The Fresnel integrals read

(1)

Consider the following integral

(2)

From this we have

(3)

(4)

Equation (2) also gives

(5)

We give an alternative derivation in the following. Let us start with

(6)

Differentiating with respect to

(7)

where we chnage the variable, .

Therefore we have

(8)

Since in the limit

(9)

we have

(10)

Solving the equation

(11)

gives (4).