What are the expression for complex Fourier series with a minus sign in the exponent? If a function $f(t)$ periodic with period $T=2\pi/\omega$ can be expanded with Fourier series, then is it possible to write the following?
$$f(t)=\sum_{-\infty}^{\infty} c_m e^{-im\omega t}\,\,\,,\,\,\,\,\,\,\,\,\,\, m \in \mathbb{N}$$
Where
$$c_m=\frac{1}{T} \int_{0}^{T}f(t) e^{i m \omega t }dt$$
Usually in texts the sign in the exponents are reversed, but are those expression correct?
Your expressions are correct.
Most textbooks will switch the minus sign for an aesthetical reason.
It's "nicer" to have your function decomposed with respect to the basis $\{f_n(t)\}_{n\in\mathbb{Z}}$ where $$f_n(t) = \cos n\omega t + i \sin n\omega t$$ instead of $f_n(t) = \cos n\omega t - i \sin n\omega t$.