Mathematical Physics Vol 1

Chapter 6. Trigonometric Fourier series. Fourier integral

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Proof Using the definition of periodicity, for T we obtain f ( x + nT )= f [ x +( n − 1 ) T + T ]= f [ x +( n − 1 ) T ]= f [ x +( n − 2 ) T + T ]= f [ x +( n − 2 ) T ]= ··· = f ( x + T )= f ( x ) . (6.2) This proves the Theorem. The smallest constant T ( T > 0), for which (6.1) is true is called the fundamental (primitive) period of the function f ( x ) , and nT , a multiple of the fundamental period (non-fundamental period). R Note 1. It is sufficient to analyze (and draw) a periodic function only on the interval of length T (e.g. from 0 to T ). The remaining parts of the function are obtained by moving this part translationally to the left (for − T , − 2 T ,. . . ) or right (for T , 2 T ,. . . ). R Note 2. If the functions f and g are periodic, with periods T f and T g , and if T f / T g = p / q , then T ∗ = qT f = pT g is a period for both functions, i.e. f ( x )= f ( x + T ∗ ) and g ( x )= g ( x + T ∗ ) . The simplest periodic process in physics is described by the function x ( t )= A sin ( ω t + ϕ ) , − ∞ < t < ∞ , (6.3) which is called a harmonic . The origin of the name comes from harmonic oscillations, which are described by such functions.

6.1.1 Properties of periodic functions

The properties that follow from the definition of the periodic function are: a) Inverse function of a periodic function is a multivalued function. b) Derivative of a periodic function is a periodic function. c) If a periodic function f ( x ) , with period T , is integrable, then 1 a + T Z a f ( x ) d x = T Z 0 f ( x ) d x .

(6.4)

d) A primitive function of a periodic function does not have to be a periodic function.

1

a + T Z a

T Z a

a + T Z T

T Z a

a Z 0

T Z 0

f ( x ) d x =

f ( x ) d x +

f ( x ) d x =

f ( x ) d x +

f ( x ) d x =

f ( x ) d x .

We have used here

a + T Z T

a + T Z T

f ( x ) d x = f ( x − T ) d x . Substituting x − T = ¯ x ⇒ d x = d¯ x and bearing in mind that the limits also change: for x 1 = T and x 2 = a + T , the corresponding values for the new variable are ¯ x 1 = 0 and ¯ x 2 = a . This finally yields a + T Z T f ( x − T ) d x = a Z 0 f ( ¯ x ) d¯ x .