Integrals in Support of the Schmidt Analysis

Basis for all the subsequent mathematics on the integrals in support of the Schmidt analysis is the solution to the following indefinite integral :

For our purposes we need its definite counterpart with the integration performed over a complete cycle :

Another integral needed is :

The given solution is actually related to the solution of integral I₁. The first step to prove this is to use the trigonometric identity :

sin(x) = sin(x+β) cos(β) - sin(β) cos(x+β)

and break up the resulting integral as follows :

The first integral on the right hand side of above equation equals 0, the second integral equals 2π and the third integral is equal to the previously evaluated integral I₁.

Lastly, we need the following integral :

The similarity between the solutions for the integrals I₂ and I₃ is no coincidence. To prove we use the variable substitution z = x + δ. Hence ,

The second form on the right hand side of above equation is exactly equivalent to the integral I₂, substituting β for β-δ.