Whenever a function is defined over an Interval (0,L) then we try/extend to define a function from (-L,0) in such a way that it becomes either even or odd on (-L,L) so as to get cosine or sine Series expansion respectively.
If u have from (0,2L) then if u want cosine series then to extend the function on (-2L,0) so that on interval (-2L,2L) function becomes even or odd and then u get cosine or sine series expansion respectively. Note that in this case, period will become 4L
Fourier series is nothing is but a linear combination of cosine and sine terms (see the starting of first lecture). Since sine and cosine terms are defined everywhere so Fourier series is also defined everywhere.
@@ameylandge425 Our function f is piecewise continuous function and a_n, b_n are nothing but integration of f(x)*cos, f(x)* sine respectively. Now you are asking whether this Integration exists or not? Answer is yes. Their is a theorem by Riemann which says if we have a function with countably many discontinuities then such functions are always integrable. Here our f is piecewise continuous hence countable discontinuity. Hence its integration always exists i.e. a_n and b_n exists.
Ngl sir ur flow is unmatched....helped me a lott!!! I can understand everything u teach....thank you for everything sir
Glad to hear that..thank you..
This lecture helped me.thanks ...
Great... happy to hear that ☺️
What will be the Fourier sine and cosine series for period 2L, where L is any
integer?
Whenever a function is defined over an Interval (0,L) then we try/extend to define a function from (-L,0) in such a way that it becomes either even or odd on (-L,L) so as to get cosine or sine Series expansion respectively.
If u have from (0,2L) then if u want cosine series then to extend the function on (-2L,0) so that on interval (-2L,2L) function becomes even or odd and then u get cosine or sine series expansion respectively. Note that in this case, period will become 4L
Sir what if any term in the fourier series is not defined ??
Fourier series is nothing is but a linear combination of cosine and sine terms (see the starting of first lecture). Since sine and cosine terms are defined everywhere so Fourier series is also defined everywhere.
@@DrMathaholic sir but if one of the coefficients is not defined for some value of n
@@ameylandge425 Our function f is piecewise continuous function and a_n, b_n are nothing but integration of f(x)*cos, f(x)* sine respectively. Now you are asking whether this Integration exists or not? Answer is yes. Their is a theorem by Riemann which says if we have a function with countably many discontinuities then such functions are always integrable. Here our f is piecewise continuous hence countable discontinuity. Hence its integration always exists i.e. a_n and b_n exists.
Any function of period p : 2L.
And this topic are same?
Yes. They are..