Functions can be expanded in powers of x (Taylor series), or in Fourier series using cos(nx) or sin(nx) as a basis (where n is an integer).
Can functions also be expanded using a basis of exponential functions e^(nx), where n is an integer.
In particular, can cos(cx), with c being a real constant, be expanded in the form:
Summation from n=-infinity to n=+infinity of (An)e^(nx)?
No. Indeed, let coscx=\sum_{n\in Z}a_ne^{nx}. Since coscx is T-periodic, it follows
that \sum_{n\in Z}a_ne^{nx}=\sum_{n\in Z}a_ne^{n(x+T)}, or
\sum_{n\in Z}a_n(1-e^{nT})e^{nx}=0. In turn, this yields that
\sum_{n\in Z}a_n(1-e^{nT})t^n=0 for all t>0. Now the theory of Laurent series implies that a_n(1-e^{nT})=0 for all n, a contradiction.
The answer is: YES but under certain restrictions on x and n, because the change exp(x)=z turns exp(nx) into the pover functions z^n.
