so welcome to the lecture series on mathematical methods and its applications so we were discussing
fourier series i told you that uh periodic function f x if satisfies some properties
like piecewise continuity and left and right hand derivative exist at each point then f
x can be expressed in terms of sine and cosine uh series that is f x will be something a
naught by two plus summation a n cos n pi x by l plus summation b n sin n pi x by l
that i already told you and we have also solved some problems based on that
now the next is complex form of fourier series what is that and how it is important let us
see ok now fourier series of a periodic function f x of period two l is given by this expression
this we already know this can be expressed in complex form which sometimes makes calculation
easier in the problems so why we need complex forms because sometimes it make our calculation
easier ok now how we can convert this form into a complex form let us see ok so what
f x is f x is nothing but f x is equal to fourier series expansion is a naught upon
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
l interval is we are taking interval from minus l to plus l ok
this interval we are taking and f x is a periodic function eta these are fourier series expansion
of the function f x now we know that uh e k power iota theta is nothing but cos theta
plus iota sin theta this we already know ok and e k power minus iota theta is nothing
but cos theta minus iota sin theta so when we add these two we already know that cos
theta is nothing but e k power iota theta plus e k power minus iota theta upon two and
sin theta is nothing but e k power iota theta minus e k power minus iota theta upon two
iota so these things we already know by the complex numbers we already know that cos theta
is nothing but e k power iota theta plus e k power minus iota theta upon two and sin
theta is given by this expression so let us apply these things over here it
is cos theta theta is nothing but n pi n pi x by l and sin theta ok so let us apply these
expressions over here so it is equal to a naught upon two plus summation n varying from
one to infinity a n so what is cos n pi x by l by this expression it is nothing but
e k power iota n pi x by l plus e k power minus iota n pi x upon l upon two and plus
b n into summation is over entire bracket so b n is nothing but b n into sin theta sin
theta is e k power iota theta is n pi x upon x upon l minus e k power minus iota n pi x
upon l upon two iota so this is this we can do ok
now this can be further written as a naught upon two plus now e k power iota n pi x by
l is here and here also let us combine these two terms and again combine these two terms
e k power minus iota n pi x by l here and here combine these two terms so what we obtain
we obtain summation n varying from one to infinity it is a n plus b n upon iota whole
divided by two ok and e k power iota n pi x upon l so we also know that uh we also know
that one upon iota is nothing but minus iota because iota square is minus one so this one
upon iota can be written as minus iota ok eta so this is nothing but minus iota this
we can easily write ok because of this property plus again it is a n a n and again one upon
iota is minus iota minus minus plus it is plus iota b n upon two time e k power minus
iota n x by n pi x by n pi x upon l ok so these are the expressions which we obtained
over here ok now let us simplify it so this is suppose
this a naught upon two is c naught ok plus summation n varying from one to infinity this
suppose this term is nothing but suppose c n e k power iota n pi x upon l plus and suppose
this term is c minus n it is simply a notation c minus n e k power minus iota pi x upon l
so what what are what is c n c naught is nothing but a naught upon two c n is nothing but half
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
can easily see that if this f x is a real function f x is a real function then these
coefficients will be real then this c n minus is nothing but c n bar one can easily
this thing because its bar is this these are the conjugate of each other ok if f is real
this is if f is real ok now this can be further written as c naught
plus summation it is c n e k power iota n pi x upon l and we can write because all the
plus values are covered from here and all the negative values are covered from here
ok because instead of n we have minus n so we can easily write this as n from minus infinity
to plus infinity because n from one to infinity is here n n from minus infinity to minus one
is here ok and c zero will also disappear because when n is zero n is zero it is nothing
but c zero c zero will also comes here so when we combine all these terms it is nothing
but n varying from minus infinity to plus infinity c n e k power n pi x by l so this
is the complex form of the fourier series so fourier series can be represented in this
way or in this way ok both ways both are equivalent ok this is a complex form of fourier series
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
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
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
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
x so this entire expression can be rewrite as one upon two l integral minus l to plus
l you can take f x common so this is nothing but cos n pi x upon l minus iota sin n pi
x upon l and whole into d x so this can be further written as one upon two l integral
minus l to plus l f x e k power power minus iota n pi x by l into d x
so this will be c n where where c n is nothing but this value and what is c minus n c minus
n if you evaluate eta that will be nothing but c minus n will be nothing but one by two
a n plus iota b n using the same uh same steps follow the same steps so this is nothing but
when we simplify it it is nothing but one by two l integral minus l to plus l f x e
k power iota n pi x by l into d x because in this we have only the positive sign here
instead of this negative this is positive this is positive this is positive this would
be positive ok so c minus n will be nothing but simply replace n by minus n in this expression
in this expression simply replace n by minus n ok and what will be c naught
c naught is nothing but a naught upon two and a naught is nothing but one by l integral
minus l to plus l f x d x so it means when you replace n by zero in this expression you
get c naught so if you find c n so it contains c minus n also when you replace n by minus
n and it contains c naught also when you replace n by zero ok so that means when we write the
fourier complex form of fourier series which is given by this expression f x is equals
to summation n varying from minus infinity to plus infinity c n e k power iota n pi x
upon l and here c n will be nothing but one by two l integral minus l to plus l f x iota
n pi x by l into d x ok so that will be the complex form of fourier series ok it contain
negative c n also which is also covered here it contains c naught also which is also covered
here so we can easily write that the fourier series complex form of fourier series is given
by this expression where c n is given by this expression ok so this is a complex form of
fourier series now let us solve few problems based on this
now let us try these two problems based on this now in the first problem f x is what
f x is e k power minus x ok and the function is defined from minus pi to plus pi of course
function is periodic ok now we have to express this function as a as the complex form of
fourier series ok so how can we do that we first find c n ok which is given by this expression
and when we substitute c n over here so that will be the complex form of the fourier series
expression of this function f x ok so what is c n c n will be given by one by
two l here l is pi one by two pi integral minus pi to plus pi f x is e k power minus
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
but one upon two pi integral minus pi to plus pi e k power minus will be outside and one
plus iota n into x into d x so this expression we will be having so this is further can be
written as one upon two pi now when we integrate this it is nothing but e k power minus one
plus iota n x upon minus one plus iota n and x is varying from minus pi to plus pi so this
will be further equal to one upon two pi you can take this term outside so this is negative
of one plus iota n upper limit minus lower limit e k power minus one plus iota n into
pi minus e k power minus minus plus one plus iota n into pi ok this you can simplify very
easily multiply this by its conjugate one minus iota n multiply and divide by one minus
iota n so we will get minus one minus iota n upon two pi into one plus n square and here
when you simplify so this is nothing but e k power minus pi into e k power minus iota
n pi minus e k power pi into e k power iota n pi ok
now what is e k power iota n pi e k power iota n pi will be nothing but cos n pi plus
iota sin n pi and it is minus one k power n and it is sin n pi is zero and similarly
when you find e k power minus iota n pi that is nothing but the conjugate of e k power
iota n pi so that will be remain minus one k power n ok so from here what we obtained
it is equal to minus one minus iota n upon two pi one plus n square when you simplify
this so this is also minus one k power n this is also minus one k power n both will come
out and what we will be having e k power minus pi minus e k power pi and whole multiplied
by minus one k power n so this negative you can get inside and upon two so this will be
nothing but when you simplify further so it is nothing but one minus iota n into minus
one k power n upon one plus n square which we are obtaining from here into pi and this
nothing but it is sine hyperbolic pi uh because sine hyperbolic pi is e k power pi minus e
k power minus pi upon two so that will be c n ok
so what will be the fourier series expression of this function the fourier series expression
of this function f x now will be nothing but f x will be equal to summation n varying from
minus infinity to plus infinity c n is what c n is one minus iota n minus one k power
n upon one plus n square into pi sin hyperbolic pi into e k power iota n x so that will be
the fourier series it is equal to e k power x ok because we find the fourier series complex
form of fourier series this function so function is e k power minus x ok and that e k power
minus x the complex form of this is nothing but this expression so that is how we can
find out the complex form of fourier series of a function f x ok
now let us solve second next problem so again we have to write down the complex form of
fourier series of this function now so what is the how the function is defined now function
is f x is equals to it is minus one when x is varying from minus pi to zero and it is
one when x is varying from zero to pi and function is periodic with period two pi ok
now again to find out the complex form of this f x first we will find c n which is given
by this expression ok and after finding c n we will substitute it here so that will
be the complex form of the fourier series of this function f x
so what is c n now c n we can find out it is one upon two pi because l is pi minus pi
to plus pi f x e k power minus iota n pi x upon l into d x that is nothing but one upon
two pi now you can split this function because function from minus pi to zero is minus one
and from zero to pi it is one and it is periodic ok one branch in the negative side of x axis
one branch in the positive side of x axis ok so this is minus pi to zero minus one e
k power minus iota n pi x upon l d x plus zero to pi it is one into e k power minus
iota n pi x upon l into d x now this can be written as one upon two pi
now you can integrate it minus e k power minus iota n pi x upon l ok here l is pi ok here
l is pi so this is pi eta pi pi cancels out ok so you can eliminate this so this is this
and this is this ok minus iota n x because l is pi ok so this is this function now this
is n x and divided by minus iota n from minus pi to zero plus it is e k power minus iota
n x upon minus iota n from zero to pi it is further equal to now negative negative cancels
out it is one upon two pi now you apply upper limit minus lower limit
the upper limit is uh one upon iota n will be outside upper limit is one minus e k power
e k power iota n pi ok and it is minus one upon iota n will be outside and it is nothing
but when it is pi it is e k power minus iota n pi minus one now it is further equal to
one upon two pi n into iota it is one minus e k power e k power iota n pi minus e k power
minus iota n pi minus minus plus so this is nothing but when we simplify this so it is
two it is two times of this two two canceled out so it is nothing but one upon pi n iota
and one minus e k power iota n pi ok ok it is not two times ok so it is it is two into
minus e k power minus iota n pi so two is remaining here ok
now e k power iota n pi is cos n pi plus iota sin n pi which is minus one k power n and
this value is again cos n pi this value is what e k power minus iota n pi will be nothing
but cos n pi minus iota sin n pi which is nothing but minus one k power n so this is
minus one k power n this is minus one k power n so now it is two times two two cancels out
and one upon iota is nothing but minus iota so it is minus iota upon n pi and it is one
minus minus one k power n so this will be the final expression for final expression
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
it is inside bracket expression the expression is at the bracket when n is even minus one
k power even is one so one minus one is zero when n is odd it is two
so what will be the complex form of fourier series for this function so for this function
the complex form is given by f x will be equal to uh it is summation n from minus infinity
to plus infinity c n is what c n is minus iota upon n pi one minus minus one k power
n into e k power iota n pi x upon l so this will be the complex form of this fourier series
so hence whether the function is continuous like the first example is continuous the second
example is discontinuous at x equal to zero so if we have functions uh that are continuous
or discontinuous or i mean piecewise continuous the second example is piecewise continuous
so if we have such problems then problems can be either converted into the sine or cosine
terms as we have uh done before or in the complex form of fourier series which is given
by this expression where c n is given by this expression ok here the benefit is we have
to find only c n and there we have to find a naught a n and b n the three coefficients
ok so that that is how we can find out the complex form of uh fourier series of any function
f x so so thank you