WEBVTT
Kind: captions
Language: en
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so welcome to the lecture series on mathematical
methods and its applications so we were discussing
00:00:24.609 --> 00:00:32.189
fourier series i told you that uh periodic
function f x if satisfies some properties
00:00:32.189 --> 00:00:37.640
like piecewise continuity and left and right
hand derivative exist at each point then f
00:00:37.640 --> 00:00:44.470
x can be expressed in terms of sine and cosine
uh series that is f x will be something a
00:00:44.470 --> 00:00:50.780
naught by two plus summation a n cos n pi
x by l plus summation b n sin n pi x by l
00:00:50.780 --> 00:00:55.149
that i already told you and we have also solved
some problems based on that
00:00:55.149 --> 00:01:02.109
now the next is complex form of fourier series
what is that and how it is important let us
00:01:02.109 --> 00:01:09.630
see ok now fourier series of a periodic function
f x of period two l is given by this expression
00:01:09.630 --> 00:01:16.020
this we already know this can be expressed
in complex form which sometimes makes calculation
00:01:16.020 --> 00:01:21.840
easier in the problems so why we need complex
forms because sometimes it make our calculation
00:01:21.840 --> 00:01:29.739
easier ok now how we can convert this form
into a complex form let us see ok so what
00:01:29.739 --> 00:01:37.039
f x is f x is nothing but f x is equal to
fourier series expansion is a naught upon
00:01:37.039 --> 00:01:52.319
two plus summation n varying from one to infinity
a n cos n pi x by l plus b n sin n pi x by
00:01:52.319 --> 00:02:00.300
l interval is we are taking interval from
minus l to plus l ok
00:02:00.300 --> 00:02:06.619
this interval we are taking and f x is a periodic
function eta these are fourier series expansion
00:02:06.619 --> 00:02:16.020
of the function f x now we know that uh e
k power iota theta is nothing but cos theta
00:02:16.020 --> 00:02:23.660
plus iota sin theta this we already know ok
and e k power minus iota theta is nothing
00:02:23.660 --> 00:02:30.849
but cos theta minus iota sin theta so when
we add these two we already know that cos
00:02:30.849 --> 00:02:37.250
theta is nothing but e k power iota theta
plus e k power minus iota theta upon two and
00:02:37.250 --> 00:02:44.340
sin theta is nothing but e k power iota theta
minus e k power minus iota theta upon two
00:02:44.340 --> 00:02:52.409
iota so these things we already know by the
complex numbers we already know that cos theta
00:02:52.409 --> 00:02:56.940
is nothing but e k power iota theta plus e
k power minus iota theta upon two and sin
00:02:56.940 --> 00:03:01.069
theta is given by this expression
so let us apply these things over here it
00:03:01.069 --> 00:03:09.249
is cos theta theta is nothing but n pi n pi
x by l and sin theta ok so let us apply these
00:03:09.249 --> 00:03:15.629
expressions over here so it is equal to a
naught upon two plus summation n varying from
00:03:15.629 --> 00:03:24.480
one to infinity a n so what is cos n pi x
by l by this expression it is nothing but
00:03:24.480 --> 00:03:38.639
e k power iota n pi x by l plus e k power
minus iota n pi x upon l upon two and plus
00:03:38.639 --> 00:03:47.579
b n into summation is over entire bracket
so b n is nothing but b n into sin theta sin
00:03:47.579 --> 00:03:57.000
theta is e k power iota theta is n pi x upon
x upon l minus e k power minus iota n pi x
00:03:57.000 --> 00:04:08.279
upon l upon two iota so this is this we can
do ok
00:04:08.279 --> 00:04:18.190
now this can be further written as a naught
upon two plus now e k power iota n pi x by
00:04:18.190 --> 00:04:23.280
l is here and here also let us combine these
two terms and again combine these two terms
00:04:23.280 --> 00:04:29.190
e k power minus iota n pi x by l here and
here combine these two terms so what we obtain
00:04:29.190 --> 00:04:39.780
we obtain summation n varying from one to
infinity it is a n plus b n upon iota whole
00:04:39.780 --> 00:04:54.800
divided by two ok and e k power iota n pi
x upon l so we also know that uh we also know
00:04:54.800 --> 00:05:00.930
that one upon iota is nothing but minus iota
because iota square is minus one so this one
00:05:00.930 --> 00:05:06.560
upon iota can be written as minus iota ok
eta so this is nothing but minus iota this
00:05:06.560 --> 00:05:19.590
we can easily write ok because of this property
plus again it is a n a n and again one upon
00:05:19.590 --> 00:05:28.840
iota is minus iota minus minus plus it is
plus iota b n upon two time e k power minus
00:05:28.840 --> 00:05:43.870
iota n x by n pi x by n pi x upon l ok so
these are the expressions which we obtained
00:05:43.870 --> 00:05:52.500
over here ok
now let us simplify it so this is suppose
00:05:52.500 --> 00:06:01.900
this a naught upon two is c naught ok plus
summation n varying from one to infinity this
00:06:01.900 --> 00:06:12.370
suppose this term is nothing but suppose c
n e k power iota n pi x upon l plus and suppose
00:06:12.370 --> 00:06:23.430
this term is c minus n it is simply a notation
c minus n e k power minus iota pi x upon l
00:06:23.430 --> 00:06:32.380
so what what are what is c n c naught is nothing
but a naught upon two c n is nothing but half
00:06:32.380 --> 00:06:48.620
of a n minus iota b n and c minus n is nothing
but one by two a n plus iota b n ok so one
00:06:48.620 --> 00:06:58.090
can easily see that if this f x is a real
function f x is a real function then these
00:06:58.090 --> 00:07:07.860
coefficients will be real then this c n
minus is nothing but c n bar one can easily
00:07:07.860 --> 00:07:18.080
this thing because its bar is this these are
the conjugate of each other ok if f is real
00:07:18.080 --> 00:07:26.409
this is if f is real ok
now this can be further written as c naught
00:07:26.409 --> 00:07:37.669
plus summation it is c n e k power iota n
pi x upon l and we can write because all the
00:07:37.669 --> 00:07:44.289
plus values are covered from here and all
the negative values are covered from here
00:07:44.289 --> 00:07:52.050
ok because instead of n we have minus n so
we can easily write this as n from minus infinity
00:07:52.050 --> 00:07:59.449
to plus infinity because n from one to infinity
is here n n from minus infinity to minus one
00:07:59.449 --> 00:08:09.060
is here ok and c zero will also disappear
because when n is zero n is zero it is nothing
00:08:09.060 --> 00:08:15.680
but c zero c zero will also comes here so
when we combine all these terms it is nothing
00:08:15.680 --> 00:08:23.240
but n varying from minus infinity to plus
infinity c n e k power n pi x by l so this
00:08:23.240 --> 00:08:29.050
is the complex form of the fourier series
so fourier series can be represented in this
00:08:29.050 --> 00:08:38.599
way or in this way ok both ways both are equivalent
ok this is a complex form of fourier series
00:08:38.599 --> 00:08:48.850
now what are c n what how you define c n ok
c n is given by c n is what c n is nothing
00:08:48.850 --> 00:09:00.290
but one by two a n minus iota b n it is equal
to one by two what is a n it is one by l integral
00:09:00.290 --> 00:09:15.389
minus l to plus l f x cos n pi x by l into
d x ok this is by definition of a n and minus
00:09:15.389 --> 00:09:29.040
iota times what is b n b n is one upon l integral
minus l to l f x sin n pi x upon l into d
00:09:29.040 --> 00:09:38.100
x so this entire expression can be rewrite
as one upon two l integral minus l to plus
00:09:38.100 --> 00:09:50.160
l you can take f x common so this is nothing
but cos n pi x upon l minus iota sin n pi
00:09:50.160 --> 00:09:58.240
x upon l and whole into d x so this can be
further written as one upon two l integral
00:09:58.240 --> 00:10:07.510
minus l to plus l f x e k power power minus
iota n pi x by l into d x
00:10:07.510 --> 00:10:17.380
so this will be c n where where c n is nothing
but this value and what is c minus n c minus
00:10:17.380 --> 00:10:22.810
n if you evaluate eta that will be nothing
but c minus n will be nothing but one by two
00:10:22.810 --> 00:10:34.630
a n plus iota b n using the same uh same steps
follow the same steps so this is nothing but
00:10:34.630 --> 00:10:41.470
when we simplify it it is nothing but one
by two l integral minus l to plus l f x e
00:10:41.470 --> 00:10:50.070
k power iota n pi x by l into d x because
in this we have only the positive sign here
00:10:50.070 --> 00:10:53.500
instead of this negative this is positive
this is positive this is positive this would
00:10:53.500 --> 00:11:03.550
be positive ok so c minus n will be nothing
but simply replace n by minus n in this expression
00:11:03.550 --> 00:11:09.449
in this expression simply replace n by minus
n ok and what will be c naught
00:11:09.449 --> 00:11:17.230
c naught is nothing but a naught upon two
and a naught is nothing but one by l integral
00:11:17.230 --> 00:11:25.670
minus l to plus l f x d x so it means when
you replace n by zero in this expression you
00:11:25.670 --> 00:11:34.440
get c naught so if you find c n so it contains
c minus n also when you replace n by minus
00:11:34.440 --> 00:11:42.540
n and it contains c naught also when you replace
n by zero ok so that means when we write the
00:11:42.540 --> 00:11:51.090
fourier complex form of fourier series which
is given by this expression f x is equals
00:11:51.090 --> 00:11:59.970
to summation n varying from minus infinity
to plus infinity c n e k power iota n pi x
00:11:59.970 --> 00:12:16.290
upon l and here c n will be nothing but one
by two l integral minus l to plus l f x iota
00:12:16.290 --> 00:12:28.100
n pi x by l into d x ok so that will be the
complex form of fourier series ok it contain
00:12:28.100 --> 00:12:32.759
negative c n also which is also covered here
it contains c naught also which is also covered
00:12:32.759 --> 00:12:36.769
here so we can easily write that the fourier
series complex form of fourier series is given
00:12:36.769 --> 00:12:45.240
by this expression where c n is given by this
expression ok so this is a complex form of
00:12:45.240 --> 00:12:50.959
fourier series
now let us solve few problems based on this
00:12:50.959 --> 00:13:01.009
now let us try these two problems based on
this now in the first problem f x is what
00:13:01.009 --> 00:13:12.019
f x is e k power minus x ok and the function
is defined from minus pi to plus pi of course
00:13:12.019 --> 00:13:18.769
function is periodic ok now we have to express
this function as a as the complex form of
00:13:18.769 --> 00:13:28.240
fourier series ok so how can we do that we
first find c n ok which is given by this expression
00:13:28.240 --> 00:13:33.509
and when we substitute c n over here so that
will be the complex form of the fourier series
00:13:33.509 --> 00:13:40.680
expression of this function f x ok
so what is c n c n will be given by one by
00:13:40.680 --> 00:13:47.750
two l here l is pi one by two pi integral
minus pi to plus pi f x is e k power minus
00:13:47.750 --> 00:13:59.310
x into e k power minus iota n pi x by pi into
d x so pi pi cancels out ok so this is nothing
00:13:59.310 --> 00:14:08.509
but one upon two pi integral minus pi to plus
pi e k power minus will be outside and one
00:14:08.509 --> 00:14:20.040
plus iota n into x into d x so this expression
we will be having so this is further can be
00:14:20.040 --> 00:14:25.540
written as one upon two pi now when we integrate
this it is nothing but e k power minus one
00:14:25.540 --> 00:14:36.850
plus iota n x upon minus one plus iota n and
x is varying from minus pi to plus pi so this
00:14:36.850 --> 00:14:44.750
will be further equal to one upon two pi you
can take this term outside so this is negative
00:14:44.750 --> 00:14:52.529
of one plus iota n upper limit minus lower
limit e k power minus one plus iota n into
00:14:52.529 --> 00:15:04.600
pi minus e k power minus minus plus one plus
iota n into pi ok this you can simplify very
00:15:04.600 --> 00:15:12.950
easily multiply this by its conjugate one
minus iota n multiply and divide by one minus
00:15:12.950 --> 00:15:22.029
iota n so we will get minus one minus iota
n upon two pi into one plus n square and here
00:15:22.029 --> 00:15:30.050
when you simplify so this is nothing but e
k power minus pi into e k power minus iota
00:15:30.050 --> 00:15:40.259
n pi minus e k power pi into e k power iota
n pi ok
00:15:40.259 --> 00:15:48.570
now what is e k power iota n pi e k power
iota n pi will be nothing but cos n pi plus
00:15:48.570 --> 00:15:59.329
iota sin n pi and it is minus one k power
n and it is sin n pi is zero and similarly
00:15:59.329 --> 00:16:05.769
when you find e k power minus iota n pi that
is nothing but the conjugate of e k power
00:16:05.769 --> 00:16:14.699
iota n pi so that will be remain minus one
k power n ok so from here what we obtained
00:16:14.699 --> 00:16:23.220
it is equal to minus one minus iota n upon
two pi one plus n square when you simplify
00:16:23.220 --> 00:16:29.620
this so this is also minus one k power n this
is also minus one k power n both will come
00:16:29.620 --> 00:16:34.740
out and what we will be having e k power minus
pi minus e k power pi and whole multiplied
00:16:34.740 --> 00:16:46.269
by minus one k power n so this negative you
can get inside and upon two so this will be
00:16:46.269 --> 00:16:53.759
nothing but when you simplify further so it
is nothing but one minus iota n into minus
00:16:53.759 --> 00:17:02.759
one k power n upon one plus n square which
we are obtaining from here into pi and this
00:17:02.759 --> 00:17:11.770
nothing but it is sine hyperbolic pi uh because
sine hyperbolic pi is e k power pi minus e
00:17:11.770 --> 00:17:19.140
k power minus pi upon two so that will be
c n ok
00:17:19.140 --> 00:17:23.839
so what will be the fourier series expression
of this function the fourier series expression
00:17:23.839 --> 00:17:30.670
of this function f x now will be nothing but
f x will be equal to summation n varying from
00:17:30.670 --> 00:17:41.210
minus infinity to plus infinity c n is what
c n is one minus iota n minus one k power
00:17:41.210 --> 00:18:03.400
n upon one plus n square into pi sin hyperbolic
pi into e k power iota n x so that will be
00:18:03.400 --> 00:18:10.650
the fourier series it is equal to e k power
x ok because we find the fourier series complex
00:18:10.650 --> 00:18:17.450
form of fourier series this function so function
is e k power minus x ok and that e k power
00:18:17.450 --> 00:18:23.471
minus x the complex form of this is nothing
but this expression so that is how we can
00:18:23.471 --> 00:18:30.039
find out the complex form of fourier series
of a function f x ok
00:18:30.039 --> 00:18:35.620
now let us solve second next problem so again
we have to write down the complex form of
00:18:35.620 --> 00:18:43.180
fourier series of this function now so what
is the how the function is defined now function
00:18:43.180 --> 00:18:53.520
is f x is equals to it is minus one when x
is varying from minus pi to zero and it is
00:18:53.520 --> 00:18:59.900
one when x is varying from zero to pi and
function is periodic with period two pi ok
00:18:59.900 --> 00:19:09.600
now again to find out the complex form of
this f x first we will find c n which is given
00:19:09.600 --> 00:19:16.460
by this expression ok and after finding c
n we will substitute it here so that will
00:19:16.460 --> 00:19:21.669
be the complex form of the fourier series
of this function f x
00:19:21.669 --> 00:19:29.340
so what is c n now c n we can find out it
is one upon two pi because l is pi minus pi
00:19:29.340 --> 00:19:41.650
to plus pi f x e k power minus iota n pi x
upon l into d x that is nothing but one upon
00:19:41.650 --> 00:19:48.179
two pi now you can split this function because
function from minus pi to zero is minus one
00:19:48.179 --> 00:19:57.419
and from zero to pi it is one and it is periodic
ok one branch in the negative side of x axis
00:19:57.419 --> 00:20:06.100
one branch in the positive side of x axis
ok so this is minus pi to zero minus one e
00:20:06.100 --> 00:20:17.080
k power minus iota n pi x upon l d x plus
zero to pi it is one into e k power minus
00:20:17.080 --> 00:20:28.130
iota n pi x upon l into d x
now this can be written as one upon two pi
00:20:28.130 --> 00:20:38.100
now you can integrate it minus e k power minus
iota n pi x upon l ok here l is pi ok here
00:20:38.100 --> 00:20:49.029
l is pi so this is pi eta pi pi cancels out
ok so you can eliminate this so this is this
00:20:49.029 --> 00:21:03.780
and this is this ok minus iota n x because
l is pi ok so this is this function now this
00:21:03.780 --> 00:21:13.419
is n x and divided by minus iota n from minus
pi to zero plus it is e k power minus iota
00:21:13.419 --> 00:21:23.220
n x upon minus iota n from zero to pi it is
further equal to now negative negative cancels
00:21:23.220 --> 00:21:29.270
out it is one upon two pi
now you apply upper limit minus lower limit
00:21:29.270 --> 00:21:37.279
the upper limit is uh one upon iota n will
be outside upper limit is one minus e k power
00:21:37.279 --> 00:21:45.970
e k power iota n pi ok and it is minus one
upon iota n will be outside and it is nothing
00:21:45.970 --> 00:21:55.029
but when it is pi it is e k power minus iota
n pi minus one now it is further equal to
00:21:55.029 --> 00:22:07.570
one upon two pi n into iota it is one minus
e k power e k power iota n pi minus e k power
00:22:07.570 --> 00:22:15.620
minus iota n pi minus minus plus so this is
nothing but when we simplify this so it is
00:22:15.620 --> 00:22:21.820
two it is two times of this two two canceled
out so it is nothing but one upon pi n iota
00:22:21.820 --> 00:22:38.279
and one minus e k power iota n pi ok ok it
is not two times ok so it is it is two into
00:22:38.279 --> 00:22:45.320
minus e k power minus iota n pi so two is
remaining here ok
00:22:45.320 --> 00:22:53.279
now e k power iota n pi is cos n pi plus iota
sin n pi which is minus one k power n and
00:22:53.279 --> 00:23:01.299
this value is again cos n pi this value is
what e k power minus iota n pi will be nothing
00:23:01.299 --> 00:23:09.930
but cos n pi minus iota sin n pi which is
nothing but minus one k power n so this is
00:23:09.930 --> 00:23:16.220
minus one k power n this is minus one k power
n so now it is two times two two cancels out
00:23:16.220 --> 00:23:25.240
and one upon iota is nothing but minus iota
so it is minus iota upon n pi and it is one
00:23:25.240 --> 00:23:33.070
minus minus one k power n so this will be
the final expression for final expression
00:23:33.070 --> 00:23:45.299
for c n ok you can see when it is when n is
even it is zero and when n is odd it is two
00:23:45.299 --> 00:23:52.640
it is inside bracket expression the expression
is at the bracket when n is even minus one
00:23:52.640 --> 00:23:58.000
k power even is one so one minus one is zero
when n is odd it is two
00:23:58.000 --> 00:24:03.720
so what will be the complex form of fourier
series for this function so for this function
00:24:03.720 --> 00:24:13.820
the complex form is given by f x will be equal
to uh it is summation n from minus infinity
00:24:13.820 --> 00:24:23.649
to plus infinity c n is what c n is minus
iota upon n pi one minus minus one k power
00:24:23.649 --> 00:24:38.039
n into e k power iota n pi x upon l so this
will be the complex form of this fourier series
00:24:38.039 --> 00:24:45.380
so hence whether the function is continuous
like the first example is continuous the second
00:24:45.380 --> 00:24:52.720
example is discontinuous at x equal to zero
so if we have functions uh that are continuous
00:24:52.720 --> 00:24:57.779
or discontinuous or i mean piecewise continuous
the second example is piecewise continuous
00:24:57.779 --> 00:25:04.140
so if we have such problems then problems
can be either converted into the sine or cosine
00:25:04.140 --> 00:25:11.360
terms as we have uh done before or in the
complex form of fourier series which is given
00:25:11.360 --> 00:25:17.900
by this expression where c n is given by this
expression ok here the benefit is we have
00:25:17.900 --> 00:25:24.149
to find only c n and there we have to find
a naught a n and b n the three coefficients
00:25:24.149 --> 00:25:31.409
ok so that that is how we can find out the
complex form of uh fourier series of any function
00:25:31.409 --> 00:25:34.679
f x so so
thank you