Final answer:
The proof is based on the properties of odd functions and orthogonality of complex exponentials that represent different frequencies, which results in the integral equating to zero over the period from −π to π.
Step-by-step explanation:
The question asks us to prove that the integral ∫π−π e⁰ˡ e⁻ˡ²ˡ x x is equal to 0. This integral represents the inner product of two functions over a period, and it can be shown that when these functions are orthogonal, the result is zero.
To prove the first integral is zero, notice that eix and e−i²x are complex conjugates over the interval from −π to π. The integral of the product of complex conjugates over a symmetric interval around zero results in zero due to the properties of odd functions.
For the second integral, when n ≠ m, we can use the orthogonality of complex exponentials on the interval −π to π. Because ei nx and e−i mx represent different frequencies when n ≠ m, their product will again yield an odd function with respect to the interval, ensuring that the integral over the complete period will be zero.
The concepts employed in this proof rely on knowledge of complex numbers, integrals, and properties of functions, particularly the orthogonality of the complex exponentials which form a Fourier series.