FOURIER SERIES

PERIODIC FUNCTIONS:  (Definition)

    A function ƒ( x ) said to be periodic if for all values of  x, ƒ ( x +p ) = ƒ ( x ), p > 0 → ( 1 )

  • There are several values of ‘p’ satisfying ( 1 ) of those the least value of ‘p’ satisfying ( 1 )  is known as the period of the function. The least value of ‘p’ is known as the primitive value of the function
LIMITATION OF A FUNCTION 
  • Let ‘x’ be a variable and ‘a’ be the fixed number. Then x→a means x tends to ‘a ‘ (or) x approaches ‘ a’, what we mean is that the variable x is taking a value very close ‘a’. Let  ƒ (x) be a real function. In notation as x→a, it may be true that ƒ (x) → l for some number l. If it so happens that when x→a, ƒ (x)→l then we say that ‘l’ is the limit of ƒ (x) as x→a. We denote this by \lim_{x\to a} f(x) = l

   ONE SIDED LIMITS

  • Let  a be a real number. x→a means x takes a value which are very close to ‘a ‘. Then the values of taken by x can be on either side of a . They may be greater than ‘a’ or less than ‘a’, but are near ‘a’. Now x→a + means x takes values near  ‘a’ and greater than ‘a’. The notation x→a – on the other hand means that x takes values near a and less than ‘a’.
LEFT AND RIGHT LIMITS
  • \lim_{x\to a^+} f(x) is called the right hand limited of ƒ (x) at ‘a’
  • \lim_{x\to a^-} f(x) is called the left hand limited of ƒ (x) at ‘a’. Both of these are called the limit of ƒ (x) at ‘a’. To find the left hand limit at x = a. i.e.., ƒ (a−), replace x by  a − h in ƒ (x) and then take the limit as h → 0. Thus f(a-) = \lim_{h\to 0} f(a-h) . To find the right hand limit at x = a, i.e.,  ƒ (a +) , replace x by a + h in ƒ (x) and then the limit as h→0. Thus f(a+) = \lim_{h\to 0} f(a+h)
CONTINUOUS FUNCTION

     DEFINITION   

  •  A function ƒ ( x ) is said to be continuous at x = a if \lim_{h\to 0} f(x) = f(a). In other words we can say that ƒ (x) is continuous at x = a if ƒ (a −) = ƒ ( a +) = ƒ (a).
 DISCONTINUOUS FUNCTION

       DEFINITION

  •   A function ƒ (x) is said to be discontinuous at a point if it is not continuous at a points. In other words a function ƒ ( x ) is discontinuous at x = a if ƒ (a −) ≠ ƒ (a +) 

INFINITE DISCONTINUITY

  • The discontinuity x = a is called infinite whenever ƒ ( x ) becomes infinite as x→a.

FINITE DISCONTINUITY

  • Again the discontinuity is said to be finite or jump or ordinary left and right hand are unequal and their difference is finite.

 PIECE WISE CONTINUOUS

  DEFINITION 

  • A function which is continuous everywhere except at a finite number of jumps in a given interval is called piece wise continuous. In other words, a function is said to be a piece wise continuous if it can have finite number of finite discontinuous in a given interval.

 FOURIER SERIES

  DEFINITION

  •  Let ƒ (x) be a periodic function satisfying the Dirichlet’s  condition. The series in the form f(x) = \frac{x_{0}}{2} + \sum_{n=1}^{\infty} (a_{n} cos nx + b_{n} sin nx)……(1)
  •  The coefficient a_{{n}} , b_{{n}} , a_{{0}} are Fourier Coefficient ( i )  known as the trigonometric form of the Fourier Series.

DIRICHLET’S CONDITION

  •   Let ƒ (x) be a periodic function defined in the interval ( c, c + 2l ) can be expanded as a Fourier series of the following conditions are satisfied.
  • ƒ (x) is well defined and single valued except possibly at a finite number of points. ƒ (x) should be of piece wise continuous. 
  • ƒ (x) should have finite number of maximum and minimum.

THE EULER FORMULA

  • It is a function defined in ( c, c + 2\pi) can be expanded as a Fourier series \frac{a_{0}}{2} + \sum_{n=1}^{\infty}  a_{{n}} cos nx + 
\sum_{n=1}^{\infty}    b_{{n}} sin nx then a_{{0}} = \frac{1}{\pi}   \int_{c}^{c+2 \pi} f(x)dx ; a_{{n}} = \frac{1}{\pi}   \int_{c}^{c+2 \pi} f(x)   cos nxdx ;b_{{n}} = \frac{1}{\pi}   \int_{c}^{c+2 \pi} f(x)   sin nxdx ;

CONVERGENCE OF FOURIER SERIES

  • The Fourier series of ƒ(x) converges to ƒ (x) at all points whereƒ( x ) is continuous, I.e., ƒ (x) is continuous at x = a then the Fourier series converges to ƒ (a).
  • At a point of discontinuity, the series converges to the average value of right band and left hand limit . I.e., ƒ (x) is discontinuous at  x = a then the Fourier series converges to f(a) = \lim_{h\rightarrow 0}  \frac{1}{2} [ f (a-h) + f(a+h)]

            

       

       

       

                                   

                                                             

                                       

     

 

Leave a Reply

Your email address will not be published. Required fields are marked *