English: Graphics visualizing the symmetry properties of the Fourier series. When the real and imaginary parts of a complex function are decomposed into their even and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform:^[1]

${\displaystyle {\begin{aligned}{\mathsf {Time\ domain}}\quad &\ f\quad &=\quad &f_{RE}\quad &+\quad &f_{RO}\quad &+\quad i\ &f_{IE}\quad &+\quad &\underbrace {i\ f_{IO}} \\&{\Bigg \Updownarrow }{\mathcal {F}}&&{\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}\\{\mathsf {Frequency\ domain}}\quad &F\quad &=\quad &F_{RE}\quad &+\quad &\overbrace {i\ F_{IO}} \quad &+\quad i\ &F_{IE}\quad &+\quad &F_{RO}\end{aligned}}}$