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
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
- is called the right hand limited of ƒ (x) at ‘a’
- 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 . 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
- A function ƒ ( x ) is said to be continuous at x = a if . In other words we can say that ƒ (x) is continuous at x = a if ƒ (a −) = ƒ ( a +) = ƒ (a).
- 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 +)
- The discontinuity x = a is called infinite whenever ƒ ( x ) becomes infinite as x→a.
- 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
- 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.
- Let ƒ (x) be a periodic function satisfying the Dirichlet’s condition. The series in the form ……(1)
- The coefficient are Fourier Coefficient ( i ) known as the trigonometric form of the Fourier Series.
- 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) can be expanded as a Fourier series then
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