**Question:** Find the **Fourier series** for a function
*f* (*t*)*T*-periodic*a*,*a* + *T* ).

**Solution:** First determine the frequency
*ω* = 2π/*T*

The Fourier series is then obtained by writing

If *f* is nice enough (see
Jordan's conditions), you
can determine the sum of the resulting Fourier series. First, draw the
*T*-periodic*f* (either it was given as such or it was given on a suitable
interval, then draw its periodic extension). Then, at every point of
discontinuity, draw a dot at the level that is the average of the left and
right value there. This is the sum of the Fourier series.

Note: The coefficient *a*_{0} is actually given by the same
formula as the other *a*_{k}, so we could just specify one
formula. However, for *k* = 0*a*_{0} separately from
the other integrals anyway. Thus it is convenient to put it aside as a
special case.

**Question:** Find the **sine Fourier series** for a function
*f* (*t*)*L* ).

**Solution:** The sine series will be based on an odd extension with
period *T* = 2*L*.*ω* = 2π/*T* = π/*L*.*b*_{k} as above but with *L*
in place of *T*, set
*a*_{0} and *a*_{k} equal to zero and write
the series exactly as above.

If *f* is nice enough (see
Jordan's conditions), you
can determine the sum of the resulting Fourier series. First, draw the given
function. Second, extend the graph also to the interval
*L*,0)*T* = 2*L*-periodic

**Question:** Find the **cosine Fourier series** for a function
*f* (*t*)*L* ).

**Solution:** The cosine series will be based on an odd extension with
period *T* = 2*L*.*ω* = 2π/*T* = π/*L*.*a*_{0} and *a*_{k}
as above but with *L* in place of *T*,
set *b*_{k} equal to zero and write
the series exactly as above.

If *f* is nice enough (see
Jordan's conditions), you
can determine the sum of the resulting Fourier series. First, draw the given
function. Second, extend the graph also to the interval
*L*,0)*y*-axis).*T* = 2*L*-periodic

It is often possible to improve the resulting series by using these handy formulas.

**Example:** Find the Fourier series, the sine Fourier series and the
cosine Fourier series for the function

This function can be extended to a function with period
*T* = 4,*ω* = π/2.

Thus we get the series

We noticed that in the series, all even coefficients are zero and odd ones are

The sine and cosine series have *L* = 4,*T* = 8,*ω* = π/4.*a*_{k} = 0

Thus the sine Fourier series is

The cosine Fourier series has
*b*_{k} = 0

Thus the cosine Fourier series is

Now we determine the sums of the above three series. In the following picture there are six graphs. The first is the periodic extension of the given function and the second is the sum of the Fourier series by Jordan's criterion. The third graph is the odd periodic extension and then there is the sum of the sine Fourier series. Finally, the fifth graph is the even periodic extension and then the sum of the cosine Fourier series.

Note that the Fourier series has the form of *f*
and move it down by