WEBVTT
Kind: captions
Language: en
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so welcome to the lecture series on mathematical
methods and the applications so now we will
00:00:24.130 --> 00:00:31.100
discuss fourier half range series uh we have
already discussed in last lecture what how
00:00:31.100 --> 00:00:36.500
we can find out fourier series expansion of
even and odd functions that we have already
00:00:36.500 --> 00:00:43.280
seen in the last lecture now half range series
fourier half range series let us see half
00:00:43.280 --> 00:00:49.430
range series plays an important role in solving
several engineering and physical applications
00:00:49.430 --> 00:00:53.370
where it is required to get the fourier series
expansion of a function in an interval say
00:00:53.370 --> 00:01:00.070
zero to l so its a half range basically uh
in fourier series basically in expansion of
00:01:00.070 --> 00:01:07.080
fourier series we need interval minus l to
l ok and function must be periodic that is
00:01:07.080 --> 00:01:10.280
the condition for the fourier series of the
expansion f x
00:01:10.280 --> 00:01:16.330
now here the fourier series expansion of a
function is in the interval zero to l where
00:01:16.330 --> 00:01:23.280
l is an half of the period ok now how can
we find out fourier series expansion of such
00:01:23.280 --> 00:01:30.080
a function and these problems are important
in several engineering and physical applications
00:01:30.080 --> 00:01:36.770
ok now it is possible to extend the function
f x to the other half that is minus l to zero
00:01:36.770 --> 00:01:44.110
of minus l to plus l so that f x is either
an even or an odd function basically what
00:01:44.110 --> 00:01:53.110
we do we know the nature of the function from
zero to l and that is given to us ok and depending
00:01:53.110 --> 00:02:03.220
upon the depending upon the basically nature
of the function we find out we either extend
00:02:03.220 --> 00:02:10.420
the function uh considering function as an
even function or we either extend the function
00:02:10.420 --> 00:02:17.590
taking function as an odd function ok suppose
suppose suppose function is given to you
00:02:17.590 --> 00:02:23.720
from zero to l suppose this function f function
is given to us from zero to l ok this function
00:02:23.720 --> 00:02:28.069
is given to you from zero to l it is a half
half period ok
00:02:28.069 --> 00:02:34.590
now you can find out the even extension of
this function you can extend this function
00:02:34.590 --> 00:02:42.000
as an even function from minus l to plus l
ok and then find out the fourier series expansion
00:02:42.000 --> 00:02:48.650
of this function ok so you can take this function
as an even function whatever function is given
00:02:48.650 --> 00:02:56.060
to you from zero to l you can find out the
even extension of this function uh taking
00:02:56.060 --> 00:03:02.530
function as an even function on the other
way you can also find the suppose this is
00:03:02.530 --> 00:03:11.250
given from zero to l you can find an odd extension
of the function ok you can you can extend
00:03:11.250 --> 00:03:17.590
this function like this from zero to minus
l minus l to plus l and in the same way over
00:03:17.590 --> 00:03:27.689
here ok in the same way over here so basically
we need a fourier series expansion of a function
00:03:27.689 --> 00:03:34.489
which is given in the half range from zero
to l so we can either to find the fourier
00:03:34.489 --> 00:03:39.049
series expansion of such a function we can
either extend the function taking function
00:03:39.049 --> 00:03:44.739
as either as an odd function or taking function
either as an even function ok
00:03:44.739 --> 00:03:49.829
so in the first case it is called as even
periodic extension of f x while in the second
00:03:49.829 --> 00:03:57.419
case it is called an odd periodic extension
of the function f x if we do an even periodic
00:03:57.419 --> 00:04:01.060
extension of f x which is the first one which
is the first case it is the even extension
00:04:01.060 --> 00:04:07.730
of the function f x then f x is an even function
in minus l to plus l of course it is an even
00:04:07.730 --> 00:04:13.620
function then f x has fourier cosine series
that we already know that if function is an
00:04:13.620 --> 00:04:20.400
even function ok we extend the function as
an even periodic function then f x will be
00:04:20.400 --> 00:04:26.110
an even function and in even function we only
have a cosine series cosine terms in the fourier
00:04:26.110 --> 00:04:30.410
series expansion
now if we do an odd periodic extension of
00:04:30.410 --> 00:04:38.090
the f x then f x is an odd function in minus
l to plus l and therefore f x has fourier
00:04:38.090 --> 00:04:46.020
sine series ok suppose we extend the function
as an odd periodic function so this f x will
00:04:46.020 --> 00:04:53.460
be an odd function and we know that if function
is an odd function then it contains only sine
00:04:53.460 --> 00:05:00.130
terms the fourier series expansion of that
function will contain only sine terms ok if
00:05:00.130 --> 00:05:05.730
a function is defined on a half interval from
zero to l then we can obtain a fourier cosine
00:05:05.730 --> 00:05:13.120
or fourier sine series expansion by suitable
periodic extensions depending on the problem
00:05:13.120 --> 00:05:20.520
ok if you want a fourier cosine series expansion
that means we take an even periodic function
00:05:20.520 --> 00:05:28.410
or we or we take function as an even function
and if we want fourier sine series that means
00:05:28.410 --> 00:05:35.880
we will take as a odd extension odd periodic
extension of the function ok that depends
00:05:35.880 --> 00:05:42.810
on the problem
so suppose you want fourier sine series ok
00:05:42.810 --> 00:05:49.830
so we already know this result that if you
want fourier sine series now function is given
00:05:49.830 --> 00:05:57.300
in zero to l ok and you want an odd extension
of the function odd extension means only sine
00:05:57.300 --> 00:06:02.700
terms ok so what will be the fourier series
of that that function the fourier series of
00:06:02.700 --> 00:06:08.490
that function will be nothing but uh it will
be given by summation n from one to infinity
00:06:08.490 --> 00:06:23.090
b n sin n pi x by l and where b n is nothing
but two upon l integral zero to l f x sin
00:06:23.090 --> 00:06:36.560
n pi x by l into d x ok so this is an odd
extension of the function ok uh now if we
00:06:36.560 --> 00:06:42.530
want fourier cosine series expansion or the
function defined on a half range from zero
00:06:42.530 --> 00:06:50.950
to l so that will contain only cosine terms
so that will be given by that will be given
00:06:50.950 --> 00:07:03.920
by a naught by two plus summation a n cos
n pi x by l n varying from one to infinity
00:07:03.920 --> 00:07:12.060
and here a naught will be nothing but two
upon l integral integral zero to l f x d x
00:07:12.060 --> 00:07:24.860
and a n will be given by two upon l integral
zero to l f x cos n pi x by l into d x so
00:07:24.860 --> 00:07:32.160
this will be the even extension of the function
function is given from zero to l and we defined
00:07:32.160 --> 00:07:38.710
an even extension of a function as a cosine
terms in the form of cosine terms
00:07:38.710 --> 00:07:45.710
now let us try to solve this problem uh obtain
cosine and sine series for f x equal to x
00:07:45.710 --> 00:07:51.460
in the interval from zero to pi and hence
show this result ok so first of all function
00:07:51.460 --> 00:07:58.330
is given function is x f x equal to x so a
very simple function f x equal to x interval
00:07:58.330 --> 00:08:09.250
is zero to zero to pi now suppose you want
cosine series expansion of this function cosine
00:08:09.250 --> 00:08:16.630
series expansion means we want to extend this
function as an even function we want to extend
00:08:16.630 --> 00:08:22.670
this function as an even function so this
function is like this it is from zero to pi
00:08:22.670 --> 00:08:32.089
it is uh it is x ok f x equal to x from zero
to pi and we want to extend the function as
00:08:32.089 --> 00:08:39.789
an even periodic function so from minus pi
also if it is an even function it must be
00:08:39.789 --> 00:08:46.370
symmetrical about x axis so we will draw this
function like this we extend this function
00:08:46.370 --> 00:08:54.710
like this and similarly here also like this
and similarly here also like this and similarly
00:08:54.710 --> 00:09:08.810
here also this will be an even extension of
this function or uh or now we can write this
00:09:08.810 --> 00:09:19.050
function in the cosine series form how uh
how f x will be nothing but a naught by two
00:09:19.050 --> 00:09:30.090
plus summation a n cos n pi x by pi ok n varying
from one to infinity and what will be a naught
00:09:30.090 --> 00:09:38.450
now it is two upon pi integral zero to pi
so here pi pi cancels out uh zero to pi f
00:09:38.450 --> 00:09:50.580
x d x f x is nothing but uh f x is nothing
but x d x so it is nothing but two upon pi
00:09:50.580 --> 00:10:00.490
it is x square upon two from zero to pi so
it is nothing but two upon pi pi square by
00:10:00.490 --> 00:10:12.960
two minus zero so it is nothing but pi so
a naught is nothing but pi ok and a n what
00:10:12.960 --> 00:10:24.300
will be a n a n will be nothing but two upon
pi integral zero to pi f x cos n x d x so
00:10:24.300 --> 00:10:35.750
it is equal to two upon pi integral zero to
pi now what is f x f x is x so x cos n x d
00:10:35.750 --> 00:10:40.080
x
so now we will apply integration by parts
00:10:40.080 --> 00:10:48.130
to simplify this expression so this will be
nothing but two upon pi so first as it is
00:10:48.130 --> 00:10:56.890
integral of second sin n x upon n minus derivative
of this and integration of this so this is
00:10:56.890 --> 00:11:10.540
nothing but minus cos n x upon n square zero
to pi ok so that is further equal to two upon
00:11:10.540 --> 00:11:18.880
pi now the first term is zero when x is pi
or x is zero ok now second term is one upon
00:11:18.880 --> 00:11:26.339
n square times when x is pi it is minus one
k to the power n and when x is zero it is
00:11:26.339 --> 00:11:41.390
one so this would be this term ok so this
a n is equal to now when n is odd when n is
00:11:41.390 --> 00:11:49.210
odd so this is nothing but when n is odd it
is minus one minus one minus two minus two
00:11:49.210 --> 00:11:55.590
minus into two minus four upon pi into n square
so this will be nothing but this term when
00:11:55.590 --> 00:12:06.250
n is odd and when n is even it is nothing
but zero because minus k to the power even
00:12:06.250 --> 00:12:11.050
is one one minus one is zero ok so n is defined
like this
00:12:11.050 --> 00:12:19.170
so what will be the cosine series of this
function f x so a naught is pi ok a naught
00:12:19.170 --> 00:12:29.830
is pi so it will be nothing but pi by two
plus summation n from one to infinity it is
00:12:29.830 --> 00:12:41.290
two upon n square pi minus one k to the power
n minus one into cos n x which is further
00:12:41.290 --> 00:12:51.029
equal to pi by two plus two upon pi can come
out and when n is odd it is minus four upon
00:12:51.029 --> 00:12:59.600
it is i mean minus two will come out so when
when n is odd it is minus four upon pi into
00:12:59.600 --> 00:13:09.660
one by n square that is cos one upon one square
plus cos three x upon three square plus cos
00:13:09.660 --> 00:13:19.910
five x into five upon five square and so on
and when n is even it is zero ok so this will
00:13:19.910 --> 00:13:32.690
be the expansion of x f x is x ok so this
is an even extension of the function x
00:13:32.690 --> 00:13:41.230
so x is defined in only zero to pi we extend
this function take assuming the function as
00:13:41.230 --> 00:13:47.840
an even function ok and that will contain
only cosine terms these are the cosine series
00:13:47.840 --> 00:13:55.589
expansion of function f x equal to x now to
reduce this expression one is one upon one
00:13:55.589 --> 00:14:02.800
square one upon three square one upon five
square and so on with all plus signs ok we
00:14:02.800 --> 00:14:10.300
can simply substitute x equal to zero so put
x equal to zero both sides so when you put
00:14:10.300 --> 00:14:19.940
x equal to zero it is pi by two plus or minus
minus four upon pi it is one upon one square
00:14:19.940 --> 00:14:27.790
plus one upon three square plus one upon five
square and so on so that implies one upon
00:14:27.790 --> 00:14:34.890
one square plus one upon three square plus
one upon five square is simply equal to pi
00:14:34.890 --> 00:14:44.300
square by eight ok so that we have proved
this result also
00:14:44.300 --> 00:14:52.920
now to obtain fourier sine series of this
function f x equal to x so how you will obtain
00:14:52.920 --> 00:14:59.700
that the first part is over of this problem
now we want to obtain fourier sine series
00:14:59.700 --> 00:15:06.709
of this function that means we have to extend
the function f x equal to x as an odd function
00:15:06.709 --> 00:15:15.060
ok then only we can obtain fourier uh sine
series that is the uh series containing sine
00:15:15.060 --> 00:15:23.120
terms only ok so how can we do that we have
the function f x equal to x here ok we have
00:15:23.120 --> 00:15:30.650
the function f x equal to x here is y equal
to x ok and we have to extend this function
00:15:30.650 --> 00:15:37.190
as an odd function to obtain the fourier sine
series of this function f x equal to x so
00:15:37.190 --> 00:15:46.100
it must be like this and assuming function
as periodic function we can extend this function
00:15:46.100 --> 00:15:54.850
as an odd function ok so what will be the
fourier sine series of this function it will
00:15:54.850 --> 00:16:08.980
be summation n from one to infinity b n sin
n x ok where b n is nothing but two upon pi
00:16:08.980 --> 00:16:21.180
integral zero to pi f x sin n x d x ok so
this is nothing but is equal to two upon pi
00:16:21.180 --> 00:16:32.580
integral zero to pi f x is x sin n x d x so
this is nothing but two upon pi integral of
00:16:32.580 --> 00:16:41.089
this is x into minus cos n x upon n minus
derivative of this and integration of this
00:16:41.089 --> 00:16:48.370
will be minus sin n x upon n square from zero
to pi
00:16:48.370 --> 00:16:54.440
so after integration by parts we obtain this
thing so this is nothing but it is two upon
00:16:54.440 --> 00:17:02.130
pi when you put x equal to pi here or zero
here this term is zero ok and here when you
00:17:02.130 --> 00:17:07.970
put x equal to zero it is zero so only term
is left here so negative will come here so
00:17:07.970 --> 00:17:15.720
it is pi into minus one k power n upon n ok
so pi pi cancel out it is nothing but minus
00:17:15.720 --> 00:17:25.870
one k to the power n plus n plus one upon
n so this will be two into ok two into minus
00:17:25.870 --> 00:17:32.840
two n k to the power n plus one upon n so
this will be b n so what will be f x f x will
00:17:32.840 --> 00:17:40.450
be nothing but summation n from one to infinity
b n b n is two into minus one k power n plus
00:17:40.450 --> 00:17:51.780
one upon n into sin n x ok so this is nothing
but two when n is one it is sin x upon one
00:17:51.780 --> 00:18:05.490
minus sin two x upon two plus sin three x
upon three and so on f x is x so this will
00:18:05.490 --> 00:18:14.360
be the odd extension of the sine series expression
of the function f x in terms of sine so this
00:18:14.360 --> 00:18:20.049
will be the final solution of this problem
ok
00:18:20.049 --> 00:18:28.340
so now let us try to solve one more problem
of this function which is we find as an this
00:18:28.340 --> 00:18:36.980
function is also given from zero to one ok
and that is in half range and we want function
00:18:36.980 --> 00:18:43.870
to expand as an fourier sine series that means
we want an odd extension of this function
00:18:43.870 --> 00:18:51.430
ok so how we can expand this as an odd function
so again for the odd expansion that is we
00:18:51.430 --> 00:18:57.720
want the fourier sine series of this function
eta f x will be given by this expression so
00:18:57.720 --> 00:19:05.280
what will be b n b n will be nothing but two
upon l now here l is one ok in this problem
00:19:05.280 --> 00:19:19.570
l is one zero to one f x and here also it
is sin pi x upon one because l is one ok f
00:19:19.570 --> 00:19:31.250
x sin n pi x upon one into d x
now we have to split the function from zero
00:19:31.250 --> 00:19:40.730
to half it is one by four minus x from zero
to half ok from zero to half ok sorry zero
00:19:40.730 --> 00:19:57.290
to half it is it is one by four minus x into
sin n pi x d x plus half to one it is x minus
00:19:57.290 --> 00:20:12.960
three by four sin n pi x into d x and whole
two times ok so we can simplify this to find
00:20:12.960 --> 00:20:23.030
out the sin series expansion of this function
ok so how we can simplify simplification is
00:20:23.030 --> 00:20:29.540
simple we have to apply by parts only two
times it is nothing but one by four minus
00:20:29.540 --> 00:20:46.130
x into minus of cos of n pi x upon upon n
pi ok and minus derivative of this which is
00:20:46.130 --> 00:20:55.090
nothing but minus one and integration of this
which is nothing but minus sin n pi x upon
00:20:55.090 --> 00:21:07.100
n square pi square ok and that whole from
zero to one by two ok i simply apply integration
00:21:07.100 --> 00:21:12.960
by parts ok first as it is integration of
second minus derivative of this and integration
00:21:12.960 --> 00:21:20.020
of this which is nothing but this again for
the second part plus two times again two is
00:21:20.020 --> 00:21:25.630
here ok for second part it is x minus three
by four that you will easily solve by your
00:21:25.630 --> 00:21:31.000
own you can easily apply integration by parts
and simplify to find the fourier series expansion
00:21:31.000 --> 00:21:42.780
of this function ok so it is nothing but minus
cos n pi x upon n pi and it is minus plus
00:21:42.780 --> 00:21:55.230
one into minus sin n pi x upon n square pi
square and it is varying from half to one
00:21:55.230 --> 00:22:03.860
so so after simplification what we will get
two times ok so when x is half one by four
00:22:03.860 --> 00:22:11.500
minus half is minus one by four minus minus
plus so it is one by four into cos n pi remains
00:22:11.500 --> 00:22:23.510
same into cos n pi by two ok when x is half
here it is nothing but minus of one upon n
00:22:23.510 --> 00:22:34.150
square pi square and when n is half it is
sin n pi by two ok when x is zero it is zero
00:22:34.150 --> 00:22:42.850
when x is zero it it is one one upon one by
four so it is minus minus plus one by four
00:22:42.850 --> 00:22:52.750
into n pi ok
now for this term ok when x is one it is one
00:22:52.750 --> 00:23:00.360
minus three by four that is one by four ok
negative is here so minus one by four and
00:23:00.360 --> 00:23:10.910
pi into minus one k power n and when x is
one here it is zero now when x is half it
00:23:10.910 --> 00:23:18.290
is half minus three by four half minus three
by four is minus one by four ok it is minus
00:23:18.290 --> 00:23:28.290
one by four minus minus plus one by four ok
and minus is again here so it is minus and
00:23:28.290 --> 00:23:39.890
then it is cos n pi by two upon n pi and when
n is half here it is nothing but minus minus
00:23:39.890 --> 00:23:47.100
plus and minus again so it is sin n pi by
two upon n square pi square so you can simplify
00:23:47.100 --> 00:23:56.480
this entire expression ok to find out the
uh value of b n ok and that b n you can substitute
00:23:56.480 --> 00:24:02.350
here to find out the fourier series expansion
fourier series expansion of this function
00:24:02.350 --> 00:24:11.100
f x ok
now on the in the same lines uh we can also
00:24:11.100 --> 00:24:16.270
find out the fourier series expansion of x
sin x as a cosine series cosine series means
00:24:16.270 --> 00:24:23.320
you want an even extension of this function
so that means for this function x x sin x
00:24:23.320 --> 00:24:30.419
you will simply find a naught and a n and
a naught will be given by a naught will be
00:24:30.419 --> 00:24:40.549
given by so for this problem a naught will
be given by uh two upon pi integral zero to
00:24:40.549 --> 00:24:53.160
pi x sin x d x and a n will be given by two
upon pi integral zero to pi it will be x sin
00:24:53.160 --> 00:25:08.610
x into into cos n x d x ok so this is how
we can find a naught and a n in this expression
00:25:08.610 --> 00:25:16.260
and finally for some x we can also find out
the series expansion of this function ok so
00:25:16.260 --> 00:25:24.740
uh thank you uh for this lecture in the next
lecture we will study an identity and some
00:25:24.740 --> 00:25:30.280
problems based on that and also complex form
of fourier series expansion ok
00:25:30.280 --> 00:25:30.590
thank you