Notes On Mathematics-3

, Fourier series : _




Faeries series is an infinite series .




Representation of Periodic functions in terms of trigonometric
function sine S cosine .




5in × , Rosa ,
tan ×



~ .




g? ¥
Sim =
✗
-
+
,



ten
=
ao + ?[ an
eoslnj-itbnsinfq.TK/ ]
ao
=
i ↑ fed
c
dx
2L
-




u


an
f- 1 text cos
fn¥ )
=




,

l




{ 1 (
bn =

f- a) 5in n

, Even function odd function
f- ex) =

f- e-×) f- G)
= -



f- G)
3
: eosx ✗2 : 5in × ✗ ✗
eg eg
-




,
-


, ,
L l v




1 f- ex)d✗=2
/ f- G) dx
{ flxldx
-_
0 .




If f- IN is even
If function is odd
Tt Clo
=
O g an = O




¥ { f- G)
ao
=
dx IT

bn =
2- ffexisinxdx
flxleosxrdx
°
IT
¥
an =




bn = 0
f- ex)
=

bnsinnx
es



f- ex) ao an COSNX
=


,




QI find Fourier series of f- a)
=
× ,
-
IT < ✗ < IT

Here f- ex) is an odd function .




0
'

an =
0 90 =
a
.
•
+
,




¥¥
bn =

f- G) 5in nxdx

bn sinnxdx
¥
=
✗ .

, bn =




¥
[ ✗ fsinnxdx -



* fsinnxdx ) diff

ufvdx Idea / vdxldx
[ fund ✗ Integration by
=
<
-


.




dx
parts .




'T

¥ f- f- eosnn×_ dx
)
=
✗ eosnx
-




n o




¥ f- sing)É
=
xeosnx +

n




¥(
IT eosnt + Sinn -11
) ( o 5in
:o)
-


=
- -




2
n n n




¥( )
"
=
→ e- 11 + 0 +0
n


"


2-1-11
bn = _




f- ex) =

nÉ
,
bnsinx


FE
"

f- e- 1) sin ✗
g-
ex) = -




,




Representation in series
f- ex) -2
Resins sina.at
+ sink
]
=
+ . . .
.




3