Fresnel formulae

Proof.



The function $ \displaystyle z \mapsto e^{-z^2}$ is entire, whence by the fundamental theorem of complex analysis we have

(1)

where is the perimeter of the circular sector described in the picture. We split this contour integral to three portions:

(2)

By the entry concerning the Gaussian integral, we know that

For handling , we use the substitution

Using also de Moivre's formula we can write

Comparing the graph of the function with the line through the points and allows us to estimate downwards:

   for

Hence we obtain

and moreover

   as

Therefore

Then make to the substitution

It yields

   
     

Thus, letting , the equation (2) implies

(3)

Because the imaginary part vanishes, we infer that , whence (3) reads

So we get also the result , Q.E.D.



Contributors to this entry (in most recent order):

As of this snapshot date, this entry was owned by pahio.