Here we will look at the usual properties we ask from series expansion (how it behaves with respect to the usual operations and how it reacts to transformations of functions). Then we look at alternative ways to write Fourier series, namely the amplitude - phase angle form and the complex form.

We start with a result that is not related to the main topic of this section, but it is sometimes useful, so we quickly go through it.

Theorem.

Letdenote T> 0,ω= 2π/T.

For anyfunction T-periodicfthat is integrable on[0, the Fourier coefficientsT],a_{k}andb_{k}tend to zero.

Now we turn to investigation of operations. First one fairly obvious result that follows from the fact that coefficients of Fourier series are given by integrals, that is, by a linear process.

Theorem.

Letdenote T> 0,ω= 2π/T.

Letcbe any real number, letfandgbefunctions that are integrable on T-periodic[0, Assume thatT].Then

Now we would like to determine similar formulas for derivative and integral.
With derivative it is simple. Assume that *f* is
*T*-periodic*a*_{k} and *b*_{k}. Denote the
coefficients of the Fourier series of the derivative *f* ′ by
*A*_{k} and *B*_{k}. Just by using
periodicity it follows that

The integrals for the other coefficients can be changed into integrals
featuring *f* using
integration by parts.

Similarly we evaluate *b*_{k} and obtain the following
statement.

Theorem.

Letdenote T> 0,ω= 2π/T.

Letfbe afunction integrable on T-periodic[0, Assume thatT].Then

So we see that we can differentiate both sides of the "tilde relation"
(the Fourier series we differentiate term by term) and the relation stays
valid. How about a true equality? If we use the usual Jordan conditions (but
applied to *f* ′), we get the following implication:

Assume that

fis piecewise continuous and has the first and second derivative that are piecewise continuous. Then

That is, we can differentiate a convergent Fourier series term by term. Now
we turn to integration. We again start with formal assignment of Fourier
series, that is, assume that *f* is
*T*-periodic*a*_{k} and *b*_{k}. Assume that it
has an antiderivative *F* (which need not always exists even for a
piecewise continuous function, see for instance
this example),
denote coefficients of its Fourier series of by *A*_{k}
and *B*_{k}. Here we have a serious problem, this
antiderivative need not be *T*-periodic. To see this, consider one
special antiderivative, namely
(see The
Fundamental Theorem of Calculus)

Now consider some *t* from the interval
*T* )*k*. Using the fact that
*f* is
*T*-periodic

We see that this antiderivative *F* is
*T*-periodic*f* over the
basic interval is 0; this actually means that
*a*_{0} = 0.

Antiderivatives offareexactly if T-periodica_{0}= 0.

Under this assumption we may start asking about the Fourier series assigned
to *F*. This time coefficients are given as integrals with *F* and
we use integration by parts to pass to integrals with *f*, then we use
periodicity of *f* and also the fact that our special *F* has
*F*(*T* ) = *F*(0) = 0.

Similarly we evaluate the integral for *B*_{k}. Thus we
get the following statement.

Theorem.

Letdenote T> 0,ω= 2π/T.

Letfbe afunction continuous on T-periodic[0, Assume thatT].If

a_{0}= 0, then for the antiderivative given bywe have

This tells us about one particular antiderivative. Other antiderivatives differ by a constant, so if we consider the set of all of them (the indefinite integral), we can write the conclusion in the following way:

How about true equality instead of formal assignment? Again, we will use
Jordan's conditions. We need to guarantee that an
antiderivative exists, for which the natural assumption is that *f* is
continuous. The antiderivative *F* in the above theorem is then also
continuous, therefore *F* and its derivative *f* satisfy the
"better" version of assumptions in Jordan's theorem and we actually get
uniform convergence.

Instead of expressing this formally we will try something else, we look at
definite integral (which is in a sense more general). Since we will not work
with antiderivatives, we do not have to worry about their periodicity and
thus we need not require that
*a*_{0} = 0.

Theorem.

Letdenote T> 0,ω= 2π/T.

Letfbe afunction that is piecewise continuous and integrable on T-periodic[0, Assume thatT].Then for any

a<bwe have

Now we will look at how Fourier series reacts to transformations of
*f*. Note that if *f* is a periodic function, then the usual
transformations again yield periodic functions, and apart from scaling the
variable they even preserve the original period.

Theorem(transformations).

Letdenote T> 0,ω= 2π/T.

Letfbe afunction that is integrable on T-periodic[0, Assume thatT].Then for any non-zero real number

cwe have

The first statement we already saw above, it follows from linearity.
Note that *a*_{0} is actually the average of *f*
(multiplied by 2), so by scaling the variable (shrinking or expanding the
function along the *x*-axis)*a*_{0} in those formulas above. The other
coefficients show how important individual frequencies are in *f*,
which is something that does not change when we shift the function vertically
or scale its variable, but by scaling the values of the function we obviously
change importance of all frequencies. The last expression is more
complicated, since we shift "waves in *f*" left or right, but the
cosines and sines on the right do not get shifted, which makes things rather
difficult. However, note that if *c* is a multiple of the basic period
*T*, then in fact the above formulas show that
*f* (*t* − *c*)*f*. This is to be expected, since shifting a periodic
function by a multiple of its period does not change it at all and if the
above formula did not yield this result, it would have been wrong.

Note that this last formula looks much better in complex form, see below.

The basic Fourier series can be rearranged to better suit one's needs.
The first rearrangement that we will show here uses a trick that is fairly
popular when working with waves (signals, electrical circuits
etc.). Given numbers *a* and *b*, there is a certain angle
*φ* and a number *A*
such that for any *x* we have

The number *A* is called the **amplitude** and the angle
*φ* is called the **phase
angle**, they are given by the following formulas.

If we exchange sine and cosine in this definition of
*φ*, we get a similar
reduction, but this time with cosine on the right (the phase angle will be
now different). If we apply this to all terms in the Fourier series, we get
the following formulas.

Here we use a different trick. If we replace sines and cosines in a Fourier series by their equivalent expressions with exponentials, we get

Thus if we denote

we get the **complex form of Fourier series**

This form is actually the natural form of this series, since many formulas
become much nicer. For instance, we do not have to find
*c*_{k} by doing several cases, there is a common
formula for them all:

Indeed, for instance, for a positive integer *k* we have

Similarly, there are more convenient forms for the rules for transformations above.

In the last formula, *n* is obviously some integer.

Going a bit further in this direction would get us to the notion of Fourier transform, which is another story, so we'd better stop.