PlanetPhysics/Fresnel Formulae Result

The Fresnel formulae presented in the parent entry-- that the proof refers to-- are as follows:

(visible there only in the Tex code mode because of a current quirk/glitch with pstricks vs. Tex).

The remainder of the equations Tex mode correctly specified by pahio in the parent entry entitled Fresnel formulae is precisely quoted here as follows: ``The function \,\, is entire, whence by the fundamental theorem of complex analysis we have

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

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: Hence we obtain and moreover Therefore Failed to parse (syntax error): {\displaystyle \lim_{R\to\infty}I_2 = 0.\\}

Then make to the substitution It yields

Thus, letting\, ,\, the equation (2) implies

Because the imaginary part vanishes, we infer that\, ,\, whence (3) reads So we get also the result\, "