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![Fourier Series of a Periodic FunctionFourier Series of a Periodic Function
Definition : A Fourier series may be defined as an expansion of a
function in a series of sines and cosines such as ,
0<x<2π
The coefficients are related to the periodic function f(x)
by definite integrals:
Henceforth we assume f satisfies the following (Dirichlet)
conditions:
(1) f(x) is a periodic function;
(2) f(x) has only a finite number of finite discontinuities;
(3) f(x) has only a finite number of extrem values, maxima and
minima in the interval [0,2p].](https://image.slidesharecdn.com/160280102001c1aem-171011170020/85/160280102001-c1-aem-5-320.jpg)
![Fourier IntegralFourier Integral
If f(x) and f’(x) are piecewise continuous in every finite interval, and
f(x) is absolutely integrable on R, i.e.
converges, then
Remark: the above conditions are sufficient, but not necessary.
16
∫ ∫
∞
∞−
∞
∞−
−
=++− dwdttfeexfxf iwtiwx
)(
2
1
)]()([
2
1
π](https://image.slidesharecdn.com/160280102001c1aem-171011170020/85/160280102001-c1-aem-16-320.jpg)
Uploaded byL.D. COLLEGE OF ENGINEERING
160280102001 c1 aem
1) Fourier series and integrals are used to represent periodic functions as an infinite sum or integral of sines and cosines. They are useful for solving differential equations. 2) The Fourier series of a periodic function f(x) with period T is the sum of its coefficients multiplied by sines and cosines of integer multiples of x/T. The coefficients are calculated using integrals involving f(x). 3) Half range expansions like half range cosines (HRC) and half range sines (HRS) are used when f(x) is defined on a finite interval, by extending it to a periodic function on the entire real line.
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160280102001 c1 aem
- 2. TOPIC: FOURIERTOPIC: FOURIER SERIESANDSERIES AND FOURIER INTEGRALFOURIER INTEGRAL Guided By: Prof. Jaydev PatelGuided By: Prof. Jaydev Patel
- 3. Introduction Jean Baptiste JosephFourier (Mar21st 1768 –May16th 1830) French mathematician, physicist Main Work: (The Analytic Theory of Heat) •Any function of a variable, whether continuous or discontinuous, can be expanded in a series of sines or cosines multiples of the variable •The concept of dimensional homogeneity in equations •Proposal of his partial differential equation for conductive diffusion of heat Discovery of the "greenhouse effect“ •Fourier series is very useful in solving ordinary and partial differential equation.
- 4. Even, Odd, andPeriodic FunctionsEven, Odd, and Periodic Functions
- 5. Fourier Series ofa Periodic FunctionFourier Series of a Periodic Function Definition : A Fourier series may be defined as an expansion of a function in a series of sines and cosines such as , 0<x<2π The coefficients are related to the periodic function f(x) by definite integrals: Henceforth we assume f satisfies the following (Dirichlet) conditions: (1) f(x) is a periodic function; (2) f(x) has only a finite number of finite discontinuities; (3) f(x) has only a finite number of extrem values, maxima and minima in the interval [0,2p].
- 6. The formula fora Fourier series is: We have formulae for the coefficients (for the derivations see the course notes): One very important property of sines and cosines is their orthogonality, expressed by: ∑ ∞= = + += n n nn T xn b T xn aaxf 1 0 2 sin 2 cos)( ππ ∫ − = 2 2 0 )( 1 T T dxxf T a ∫ − = 2 2 2 cos)( 2 T T n dx T xn xf T a π ∫ − = 2 2 2 sin)( 2 T T n dx T xn xf T b π = ≠ = ∫ − mn T mn dx T xm T xn T T 2 0 2 sin 2 sin 2 2 ππ
- 7. nmdx T xm T xn T T ,allfor0 2 sin 2 cos 2 2 = ∫ − ππ These formulae areused in the derivation of the formulae for Example – Find the coefficients for the Fourier series of: To Find )()2( 0 0 )( xfxf xx xx xf =+ ≤≤ ≤≤−− = π π π 0a nn ba ,
- 8. Find ,0a ∫ − = 2 2 0 )( 1 T T dxxf T a∫− =⇒ π π π dxxfa )( 2 1 0 ∫=⇒ π π 0 0 1 xdxa 2 0 π =⇒ a f (x) is an even function so: ∫− = π π π dxxfa )( 2 1 0 ∫=⇒ π π 0 0 )( 1 dxxfa π π 0 2 0 2 1 =⇒ x a
- 9. Find na ∫ − = 2 2 2 cos)( 2 T T n dx T xn xf T a π ∫− =⇒ π π π π π dx xn xfan 2 2 cos)( 1 Sinceboth functions are even their product is even: ( )∫− = π π π dxnxxfan cos)( 1 ( )∫=⇒ π π 0 cos 2 dxnxxan nb ∫ − = 2 2 2 sin)( 2 T T n dx T xn xf T b π ∫− =⇒ π π π π π dx xn xfbn 2 2 sin)( 1 ( )∫− = π π π dxnxxfbn sin)( 1 0=⇒ nb
- 10. So we canput the coefficients back into the Fourier series formula: ∑ ∞= = + += n n nn T xn b T xn aaxf 1 0 2 sin 2 cos)( ππ ( )( ) ( )∑ ∞= = −−+=⇒ n n n nx n xf 1 2 cos11 2 2 )( π π ( ) ( ) +−+−=⇒ xxxf 3cos 9 4 0cos 4 2 )( ππ π
- 11. Summary of findingSummaryof finding coefficientscoefficients function even function odd function neither 0a na nb 0)( 1 2 2 0 == ∫ − T T dxxf T a ∫ − = 2 2 2 cos)( 2 T T n dx T xn xf T a π Though maybe easy to find using geometry ∫ − = 2 2 2 sin)( 2 T T n dx T xn xf T b π ∫ − = 2 2 2 sin)( 2 T T n dx T xn xf T b π 0 0 0)( 1 2 2 0 == ∫ − T T dxxf T a ∫ − = 2 2 2 cos)( 2 T T n dx T xn xf T a π Though maybe easy to find using geometry 0
- 12. HALF RANGE EXPANSIONSHALFRANGE EXPANSIONS It often happens in applications, especially when we solve partial differential equations by the method of separation of variables, that we need to expand a given function f in a Fourier series, where f is defined only on a finite interval. We define an “extended function”, say fext, so that fext is periodic in the domain of -∞< x < ∞, and fext=f(x) on the original interval 0<x<L. There can be infinite number of such extensions. Fourier extensions: half- and quarter- range cosine and sine extensions, which are based on symmetry or anti-symmetry about the endpoints x=0 and x=L.
- 13. HRC (HALF RANGECOSINES)HRC (HALF RANGE COSINES) fext is symmetric about x=0 and also about x=L. Because of its symmetry about x=0, fext is an even function, and its Fourier series will contain only cosines, no sines. Further, its period is 2L, so L is half the period.
- 15. HRS (HALF RANGESINE SERIES)HRS (HALF RANGE SINE SERIES)
- 16. Fourier IntegralFourier Integral Iff(x) and f’(x) are piecewise continuous in every finite interval, and f(x) is absolutely integrable on R, i.e. converges, then Remark: the above conditions are sufficient, but not necessary. 16 ∫ ∫ ∞ ∞− ∞ ∞− − =++− dwdttfeexfxf iwtiwx )( 2 1 )]()([ 2 1 π
- 17. DIFFERENT FORMS OFFOURIERDIFFERENT FORMS OF FOURIER INTEGRAL THEOREMINTEGRAL THEOREM
- 18. Complex or exponentialformComplex or exponential form
- 19.
- 20. PREPARED BY :-PREPAREDBY :- • ALAGIYA KEVALKUMAR .A (160280102001) • BACHAV AHEMEDRAZA .S (160280102002) • BHADARKA APABHAI GIGABHAI (160280102004) • BUMBADIYA ASHISH SHARAD (160280102006) • CHAUDHARY SWAPNILKUMAR (160280102007) • CHAUHAN DIGVIJAYSINH (160280102008) • CHAUHAN KARANKUMAR (160280102009) • DOMADIYA PARIMAL (160280102010)
Editor's Notes
- #17 Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans.