Fresnel Integrals

What is a Fresnel Integral?
Fresnel functions are integrals that involve quadratic equations in the sine and cosine functions. They are defined as below:

where S[x] is the Fresnel Sine function and C[x] is the Fresnel Cosine function.

What are they used for in real life?
The Fresnel integrals are used in diffraction theory.

What are the identities and interesting equations?

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How do I solve problems with them?
If you have an integral with an x^2 in the sine or cosine function, then all you have to do is isolate part of the equation that is identitcal to the definitions above.

Example

Evaluate

We have to get the equation to look something like the definition for the Fresnel Sine function. We can see that we are missing a Pi/2 inside the sine function. So, let's take a guess and say that the answer is in the form of a*S[b*x]. Now let's take the derivative of this and see if we can solve for a and b given our problem. Taking the derivative is just like any other function in math. Plug in what we have as input (b*x here), take the derivative of the function (Sin[Pi*(x^2*b^2)/2]) times the derivative of what's inside (D[x*b,x] is b). The derivative is a*b*Sin[(b^2*Pi*x^2)/2]. So, by looking at all of our coefficients, we can see that a*b == 1 and that b^2*Pi/2 == 1. Now we just solve.

So which one is right?

or ?

Well, look to the properties above. S[-x] == -S[x]. So, this would mean that the two minus signs cancel each other out and the equations are the same.

Piece of cake!

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