Half range sine and cosine series

Half Range Sine and Cosine Series

1. HALF-RANGE SINE AND COSINE SERIES
At the end of this session, students will be able to:
Find a Fourier series expansion of a function defined over a finite
interval
Expand any periodic or non periodic function ??????(??????)on an interval
[0,??????]as a series of sine or as a series of cosine series

2. MOTIVATION
•So far in the previous session, we have shown how to
represent given periodic functions by Fourier Series
•We now consider a slight variation on this theme which will
be useful in solving Partial Differential equations
•Imagine that we wish to expand a function ??????(??????)defined
for −??????≤??????≤??????in terms of a Fourier series, it might seem
that we must use the Fourier full range series
•If the function is either even or odd we can expand ??????(??????)in
the range 0≤??????≤??????with either a cosine or sine Fourier
half range series and we will get exactly the same result

3. MOTIVATION
•To be specific, suppose we define
•We shall consider the interval 0<&#3627408481;<??????to be half a period
of a 2??????periodic function
•We must therefore define f(t) for −??????<&#3627408481;<0to complete the
specification
•Let us complete the definition of the above function by
defining it over −??????<&#3627408481;<0such that the resulting functions
will have a Fourier Series t0 ,)(
2
ttf

4. MOTIVATION
•Containing only cosine terms: even periodic function,
??????&#3627408481;=&#3627408481;
2
,(−??????<&#3627408481;<0)
•Containing only sine terms: odd periodic function,
??????&#3627408481;=−&#3627408481;
2
(−??????<&#3627408481;<0)

5. MOTIVATION
•Containing both cosine and sine terms: We may define ??????(&#3627408481;)=
0in any way we please
•The point is that all 3 periodic functions ??????
&#3627409359;(&#3627408481;),??????
&#3627409360;(&#3627408481;)and ??????
&#3627409361;(&#3627408481;)will give rise
to a different Fourier Series but all will represent the function ??????(&#3627408481;)=&#3627408481;
&#3627409360;
over
0<&#3627408481;<??????
•Thus Fourier Series obtained by extending functions in this sort of way are often
referred to as half-range series
•The above considerations apply equally well for a function defined over an interval
other than 0<&#3627408481;<??????

6. HALF RANGE SINE SERIES
•Suppose that ??????(??????)satisfies Dirichlet’s condition in the interval
0<??????<??????
•????????????is an even function of period 2??????,then even periodic &#3627408442;??????
will have a convergent Fourier series representation consisting
of cosine terms only and given by
&#3627408442;??????=
??????=1
∞
&#3627408463;
??????sin
&#3627408475;????????????
??????
&#3627408463;
??????=
2
??????

0
??????
????????????&#3627408480;??????&#3627408475;
??????????????????
??????
&#3627408465;??????,(&#3627408475;=0,1,2,….)

7. HALF RANGE COSINE SERIES
•Suppose that ??????(??????)satisfies Dirichlet’s condition in the interval
0<??????<??????
•????????????is an even function of period 2??????,then even periodic &#3627408441;??????
will have a convergent Fourier series representation consisting
of cosine terms only and given by
&#3627408441;??????=
&#3627408462;
0
2
+
??????=1
∞
&#3627408462;
??????cos
&#3627408475;????????????
??????
&#3627408462;
0=
2
??????

0
??????
????????????&#3627408465;??????and &#3627408462;
??????=
2
??????

0
??????
??????(&#3627408481;)&#3627408464;&#3627408476;&#3627408480;
??????????????????
??????
&#3627408465;??????,
for(&#3627408475;=0,1,2,….)

8. EXAMPLE
•Obtain the half range sine series for ????????????=
??????
2
−??????in 0<??????<??????.
•Soln. We will write the Fourier sine series of ??????(??????)on 0,??????.The
coefficients are
&#3627408463;
??????=
2
??????

0
??????
(??????−2??????)s??????&#3627408475;
??????????????????
??????
,(integrating by parts)
=
1
??????
(??????−2??????)
0
??????
sin
??????????????????
??????
&#3627408465;??????−2
0
??????
cos
??????????????????
??????
&#3627408465;??????
=??????−2??????&#3627408464;&#3627408476;&#3627408480;&#3627408475;??????−??????.cos0=
??????
&#3627408475;??????
[−1
??????
+1]
The Fourier sine series for ??????(??????)is given by
????????????=
??????
2
−??????=
??????
&#3627408475;??????
[−1
??????
+1]sin
&#3627408475;????????????
??????

9. EXAMPLE
•Obtain the half range cosine series for ????????????=??????in 0<??????<2.
•Soln. We will write the Fourier cosine series of ??????(??????)on 0,2.The
coefficients are
&#3627408462;
0=
2
2

0
2
??????&#3627408465;??????=
??????
2
2
0
2
=2.
a
n=
2
2

0
2
??????&#3627408464;&#3627408476;&#3627408480;
??????????????????
2
&#3627408465;??????
=
2
????????????
??????sin
??????????????????
2
−
2
????????????

0
2
sin
??????????????????
2

10. EXAMPLE…
•Therefore
=
2
&#3627408475;??????
2
cos
&#3627408475;????????????
2
0
2
=
4
??????
2
&#3627408475;
2
cos&#3627408475;??????−1
????????????=1+
4
??????
2

??????=1
∞
−1
??????
−1
&#3627408475;
2
cos
&#3627408475;????????????
2
.

11. SESSION SUMMARY
Let ??????(??????)be a function defined and integrableon 0,??????.Set
??????
1??????=
−??????−??????,−??????≤??????<0
????????????,0≤x≤l
and ??????
2??????=
??????−??????,−??????≤??????<0
????????????,0≤x≤l
We have to check that these two functions are defined on [−??????,??????]and
are equal to ??????(??????)on [0,??????]

12. SESSION SUMMARY
The Fourier series of ??????
1(??????)is called Fourier sine series of ????????????,i.e.,
??????
1(??????)=
??????=1
∞
&#3627408463;
??????sin
??????????????????
??????
, where &#3627408463;
??????=
2
??????

0
??????
??????&#3627408481;&#3627408480;??????&#3627408475;
??????????????????
??????
&#3627408465;&#3627408481;
The Fourier series of ??????
2(??????)is called Fourier cosine series of ????????????,i.e.,
??????
2??????=
??????0
2
+
??????=1
∞
&#3627408462;
??????cos
??????????????????
??????
, where
&#3627408462;
0=
−??????
??????
??????&#3627408481;&#3627408465;&#3627408481;and&#3627408462;
??????=
2
??????

0
??????
??????(&#3627408481;)&#3627408464;&#3627408476;&#3627408480;
&#3627408475;??????&#3627408481;
??????
&#3627408465;&#3627408481;.