Solving Complex Integral: Cos(x^2) + Sin(x^2)

In summary, the conversation discusses the integral \int_{-\infty}^{\infty} \cos(x^2) + \sin(x^2) \, \mathrm{d}x and how it can be evaluated using complex analysis. One approach is to use the even symmetry of the function and split the integral into two parts, while another approach involves using the identity e^{-ix^2} = \cos(x^2) - i \sin(x^2). The conversation also mentions agomez's calculations and the question of whether the minus sign in the identity changes anything.
  • #1
Nebuchadnezza
79
2
I read in some text or book that the integral

[tex] \int_{-\infty}^{\infty} \cos(x^2) + \sin(x^2) \, \mathrm{d}x = \sqrt{2\pi}[/tex]

I was wondering how this is possible. I read on this site that one such possible way was to start by integrating

[tex] e^{-i x^2} = \cos(x^2) - i \cos(x^2)[/tex]

My knowledge about complex analysis is rather limited. Could anyone expain to me how the integral at the top is evaluated? (I know one could start of by noticing the symmetry about the y-axis. )

https://www.physicsforums.com/showthread.php?t=139465

Agomez, shows one way to do it. But it is not exactly the same as the integral above. Sigh, I feel stupid for not seeing this one...
 
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  • #2
Because cos(x^2)+sin(x^2) is an even function, we have
[tex]\int_{-\infty}^\infty \cos(x^2)+\sin(x^2)\, dx = 2 \int_0^\infty \cos(x^2)+\sin(x^2) \,dx.[/tex]
Now use the results from agomez's calculations.
 
  • #3
My main problem is that

$$ e^{-ix^2} = \cos(x^2) - i \sin(x^2) $$

Whilst this integral is [itex] \cos(x^2) + \sin(x^2) [/itex]. Does the minus sign change anything?
 
  • #4
It changes nothing. All I'm saying is that you can use agomez's evaluation of [itex]\int_0^\infty \cos(x^2)\, dx[/itex] and [itex]\int_0^\infty \sin(x^2)\, dx[/itex].
 

1. What is a complex integral?

A complex integral is a mathematical concept that involves finding the area under a curve in the complex plane. It is similar to a regular integral, but takes into account the imaginary numbers involved in the function.

2. How do you solve a complex integral?

To solve a complex integral, you can use techniques such as contour integration, residue theorem, or Cauchy's integral formula. These methods involve manipulating the function and integrating along a chosen path in the complex plane.

3. What is the function cos(x^2) + sin(x^2)?

The function cos(x^2) + sin(x^2) is a combination of trigonometric functions where the argument is squared. This function is commonly used in physics and engineering to describe wave phenomena.

4. Is there a general formula for solving complex integrals?

No, there is no one general formula for solving complex integrals. The method used to solve a complex integral depends on the specific function and the chosen path of integration.

5. What are some applications of solving complex integrals?

Complex integrals have various applications in mathematics, physics, and engineering. They are used to calculate areas, volumes, and work done in complex systems. They are also used in quantum mechanics, signal processing, and image processing.

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