WEBVTT
Kind: captions
Language: en
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so welcome to the lecture series on the mathematical
methods and its applications so we have discussed
00:00:25.050 --> 00:00:29.650
in the last two lectures basically we have
discussed fourier series and its convergence
00:00:29.650 --> 00:00:37.250
that how we can expand a function f x periodic
function f x in fact uh in the series form
00:00:37.250 --> 00:00:42.430
and the series of trigonometric functions
of sine and cosine ok that we have seen and
00:00:42.430 --> 00:00:49.050
when a series when a x or function is continuous
and having first and second derivatives are
00:00:49.050 --> 00:00:56.650
continuous then uh function f x is convergent
to that series of sine and cosine not even
00:00:56.650 --> 00:01:01.559
continuity if we have piecewise continuous
function piecewise periodic continuous function
00:01:01.559 --> 00:01:09.700
in fact then also that f x will converge to
the infinite series of sine and cosine uh
00:01:09.700 --> 00:01:14.810
if left and right hand derivatives exist at
each point that we have already seen
00:01:14.810 --> 00:01:21.610
now we will see fourier series of even and
odd functions ok we already know what even
00:01:21.610 --> 00:01:26.800
function is even function means f of minus
x equals to f x that function is called an
00:01:26.800 --> 00:01:33.660
even function and if f of minus x is minus
of f x that function is called as odd function
00:01:33.660 --> 00:01:38.700
we already know that mod x x square e k to
the power minus x square cos x etcetera are
00:01:38.700 --> 00:01:44.789
all even function because if we replace x
by minus x in these functions then the functions
00:01:44.789 --> 00:01:50.740
remain as it is that is f of minus x is same
as f x then so therefore these functions as
00:01:50.740 --> 00:01:56.849
even functions however if you see x cube x
sin x or minus cos x then these functions
00:01:56.849 --> 00:02:03.390
odd functions because if we replace x by minus
x then the function will be equal to minus
00:02:03.390 --> 00:02:09.390
of f x that means f of minus x is minus of
f x therefore these functions are odd functions
00:02:09.390 --> 00:02:16.310
so we have functions which are neither even
nor odd also like like x minus x square which
00:02:16.310 --> 00:02:24.459
is neither even nor odd ok now we already
know this result that if f is an even function
00:02:24.459 --> 00:02:30.130
then integration minus l to plus l f x d x
is nothing but two times zero to l f x d x
00:02:30.130 --> 00:02:36.440
that we already know and if f is an odd function
then minus l to l f x d x is simply zero this
00:02:36.440 --> 00:02:41.540
is by the properties of definite integral
we already know this now come to fourier series
00:02:41.540 --> 00:02:48.030
representation ok now we already know that
fourier series representation of any f x is
00:02:48.030 --> 00:02:58.790
given by a naught upon two plus summation
n from one to infinity a n cos n pi x by l
00:02:58.790 --> 00:03:07.129
plus summation n from one to infinity b n
sin n pi x by l
00:03:07.129 --> 00:03:13.680
so this is when interval is minus l to plus
l and the function is periodic with period
00:03:13.680 --> 00:03:20.739
two l ok now here a naught is given by uh
that is one by l integration minus l to plus
00:03:20.739 --> 00:03:32.319
l f x d x a n is given by one by l integral
minus l to plus l f x cos n pi x by l into
00:03:32.319 --> 00:03:45.129
d x and b n is given by one by l integral
minus l to plus l f x sin n pi x by l into
00:03:45.129 --> 00:03:51.810
d x now suppose function which is given to
us whose fourier expansion is to find out
00:03:51.810 --> 00:04:00.090
is an even function ok so if it is even function
then minus l to plus l f x d x will be nothing
00:04:00.090 --> 00:04:08.700
but two times by the property of definite
integral
00:04:08.700 --> 00:04:15.500
and the cos cos is an even function it is
cosine ok c o s it is cosine now cosine is
00:04:15.500 --> 00:04:21.780
an even function and f x is even function
so even into even is an even function ok so
00:04:21.780 --> 00:04:31.330
minus l to plus l it will again gives two
upon l integral zero to l f x into cos n pi
00:04:31.330 --> 00:04:38.460
x by l into d x
now if we see this function f x is an even
00:04:38.460 --> 00:04:46.610
function and sine is an odd function even
into odd is odd so this value will be zero
00:04:46.610 --> 00:04:52.419
so hence if we have an even function so we
have only cosine terms in the fourier series
00:04:52.419 --> 00:04:58.300
expansion ok because b n is zero b n is zero
means no sine terms only cosine terms will
00:04:58.300 --> 00:05:05.210
be there ok so if function is an even function
we have the fourier series representation
00:05:05.210 --> 00:05:12.710
of that function will contain only cosine
terms ok now suppose now suppose we want to
00:05:12.710 --> 00:05:23.509
expand an odd periodic function ok if it is
odd suppose it is odd ok now suppose function
00:05:23.509 --> 00:05:34.080
is odd if it is odd minus l to plus l this
will be zero and f x is odd cos is even odd
00:05:34.080 --> 00:05:43.410
into even is again odd it will be zero and
this is odd sine is also odd odd into odd
00:05:43.410 --> 00:05:48.620
is even this is an even function so when it
is even this will be nothing but two upon
00:05:48.620 --> 00:06:00.639
l times zero to l f x sin n pi x by l that
means that if we have an odd function then
00:06:00.639 --> 00:06:05.599
there will be no cosine term in that series
expansion fourier series expansion
00:06:05.599 --> 00:06:11.380
we will be having only sine terms because
there is only b n a n a naught and n all are
00:06:11.380 --> 00:06:20.570
zero ok so that is how one can find even and
odd extension odd i mean even and odd fourier
00:06:20.570 --> 00:06:28.840
series expansion for even and odd functions
ok now let us solve two examples on this suppose
00:06:28.840 --> 00:06:34.180
this function is given to us ok now what is
this function what is this function this function
00:06:34.180 --> 00:06:43.490
is f x equal to minus x plus one when minus
pi less than equals to x less than equal to
00:06:43.490 --> 00:06:52.710
zero ok and x plus one when zero less than
equals to x less than equals to pi of course
00:06:52.710 --> 00:06:58.379
function is periodic ok with period two pi
so we have to first check whether function
00:06:58.379 --> 00:07:05.909
is even and odd ok so to check whether function
is even and odd we will find f of minus x
00:07:05.909 --> 00:07:11.199
if it is equal to f x so it will be an even
function and if it is equal to minus of f
00:07:11.199 --> 00:07:18.789
x then we say it is an odd function ok
so you find f of minus x so you replace x
00:07:18.789 --> 00:07:23.920
by minus x so it is x plus one when minus
x less than equal to zero greater than equal
00:07:23.920 --> 00:07:30.419
to minus pi and minus x plus one when minus
x less than equal to pi greater than equal
00:07:30.419 --> 00:07:35.590
to zero so this will be nothing but is equals
to x plus one when x is less than greater
00:07:35.590 --> 00:07:42.259
than equal to zero less than equals to pi
and it is minus x plus one when x is greater
00:07:42.259 --> 00:07:48.650
than equals to minus pi and less than equal
to zero so from minus pi to zero it is minus
00:07:48.650 --> 00:07:53.639
x plus one which is same as this function
and from zero to pi it is x plus one which
00:07:53.639 --> 00:08:01.720
is same which is same x plus one over here
that means it is nothing but f of x so hence
00:08:01.720 --> 00:08:09.699
it's an even function ok so one can also see
graphically because if it is an even function
00:08:09.699 --> 00:08:15.939
then it is symmetrical about it is symmetrical
about y axis so you can just plot the function
00:08:15.939 --> 00:08:19.879
and you can see the symmetry if it is symmetrical
about y axis then you can simply say it is
00:08:19.879 --> 00:08:24.150
an even function ok
now you have to find the fourier series expansion
00:08:24.150 --> 00:08:33.229
of this function now fourier series representation
can be find out because its an even function
00:08:33.229 --> 00:08:39.349
which we have already seen so it will contain
only cosine terms ok no sine terms that is
00:08:39.349 --> 00:08:45.160
b n will be zero that we have already seen
so what will be a naught this we have already
00:08:45.160 --> 00:08:52.600
seen it is two upon pi zero to pi f x d x
so that will be nothing but two upon pi two
00:08:52.600 --> 00:09:00.800
upon l that is zero to pi ok l is pi here
so it is two upon pi x plus one d x and this
00:09:00.800 --> 00:09:08.230
is nothing but two upon pi it is x square
by two plus x from zero to pi and which is
00:09:08.230 --> 00:09:17.660
nothing but zero to pi it is pi square by
two plus pi which is equals to pi plus two
00:09:17.660 --> 00:09:24.080
because two two cancels out it is pi and it
is two yeah it is pi plus two ok
00:09:24.080 --> 00:09:32.641
now if you compute a n a n is nothing but
two upon pi integral zero to pi f x cos n
00:09:32.641 --> 00:09:40.970
x d x it is again equal to two upon pi integral
zero to pi what is f x f x is x plus one into
00:09:40.970 --> 00:09:50.090
cos n x d x and when you integrate it it is
two upon pi it is first as it is integral
00:09:50.090 --> 00:09:57.490
of second is sin n x upon n we will apply
integration by parts and derivative of this
00:09:57.490 --> 00:10:11.020
is one integration of this is minus cos n
x upon n square from zero to pi ok and this
00:10:11.020 --> 00:10:18.060
is nothing but two upon pi now it is zero
when x is pi or zero
00:10:18.060 --> 00:10:27.300
now here it is minus minus plus one by n square
it is minus one k power n minus one ok so
00:10:27.300 --> 00:10:31.480
this will be a n and this will be a naught
b n is of course zero because it is an even
00:10:31.480 --> 00:10:37.240
function so what will be the fourier sine
fourier representation of this function so
00:10:37.240 --> 00:10:48.630
a naught is pi plus two so f x will be nothing
but pi plus two a naught by two ok a naught
00:10:48.630 --> 00:11:01.120
by two plus summation n varying from one to
infinity a n a n is two upon pi n square minus
00:11:01.120 --> 00:11:12.660
one k power n minus one into cos n x ok this
term come here now so this will be fourier
00:11:12.660 --> 00:11:17.060
representation of this function now we have
determined the value of this series one upon
00:11:17.060 --> 00:11:21.230
one square plus one upon three square plus
one upon five square and so on so what the
00:11:21.230 --> 00:11:28.110
series is basically it is pi by two plus one
plus two upon pi
00:11:28.110 --> 00:11:38.510
now when you open this summation when n is
one it is minus two it is minus two cos x
00:11:38.510 --> 00:11:45.270
upon one square now when n is two minus one
k to the power two is one one minus one is
00:11:45.270 --> 00:11:54.520
zero ok now when n is three it is again minus
two minus two cos three x upon three square
00:11:54.520 --> 00:12:05.490
again when n is four it is zero minus two
cos five x upon five square and so on so it
00:12:05.490 --> 00:12:13.540
only contain odd terms i mean one three five
like this ok so and we have we have this series
00:12:13.540 --> 00:12:19.420
ok now we want value of this series so we
want all these terms to be one so substitute
00:12:19.420 --> 00:12:28.620
x equal to zero so put x equal to zero so
it is f zero will be equal to pi plus two
00:12:28.620 --> 00:12:36.100
plus one minus four upon pi times one upon
one square plus one upon three square plus
00:12:36.100 --> 00:12:42.950
one upon five square and so on
now what is f zero is the function continuous
00:12:42.950 --> 00:12:48.320
at zero yes it is continuous at zero because
from both the ends value is one we can easily
00:12:48.320 --> 00:12:56.520
check so the value of this will be one only
ok if it is not continuous then we will find
00:12:56.520 --> 00:13:03.990
in the same way like half of f zero plus plus
f zero minus ok if it is not continuous at
00:13:03.990 --> 00:13:11.210
zero here it is continuous at zero so it will
be one only ok it is pi by two plus one minus
00:13:11.210 --> 00:13:21.880
four upon pi times this expression so what
this value is one one cancel out this is minus
00:13:21.880 --> 00:13:31.680
pi by two is equals to minus four by pi times
this expression so hence this value is nothing
00:13:31.680 --> 00:13:39.600
but pi square by eight this value is from
here we can say that this value is nothing
00:13:39.600 --> 00:13:56.630
but pi square by eight so hence we got this
value ok because it's an even function calculation
00:13:56.630 --> 00:14:01.860
become easy otherwise we have to split the
integral from minus pi to zero and zero to
00:14:01.860 --> 00:14:08.480
pi ok
now see this problem now f x is minus k when
00:14:08.480 --> 00:14:14.700
between x lying between minus pi to zero and
k when it is lying between zero to pi first
00:14:14.700 --> 00:14:19.430
let us check whether it is even function odd
function or neither so we have to verify this
00:14:19.430 --> 00:14:30.570
first so let us see it is minus k when x lying
between minus pi to zero and it is k when
00:14:30.570 --> 00:14:38.060
x lying between zero to pi and it is a periodic
function with period two pi ok so first we
00:14:38.060 --> 00:14:43.550
will find first we will see whether it is
even function odd function so again we will
00:14:43.550 --> 00:14:49.200
find f of minus x if it is equal to f x it
is an even function if it is equal to f of
00:14:49.200 --> 00:14:54.730
minus of f x it means it is an odd function
so you replace x by minus x so it will be
00:14:54.730 --> 00:15:02.240
minus k when minus x lying between zero and
minus pi and it is k when minus x lying pi
00:15:02.240 --> 00:15:11.031
and zero so it is nothing but minus k when
x is greater than zero less than pi and it
00:15:11.031 --> 00:15:17.080
is k when x is less than minus pi greater
than minus pi and less than zero
00:15:17.080 --> 00:15:24.410
so if we see this and this function from minus
pi to zero it is k from minus pi to zero it
00:15:24.410 --> 00:15:30.570
is minus k from zero to pi it is minus k and
from zero to pi it is k that is this is nothing
00:15:30.570 --> 00:15:39.430
but minus of f x so hence this function is
an odd function now in odd function so it
00:15:39.430 --> 00:15:47.750
will contain only sine terms that is a naught
and a n are zero we already know this so this
00:15:47.750 --> 00:15:52.620
will only contain fourier series expansion
of this function it will only contain sine
00:15:52.620 --> 00:16:00.790
terms so how you find b n so only b n will
exist so b n will be nothing but two upon
00:16:00.790 --> 00:16:13.380
pi integral zero to pi k sin x sin n x d x
f x f x is k ok and it is nothing but two
00:16:13.380 --> 00:16:23.640
upon pi k is outside integral of sine is minus
cos n x upon n from zero to pi minus will
00:16:23.640 --> 00:16:29.580
come outside two k upon pi and it is nothing
but when you integrate this what you will
00:16:29.580 --> 00:16:40.050
get find the apply the limits minus one power
n minus one so which is which is equals to
00:16:40.050 --> 00:16:48.880
this is equal to when n is odd when n is odd
it is minus one minus one minus two minus
00:16:48.880 --> 00:16:58.860
two into minus two is four that is four k
upon pi ok so n is also there ok n is also
00:16:58.860 --> 00:17:07.010
there it is n it will come here ok
because n is also there now when n is odd
00:17:07.010 --> 00:17:16.880
when n is even it is minus plus one minus
one zero so when n is even it is zero ok so
00:17:16.880 --> 00:17:21.130
in this way we can define b n now what will
be the fourier series of this function then
00:17:21.130 --> 00:17:27.699
fourier series of this function will be nothing
but summation b n with b n is b n is this
00:17:27.699 --> 00:17:38.000
term it is minus two k upon pi n minus minus
one k power n minus one sin n x and when we
00:17:38.000 --> 00:17:46.159
open this when n is odd it is and when n is
even it is zero so you can substitute values
00:17:46.159 --> 00:17:58.720
of n when n is when n is one so it is four
k upon four k upon pi will come outside it
00:17:58.720 --> 00:18:08.980
is sin one sin x upon one ok plus sin three
x upon three plus sin five x upon five and
00:18:08.980 --> 00:18:19.080
so on because when n is even it is zero we
have the existence of b n only when n is odd
00:18:19.080 --> 00:18:23.220
ok
so we have this series expansion now we want
00:18:23.220 --> 00:18:30.419
to compute this expression summation n from
one to infinity minus one k power n plus one
00:18:30.419 --> 00:18:37.820
two n minus one equal to pi by four this we
have to show ok so because it is minus one
00:18:37.820 --> 00:18:44.480
k power n plus one that means we have alternate
negative positive sign in the series of this
00:18:44.480 --> 00:18:55.289
so to have alternate sign of plus minus sign
over here so put x equals to pi by two so
00:18:55.289 --> 00:19:03.580
it will be f pi by two will be equals to four
k upon pi sin pi by two is one one by one
00:19:03.580 --> 00:19:08.860
sin three pi by two is minus one so minus
one by three sin five pi by two is plus one
00:19:08.860 --> 00:19:17.419
and so on and what is sin pi by two f pi by
two f pi by two comes from here it is k so
00:19:17.419 --> 00:19:26.690
this expression will be nothing but k is equals
to four k upon pi one minus one by three plus
00:19:26.690 --> 00:19:34.659
one by five and so on so hence this implies
one minus one by three plus one by five minus
00:19:34.659 --> 00:19:41.890
one by seven and so on will be nothing but
pi by four so this you have to derive yeah
00:19:41.890 --> 00:19:52.669
ok so it is pi by four it comes pi by four
so hence we have proved the result ok
00:19:52.669 --> 00:19:59.759
so this is simple problem fourier series expansion
four minus x square is an even function one
00:19:59.759 --> 00:20:10.240
can easily see and it has a period four so
we have only cosine terms in the series so
00:20:10.240 --> 00:20:16.360
how can we how can we find the fourier series
expansion of this function you using the same
00:20:16.360 --> 00:20:27.730
technique same expressions so f x is four
minus x square x lying between minus two to
00:20:27.730 --> 00:20:35.749
plus two and it is a periodic function with
period four ok so again we know that it will
00:20:35.749 --> 00:20:42.960
contain only cosine terms because it is an
even function so what will be a naught a naught
00:20:42.960 --> 00:20:53.950
will be two upon two because period is two
zero to two f x d x so this will be nothing
00:20:53.950 --> 00:21:01.499
but four x minus x cube by three from zero
to two so this value we can compute it is
00:21:01.499 --> 00:21:09.559
eight minus eight by three that is nothing
but eight into two by three that is sixteen
00:21:09.559 --> 00:21:18.990
by three ok ok
now a n a n will be nothing but two by two
00:21:18.990 --> 00:21:32.580
integral zero to two four minus x square cos
n pi x by two into d x because l is two ok
00:21:32.580 --> 00:21:40.399
so we can easily integrate this there is no
problem in this it is four minus x square
00:21:40.399 --> 00:21:50.940
integration of this will be sin n pi x by
two upon n pi by two minus derivative of this
00:21:50.940 --> 00:21:59.690
and integration of this will be minus cos
n pi x by two upon n square pi square by four
00:21:59.690 --> 00:22:08.850
plus derivative of this and integration of
this will be nothing but minus sin n pi x
00:22:08.850 --> 00:22:20.899
by two upon n cube pi cube by eight and whole
multiplied by zero to two limit from zero
00:22:20.899 --> 00:22:25.190
to two
now when you take when you take limits two
00:22:25.190 --> 00:22:33.090
here it is zero and when x is zero it is zero
so it is zero now here also when x is two
00:22:33.090 --> 00:22:39.139
it is n pi it is zero when it is zero it will
be zero so only this term will there ok so
00:22:39.139 --> 00:22:49.059
it is minus minus minus it is minus two comes
outside when x is two so it four into it is
00:22:49.059 --> 00:22:57.279
four upon n square pi square ok it will also
come outside four upon this so it is this
00:22:57.279 --> 00:23:07.220
will come outside two will also outside so
it is two into cos n pi and at zero it is
00:23:07.220 --> 00:23:15.360
zero so it is nothing but minus sixteen upon
n square pi square into minus one k power
00:23:15.360 --> 00:23:23.399
n ok it is minus sixteen upon this so so fourier
series expansion of this function will be
00:23:23.399 --> 00:23:30.370
nothing but sixteen upon three into two a
naught by two plus summation n from one to
00:23:30.370 --> 00:23:41.419
infinity sixteen upon n square pi square minus
one k to the power n plus one cos n x so this
00:23:41.419 --> 00:23:44.889
should be the fourier series this expression
ok so
00:23:44.889 --> 00:23:46.669
thank you