The Bessel functions of the first kind may be defined as the
coefficients
in the two-sided Laurent expansion
of the so-called generating function [82, p. 14],^{4.10}

(4.6)

where is the integer order of the Bessel function, and is its argument (which
can be complex, but we will only consider real ).
Setting
, where will interpreted as the
FM modulation frequency and as time in seconds, we obtain

(4.7)

The last expression can be interpreted as the Fourier superposition of the
sinusoidalharmonics of
, i.e., an
inverse Fourier series sum. In other words,
is
the amplitude of the th harmonic in the Fourier-series expansion of
the periodic signal .

Note that
is real when is real. This can be seen
by viewing Eq. (4.6) as the product of the series expansion for
times that for
(see footnote
pertaining to Eq. (4.6)).

Figure 4.15 illustrates the first eleven Bessel functions of the first
kind for arguments up to . It can be seen in the figure
that when the FM index is zero, and for
all . Since
is the amplitude of the carrier
frequency, there are no side bands when . As the FM index
increases, the sidebands begin to grow while the carrier term
diminishes. This is how FM synthesis produces an expanded, brighter
bandwidth as the FM index is increased.

Figure 4.15:
Bessel functions of the first kind
for a range of orders and argument .