Faulted demonstration of orthonormality in my textbook?

[LeFH18]

Homework Statement

Show that the functions:

ψ_m(θ)= (2π)^(-1/2) e(imθ), m = 0, +- 1, +-2, ...

form an orthonormal set over the interval (0,2π)

Relevant Equations

Function set:
ψ_m(θ)= (2π)^(-1/2) e(imθ), m = 0, +- 1, +-2, ...

Orthonormality condition:

int( ψ*_m ψ_n dx) = δ_nm

where δ is Kronecker's delta

The textbook, "Mathematics for Physical Chemistry - Opening doors" (McQuarrie), solves this example excercise as follows:

And is the case for n =/= m which I'm troubled with, because, if I solve the integral instead of using the cycles argument, I get that:

And I can't see how this is equal to zero. Any leads on why this must be zero for n unequal of m?

 

[fresh_42]

Euler's identity says $e^{i \varphi}=\cos \varphi + i \sin \varphi.$ Now substitute $\varphi = 2\pi (n-m) .$ Orthogonality of two vectors $v_m, v_n$ means $\bigl\langle v_m,v_n \bigr\rangle =\delta_{mn}.$ It's the definition, whether $v$ are vectors or functions, and whatever the inner product is defined as, in this case by an integral.

 

[LeFH18]

Thank you so much!
I did not realize that I could do that substitution. Doing that, I get the following:
$= \frac{i}{\varphi}(1-e^{\varphi i})$
Expanding it into cosine and sine, and noting that $(n-m)$ is always an integer because $n$ and $m$ also are integers, it leads to the inner product being zero.