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jkh4
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How to do integral for Cos(x^2)dx? Is there a chain rule for integral? Coz sometimes when I approach questions like that i just don't know how to approach...thanks!
But the major task here is to show that your result holds for [tex]\alpha[/tex] imaginary. This requires complex analysis.agomez said:[tex] e^{i \theta} = \cos (\theta) + i \sin (\theta) [/tex] & the Gaussian Integral: [tex] \int_{-\infty}^{\infty} e^{-\alpha x^2} dx = \sqrt{\frac{\pi}{\alpha}}[/tex]
Anthony said:But the major task here is to show that your result holds for [tex]\alpha[/tex] imaginary. This requires complex analysis.
Not easy enough, it would seem!Mute said:It's not that major a task. It's pretty easy once you have the right contour. ;)
You need Jordan's lemma type argument - your current version doesn't hold water, I'm afraid.Mute said:Now we take the limit as [itex]R \rightarrow \infty[/itex]. The integral on C2 satisfies the inequality
[tex]\left|iR\int_0^{\pi/4}d\theta e^{i\theta} e^{iR^2\cos(2\theta)}e^{-R^2\sin(2\theta)}\right| \leq R\int_0^{\pi/4}d\theta e^{-R^2\sin(2\theta)}[/tex]
Because [itex]\sin(2\theta) > 0[/itex] when [itex]0 < 2\theta < \pi/4[/itex], the RHS of the inequality tends to zero as R grows large and hence doesn't contribute to the integral.
Anthony said:Not easy enough, it would seem!You need Jordan's lemma type argument - your current version doesn't hold water, I'm afraid.
Excellent, well done!Mute said:If you insist on more rigor, it is easily supplied: on the integration contour the inequality
[tex]\sin(2\theta) \geq \frac{4\theta}{\pi}[/tex]
holds. (Due to concavity of the sine function on this interval).
Hence,
[tex] R\int_0^{\pi/4}d\theta e^{-R^2\sin(2\theta)} \leq R\int_0^{\pi/4}d\theta e^{-4R^2\theta/\pi} = -\frac{4}{\pi R}\left(e^{-R^2} - 1\right)[/tex]
which tends to zero as R tends to infinity.
You might not need complex analysis to evaluate integrals of this form, but it's the standard treatment. I'm sure there's actually a book containing lots of integrals that are usually done via complex analysis, but explicitly calculates them using real analysis. I don't really see the point (why rub sticks together if you have a lighter), but some are fond of such things.snipez90 said:Don't know what Jordan's lemma is (will find out soon enough), but I'm pretty sure you can just integrate by parts to get the estimate for
[tex]R\int_0^{\pi/4}d\theta e^{-R^2\sin(2\theta)}.[/tex]
Also you do not need complex analysis to evaluate the Fresnel integrals. You can use somewhat contrived arguments to compute the integrals via multivariable calculus. Since I'm more fond of real analysis at this point, I prefer real-analytic solutions.
murshid_islam said:or you can expand [tex]\cos (x^{2})[/tex] into a series and then integrate each term (and thus get an approximation).
by the way, how can you prove this:
[tex] \int_{0}^{\infty} \cos (x^{2}) dx = \int_{0}^{\infty} \sin (x^{2}) dx = \frac{1}{2}\sqrt{\frac{\pi}{2}} [/tex]
The integral of cos(x²), often written as ∫cos(x²) dx, is notable because it does not have a solution in terms of elementary functions (like polynomials, exponential functions, trigonometric functions, etc.). This means it cannot be expressed in a simple closed formula using basic functions.
Although the integral of cos(x²) cannot be expressed with elementary functions, it can be evaluated using numerical methods or special functions. For instance, computer algorithms or numerical integration techniques like Simpson's rule or the trapezoidal rule can be used to approximate its value over a specific interval.
Yes, the integral of cos(x²) is closely related to the Fresnel integrals, which are used in wave optics and other fields. The Fresnel integrals are defined as special functions and are often used to study the properties of ∫cos(x²) dx.
A series expansion of cos(x²) can be used to approximate the integral. By expanding cos(x²) into a Taylor series and then integrating term by term, one can obtain an approximate series representation of the integral. However, this method may require many terms to achieve a high degree of accuracy, especially for large values of x.
Graphical methods, like plotting the function cos(x²) over a certain interval, can provide a visual understanding of the integral. While it doesn't give a numerical answer, it can help in comprehending the behavior of the integral, such as its symmetry and periodicity.
While ∫cos(x²) dx may not have a straightforward elementary function solution, it finds applications in physics, particularly in optics and wave theory. The function is significant in the study of diffraction patterns and wave propagation, where Fresnel integrals are commonly applied.