute value, this equals
[tex]
\int_0^{\pi/4}exp\{-R^2 \sin(2\theta)\}R d\theta
[/tex]
Now find the maximum of [tex] g(\theta) = exp\{-R^2 \sin(2\theta)\} [/tex] by
differentiation (check it's a max by second derivative test).
This occurs at [tex] \theta = \pi/4 [/tex].
Then the above is less than or equal to
[tex]
R \cdot exp\{-R^2\} \frac{\pi}{4}
[/tex]
Take the limit as [tex] R\to \infty [/tex] to get 0.
Absolute value to zero, original to zero.
The Fresnel Integral is a mathematical tool used in physics and engineering to calculate the diffraction of waves, such as light or sound, when they pass through an aperture or around an obstacle.
To evaluate the Fresnel Integral, the function sin(x^2) from 0 to infinity, it is necessary to use numerical methods such as Simpson's rule, Gaussian quadrature, or Monte Carlo integration.
The Fresnel Integral has numerous applications in optics, acoustics, and signal processing. It is commonly used in the design of lenses, antennas, and diffraction gratings.
No, there is no known closed-form solution for the Fresnel Integral. It can only be evaluated using numerical methods.
Yes, the Fresnel Integral can be evaluated for other functions, such as cos(x^2), e^(-x^2), and many others. However, the method of evaluation may vary depending on the specific function.