determination of the Fourier Coefficients for symmetric functions

The determination of the Fourier-coefficients a_n und b_n from the equations (13), (14) is much simpler if the symmetriy properties are not only known for the sine- and cosine-timefunctions but also for the function f(t).
An axially symmetrical (even) function has the characteristic f(t) = f(-t) (for example :rectangular pulse train).
A (uneven) function symmetrical to the origin has the characteristic f(t) = -f(-t) (for example:saw-tooth function).
A half wave symmetry of the first kind exsists for f(t + T/2) = -f(t) (for example: rectangular-function)
A function as the product of two functions with symmetry property (e.g. g(t) = f(t) cos(nω t), g(t) = f(t) sin(nω t) has symmetry property too. (symmetry properties of products of symmetric functions).
Integrals of functions with symmetry properties can be calculated easily:
For an axially symmetrical function f(t) the equations (13), (14), (18) for the calculation of the Fourier coeficiens simlify :

If the function is half wave symmetrical (1st kind) it follows:

To visualize the circumstances above the integrands of the equations (13), (14) can be displayed for various signal functions and different n.