Fourier series in continous time domain
Under certain conditions, a $T$-periodic function can be represented by its Fourier series

$$x(t)=\sum_{k=-\infty}^{\infty}a_ke^{jk\omega_0t}\tag{1}$$

with $\omega_0=2\pi/T$. The function $x(t)$ can be complex-valued, i.e. have non-zero real and imaginary parts. Note that generally the Fourier coefficients $a_k$ are also complex-valued (even for real-valued $x(t)$).

Now if $x(t)$ is real-valued, i.e., $x(t)=x^*(t)$, you get

$$x(t)=x^*(t)=\sum_{k=-\infty}^{\infty}a^*_ke^{-jk\omega_0t}=
\sum_{k=-\infty}^{\infty}a^*_{-k}e^{jk\omega_0t}\tag{2}$$

Comparing $(2)$ with $(1)$ you see that for real-valued $x(t)$ you get the condition $a_k=a^*_{-k}$, i.e. the Fourier coefficients show Hermitian symmetry. This condition is not satisfied for general complex-valued functions $x(t)$. However, the representation $(1)$ is valid no matter if $x(t)$ is real-valued or complex-valued.