The complex form of the Fourier series
D. Craig
April 3, 2011
In addition to the “standard” form of the Fourier series, there is a form using complex exponentials instead of the sine
and cosine functions. This form is in fact easier to derive, since the integrations are simpler, and the process is also similar
to the complex form of the Fourier integral.
1
Derivation of the complex form
Begin with a real periodic function F (t) with period T :
F (t + T ) = F (t).
(1)
We are going to write this as a series in complex exponentials
F (t) = a0 + a1 eiωt + a2 e2iωt + · · · + an eniωt + · · ·
+ a−1 e−iωt + a−2 e−2iωt + · · · + a−n e−niωt + · · ·
(2)
using both positive and negative indices, so we can write
F (t) =
n=+∞
X
an einωt .
(3)
n=−∞
Here ω = 2π/T . Note that since we have assumed F (t) is real, the complex coefficients on the right side above must add
up to a real result. Now we will integrate over a period T = 2π/ω :
!
Z 2π/ω
Z 2π/ω n=+∞
X
inωt
F (t) dt =
an e
dt
(4)
0
0
n=−∞
and assuming we can interchange integration and summation:
=
n=+∞
X
Z
2π/ω
an
n=−∞
einωt dt
(5)
0
Now consider the integral in the right-hand side, for various values of n.
Z
0
2π/ω
1
(e2πin − 1) = 0
iωn
Z 2π/ω
=
dt = 2π/ω = T
einωt dt =
for n 6= 0, and
(6)
for n = 0.
(7)
0
This gives us
Z
T
F (t) dt = a0 T,
(8)
0
1
a0 =
T
Z
T
F (t) dt.
(9)
0
which is the average value of F .
Now we will obtain the other coefficients. Start over again from
F (t) =
m=+∞
X
m=−∞
1
am eimωt .
(10)
Note that for clarity further below, I am using a different label for the indices: m. This index is a “dummy,” like an
integration variable, so we may call it whatever we like. Multiply both sides of this by e−inωt :
!
m=+∞
X
−inωt
−inωt
imωt
e
F (t) = e
am e
,
(11)
m=−∞
and again integrate over a period, exchanging integration and summation. In essence, we are symbolically doing this for
many particular n’s:
Z 2π/ω
m=+∞
X Z 2π/ω
e−inωt F (t) dt =
ei(m−n)ωt dt.
(12)
0
m=−∞
Now note that because the complex exponential is periodic,
Z 2π/ω
eiω(m−n)t dt = 0
0
except when m = n
(13)
0
and e0 = 1 when m = n. So we have
Z
2π/ω
e−inωt F (t) dt = an T.
(14)
0
The coefficients of the series are thus
an =
a−n =
We also see that
1
T
Z
1
T
Z
2π/ω
e−inωt F (t) dt,
(15)
e+inωt F (t) dt.
(16)
0
2π/ω
0
a−n = (an )∗ ,
the negative index coefficients are the complex conjugates of the positive ones.
2 Back to the real form
To return to nonnegative indices and real sines and cosines is just a matter of rewriting now. Write the sum in eq. 3 in
separate parts:
n=−1
X
F (t) =
=
an einωt + a0 +
n=+∞
X
an einωt
n=−∞
n=1
n=1
X
n=+∞
X
a−n e−inωt + a0 +
n=∞
= a0 +
an einωt
(17)
(18)
n=1
n=∞
X
(an einωt + a−n e−inωt )
(19)
n=1
Using Euler’s formula eiφ = cos φ + i sin φ,
F (t) = a0 +
n=∞
X
n=∞
X
n=1
n=1
(an + a−n ) cos nωt +
i(an − a−n ) sin nωt.
(20)
We now let
A0
= a0 ,
2
(21)
n=∞
n=∞
X
X
A0
+
An cos nωt +
Bn sin nωt,
F (t) =
2
n=1
n=1
(22)
An = an + a−n ,
Bn = i(an − a−n ),
and get
the usual real form of the Fourier series. We can work out the coefficient explicitly to see where all the constants come from:
Z
1 T
An = an + a−n =
F (t)(e−inωt + einωt )
(23)
T 0
Z
2 T
=
F (t) cos nωt,
(24)
T 0
2
and
Bn = i(an − a−n ) =
2
=
T
Z
1
T
T
Z
F (t)i(e−inωt − einωt )
(25)
0
T
F (t) sin nωt.
(26)
0
Note that the A0 /2 term for the series (eq. 22) will be obtained by getting A0 from the n = 0 coefficient of the cosine
coefficients (24). The constant term in a Fourier series is always equal to the mean value of the function.
3
Magnitude-angle form
Yet another form of the Fourier series can be obtained from eq. 22 by writing
An cos nωt + Bn sin nωt = Cn cos(nωt − φn )
= Cn cos nωt cos φn + Cn sin nωt sin φn .
(27)
Equate the coefficients of like terms
An = Cn cos φn ,
Bn = Cn sin φn ,
(28)
and so
Cn =
p
n
φn = arctan B
An .
A2n + Bn2 ,
(29)
The magnitude-angle form of the series is thus
F (t) =
or
F (t) =
n=∞
X
A0
Cn cos(nωt − φn )
+
2
n=1
n=∞
X
A0
+
Cn sin(nωt +
2
n=1
π
2
− φn ).
(30)
(31)
The complex form of the Fourier series has many advantages over the real form. For example, integration and differentiation term-by-term is much easier with exponentials. The trigonometric functions and phase angles do not appear
explicitly but are contained in the complex coefficients.
4
Reference
This material (with minor changes) is closely based on
Applied Mathematics for Engineers and Physicists, Third ed. by Pipes and Harvill, McGraw-Hill Publishers,
1970, Appendix C, section 22.
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