# Inequality quick question context cauchy fresnel integral

• binbagsss
In summary, the conversation is about a problem involving inequalities and integration, with the goal of proving a certain result. The individual discussing the problem has made some progress and believes they have a proof, but they are still unsure and request feedback. They then continue to discuss their approach and make revisions to their proof.
binbagsss

## Homework Statement

please see attached, I am stuck on the second inequality.

attached

## The Attempt at a Solution

I have no idea where the ##2/\pi## has come from, I'm guessing it is a bound on ##sin \theta ## for ##\theta## between ##\pi/4## and ##0## ?

I know ##sin \theta \approx \theta ## for small ##\theta## so we could bound it to ##\pi/4## since ##sin ## increases in this range ## 0, \pi /4 ##, however that would be a stronger approximation , loosing the actual integration over ##\theta ## which is clearly not what has been done.

Many thanks in advance.

#### Attachments

• ahem.png
10.1 KB · Views: 424
I think I managed to prove it. Let ## x=\frac{\pi}{4}-x' ##. Starting with ## sin(\frac{\pi}{4}-x') \geq \frac{\frac{\pi}{4}-x'}{\frac{\pi}{2}} ## ==>> (This must be proven to be the case), This leads to ## cos(x')-sin(x')-\sqrt{2}(\frac{1}{2}-(\frac{2}{\pi})x') \geq 0 ## for ## 0<x'< \frac{\pi}{4} ##. The function (on the left side of the second inequality) can be shown to be monotonically decreasing from ## 1 -\frac{\sqrt{2}}{2} ## to ## 0 ## as ## x' ## goes from ## 0 ## to ## \frac{\pi}{4} ## by looking at the first derivative. You can try to work through what I did and see if you agree. ## \\ ## Editing: I found a sign error when I took the derivative=scratch the proof=it is incomplete at present... ## \\ ## Additional editing: A little algebra on the derivative shows it indeed is monotonically decreasing=the proof appears to work. The proof of the first derivative being less than zero on the interval ## 0<x'< \frac{\pi}{4} ## has one step to it that may not be so obvious, but I believe it works. I would be happy to show you a couple of the steps if you get stuck.

Last edited:
The function (on the left side of the second inequality) can be shown to be monotonically decreasing from ## 1 -\frac{\sqrt{2}}{2} ## to ## 0 ## as ## x' ## goes from ## 0 ## to ## \frac{\pi}{4} ## by looking at the first derivative.
mmm okay thanks.

I am stuck on making the final conclusion from what has been done,

The first derivative is constantly negative over the interval and at the very right (## x'=\frac{\pi}{4} ##) the function has the value zero. The function is thereby positive for the interval ## 0<x'< \frac{\pi}{4} ##. The first derivative is ## f'(x')=-sin(x')-cos(x')+2 \sqrt{2}/\pi ##. To show this is less than zero over the interval ## 0<x'< \frac{\pi}{4} ##, it helps to see that ## sin(x')+cos(x')=\sqrt{2} cos(x'-\frac{\pi}{4}) ##. Then the point ## x'=0 ## is the point of most concern, because that is where ## cos(x'-\frac{\pi}{4}) ## has its minimum value on the interval ## 0<x'<\frac{\pi}{4} ##..

## 1. What is the Cauchy Fresnel integral?

The Cauchy Fresnel integral is a mathematical formula used to calculate the diffraction of light through a small aperture. It was first derived by French mathematician Augustin-Louis Cauchy and physicist Augustin-Jean Fresnel in the 19th century.

## 2. How is the Cauchy Fresnel integral used in the context of inequality?

In the context of inequality, the Cauchy Fresnel integral is often used to study and analyze the distribution of wealth and income. It helps to understand the disparities between different groups and can provide insights on potential solutions to reduce inequality.

## 3. What are some limitations of using the Cauchy Fresnel integral to study inequality?

One limitation is that the Cauchy Fresnel integral assumes a normal distribution of data, which may not always be the case in real-world scenarios. Additionally, it does not take into account factors such as race, gender, and socio-economic status, which can also contribute to inequality.

## 4. Can the Cauchy Fresnel integral be used to measure inequality in other areas besides wealth and income?

Yes, the Cauchy Fresnel integral can be applied to measure inequality in various fields such as education, health care, and access to resources. It can also be used to study inequality on a global scale, between different countries and regions.

## 5. How can the Cauchy Fresnel integral be used to address inequality?

The Cauchy Fresnel integral can provide data and insights that can inform policies and initiatives aimed at reducing inequality. It can also be used to track changes in inequality over time and assess the effectiveness of interventions. However, it should be used in conjunction with other measures and considerations to fully address the complex issue of inequality.

Replies
5
Views
1K
Replies
32
Views
2K
Replies
13
Views
1K
Replies
9
Views
2K
Replies
5
Views
3K
Replies
4
Views
1K
Replies
6
Views
2K
Replies
2
Views
1K
Replies
5
Views
2K
Replies
3
Views
3K