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particular harmonic order n using trigonometric identities.
∴
=
+
−
=
=
+
=
=
∞
∑
v t
c
c
n t
c
v
c
a
b
v v
n
o
n
n
o
n
n
n
n n
( )
cos(
)
,
o
o
where
w
f
1
2
2
2
**
| |,
tan
, ,
=
=
= −∠
=
−
2
1 2 3
1
…
v
b
a
v
n
n
n
n
n
n
f
for 
(9.3-2)
9.4 
condItIons for exIstence of fourIer serIes
The exponential and trigonometric Fourier series exists for all v(t) which satisfy a set of conditions 
known as Dirichlet’s conditions.
(i) The function v(t) must be single-valued everywhere.
(ii) The integral 
| ( )|
v t dt
t
t
T
0
0
+
∫
is finite for any t
0
. This condition ensures that all Fourier series 
coefficients are finite-valued.
(iii) v(t) has only a finite number of maxima and minima in any one period.
(iv) v(t) has a only a finite number of discontinuities in any one period. Moreover, each discontinuity 
is a finite discontinuity.
There are functions that violate one or more of Dirichlet’s conditions. But they do not usually 
come up in electrical circuits. Hence we can safely assert that all waveforms we encounter in physical 
circuits will satisfy these conditions.

---

Waveform Symmetry and Fourier Series Coefficients 
9.11
If v(t) satisfies all the Dirichlet’s conditions, its Fourier series is guaranteed to exist and converge 
to the function value except at the points of discontinuity. At the points of discontinuity, the Fourier 
series will converge to the average value – i.e. to half the sum of value of v(t) at the left and right of 
the discontinuity.
This implies that, for a v(t) that satisfies all the four conditions, the partial sum of its Fourier 
series tends to approach the value of v(t) as the number of series terms included in the partial sum 
approaches infinity.
lim
( )
( )
N
n
jn t
n
N
n
v e
v t
v t
t
o
→∞ =−
=
=

w
0
0
0
if 
is continuous at and
N
N
N
n
jn t
v e
v t
v t
v t
o
∑
→∞
−
+
=
+
lim
(
)
(
)
( )

w
0
0
0
2
if 
is discontinuous aat 
t
n
N
n N
0
=−
=
∑
(9.4-1)
All the terms in a Fourier series are continuous functions of time. Eqn. 9.4-1 seems to suggest 
that the Fourier series converges to a discontinuity – i.e. the series converges to 
v t
(
)
0
-
at t 
= 
t
0
-
, to 
0 5
0 5
0
0
. (
)
. (
)
v t
v t
−
+
+
at t 
= 
t
0
and to 
v t
(
)
0
+
at t 
= 
t
0
+
.
How can a sum of continuous functions produce 
jump discontinuity? In fact it doesn’t. The partial sum of Fourier series in Eqn. 9.4-1 oscillates with 
time around the point of discontinuity. As N is increased, these oscillations in partial sum value get 
crowded more and more towards a small neighborhood of t
0
. The oscillations never really disappear.