Formulas for Complex Form of Fourier Series – Fourier Series – Engineering Mathematics 3


Hey Student so today we are gonna start with very interesting topic of engineering mathematics and it is one of the most difficult topic from student’s point of view in engineering mathematics and the name of the topic is complex form of Fourier series so guys you must have seen my previous video of Fourier series so in that I’ve explained you what is Fourier series what is the formula fourier series how do we find the Fourier series for a given function so now we have want to learn complex form of Fourier series the formula for the complex form of Fourier series and how to apply that complex mono Fourier series in different numericals so guys we know the Fourier series formula which is f of X equal to a not plus summation a cos n Pie X by L plus summation be in sine n Pie X by L so in that Fourier series if you replace cos and sine by the Euler’s formula so you know the formula or Euler’s that is cos of X is equal to e to the power X plus e to the power minus ikz upon 2 so if we replace that cos and sine by the Euler’s formula then we get the complex form of Fourier series so you are not gonna cover the derivation for complex form of Fourier series will be covering the formulae which are required for solving numerical so first of all let’s see the generalized formula for complex form of Fourier series which is applicable in almost all numericals adding that formula via wanna change the range of integration depending upon the question so let’s start so the formula for complex form of Fourier series is this so it is given by f of X equal to summation n is from negative infinity to positive infinity CN e to the power I and Phi X upon L now guys your if you see then there is only one Fourier coefficient and which is : CN but whereas in complex the novel Fourier series we have seen three Fourier coefficient that is a not an and BN so here if I want to find out the complex from furious it is then I have to find out the value of CN so how to get this here so definitely I am going to use the integration that is the definite integration to find out the value of CL so the formula for CN is it is 1 upon 2 L integration from C to C plus 2 L f of X into e to the power negative I n PI X upon L DX so this is what the formula to find out if we efficiency n we are going to substitute this value of CN in f of X to get the complex form of Fourier series now you must be knowing that why do we call it as complex form because here we get this imaginary term so whenever we get any number in the form of a plus IB or any number which is having the imaginative that number is called as complex number so since this series is having that imaginary term this series will be called as complex form of Fourier series so guys this is what the general formula for complex form of Fourier series in the range C to C plus 2 n now let’s derive the formula for particular intervals or particular ranges so in engineering mathematics we have honest awful numericals based on four different intervals so the first interval is 0 to 2 n so I will derive the series for the interval 0 to 12 nowadays while did I think this is for 0 to 2 L we have to use the generalized form now let’s also the general form in general form we have the integration from C to C plus 2 L now if I want to convert it into 0 to 2 L what should I do and it’s simple that I just have to substitute C equal to 0 so if I put Z equal to 0 here I will get 0 as a lower limit and the upper limit will be 0 plus 2 and that is 2l and guys I will get 0 to 2 it it means in the formula for the range 0 to 2 L I am just going to substitute C as 0 so that in this series that is in the complex form of Fourier series f of X there is no term policy so this f of X will not change so there will be only one difference and that 2 will be the same so there the integration will change so here I’ll say that f of X will remain same so f of X equal 2 now guys you have to observe that summation as well because the separation we are taking the values of n from negative infinity to infinity and in the Fourier series we used to take that range from 1 to infinity so that is one of the difference now let’s change the a so CN will be 1 by 2 where here I put CN 0 so this will become 0 to 2 N and the remaining terms are as it is so guys this is the formula for the range 0 to 2 n now similarly I will find out the complex form of Fourier series for the interval negative L to positive L because that is the second type of range or the interval that we get in the numerical so nowadays to get the series 4 negative 8 2 and what we have to do so let’s come back to the general form so in the general form if I put C equal to negative L we will get here negative L and here we will get minus L plus 2l which is L so we are automatically getting minus L to L so it means in the general formula I am going to substitute the value of C as negative L now guys here in the effect there is no C so f of X is gonna be exactly same so now I will not write down that f of X once again I will just write down the formula for saying which is going to change so here this formula will be seen equal to 1 by 2 L integration from negative L to L well I am putting C is negative L and the remaining function as it is which is now adapt to the second race what is the third rain that we get in the numerator so third range is 0 to 2 pi so you write down for you now it’s 43 what I’m gonna do is I have a formula for the range 0 to 2 L now in this formula I am going to substitute LS PI so by putting n equal to PI I will get the formula for 0 to 2 pi so your this n will become pi so pi and pi will be cancelled and here we will get e to the power I and X so guys f of X is similarly by finding the value of CN I am gonna substitute LS pi so this will become 1 by 2 pi 0 to 2 pi and here we will get pi so pi and pi cancel so e to the power negative I and X so we will get and now guys after this we are gonna start with the final series how I would say the final interval which is negative PI to PI so now guys for the range negative PI to PI what I am gonna do is I have range minus L to L so in this age I’ll be putting I will be substituting L equal to 5 so this will be negative PI to 5 so you will say so let’s substitute n equal 2 pi so guys this CN is gonna be 1 by 2 pi this will become negative pi 2 pi f of X and if we put LS pi this pi and pi will be removed or cancel and we will get e to the power negative ion X DX so guys this is the formula for CH but with this en the value of f of X will also change because we are substituting LS PI so guys this is the value of f of X and in this if I put LS 5 PI and PI will be canceled and we will get e to the power I and X so this will be so guys in short we have derived the formula for all possible intervals of complex form of Fourier series and we are going to use this for intervals and this formula to solve the numerical on complex form of Fourier series so and 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