Deriving Fresnel Integrals

Can someone help me?In summary, the Fresnel integrals can be obtained by integrating e^{iz^2} around the contour C, which consists of three parts: z = x, z = Re^{i\theta}, and z = te^{i\pi/4}. However, evaluating the integrals in the middle and last terms may be challenging and further assistance may be needed.
  • #1
1,357
0
Homework Statement
Integrate [itex]e^{iz^2}[/itex] around the contour C to obtain the Fresnel integrals:

[tex]\int_0^\infty \cos(x^2) \, dx = \int_0^\infty \sin(x^2) \, dx = \frac{\sqrt{2\pi}}{4}[/tex]

The contour consists of three parts:

  1. z = x, [itex]0 \le x \le R[/itex]
  2. z = [itex]Re^{i\theta}[/itex], [itex]0 \le \theta \le \pi/4[/itex]
  3. z = [itex]te^{i\pi/4}[/itex], [itex] R \ge t \ge 0[/itex]

The attempt at a solution
I'm stumped because I don't know how to evaluate integrals of the form [itex]e^{iz^2}[/itex]. How would I do this?
 
Physics news on Phys.org
  • #2
My attempt:\int_C e^{iz^2} \, dz = \int_0^R e^{ix^2} \, dx + \int_0^{\pi/4} Re^{iRe^{i\theta}^2} \, iRe^{i\theta} \, d\theta + \int_R^0 te^{i\pi/4}e^{it^2e^{i\pi/2}} \, ite^{i\pi/4} \, dt I'm not sure how to evaluate the integrals in the middle and last terms.
 

What are the Fresnel integrals and what do they represent?

The Fresnel integrals, denoted by S(x) and C(x), are special functions used in optics to describe the amplitude and phase of a wave that is partially transmitted and partially reflected at an interface between two media.

How are the Fresnel integrals derived?

The Fresnel integrals are derived by solving the differential equations that describe the electric and magnetic fields at the interface between two media. This involves using complex numbers and applying boundary conditions.

What are the applications of the Fresnel integrals?

The Fresnel integrals have many applications in optics and physics, including describing the diffraction patterns of light, calculating the intensity of light at different points in an optical system, and analyzing the polarization of light.

Are there any limitations or approximations to using the Fresnel integrals?

Yes, there are limitations and approximations when using the Fresnel integrals. They assume that the interface between two media is flat and that the media have uniform properties. In some cases, these assumptions may not hold and more complex equations may need to be used.

Are there any practical methods for calculating the Fresnel integrals?

Yes, there are several methods for calculating the Fresnel integrals, including numerical integration, power series expansions, and asymptotic approximations. The choice of method depends on the specific application and desired level of accuracy.

Suggested for: Deriving Fresnel Integrals

Replies
14
Views
900
Replies
6
Views
947
Replies
9
Views
663
Replies
3
Views
506
Replies
9
Views
609
Replies
3
Views
441
Replies
7
Views
802
Replies
3
Views
645
Replies
16
Views
959
Replies
34
Views
1K
Back
Top