Engineering Mathematics (21)

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1 Egieerig Mathematics () Zhag, Xiyu Departmet of Computer Sciece ad Egieerig, Ewha Womas Uiversity, Seoul, Korea

2 Fourier Series Sectios.-3 (8 th Editio) Sectios.- (9 th Editio)

3 Fourier Series Trigoometric Series If it coverges, the fuctio will be a fuctio of period Periodic Fuctio where f() a a cos b si a,, a b f() cos a a b si are real costats. Ewha Womas Uiversity 3

4 Itegral Idetities si d si cos md cos d cos cos md si si md m m where,, Ewha Womas Uiversity 4 m m m

5 Determiatio of a f ( ) a a cos b si Itegrate o both sides: f ( ) d a acos bsi d a d a cosd b sid a a b a a Ewha Womas Uiversity 5 f ( ) d

6 Determiatio of a Multiply both sides by Itegrate o both sides: f ( ) a a cos b si cosm f ( )cos m a cos m a cos cos m b si cos m f ( )cos md a cos m acos cos m bsi cos m d Ewha Womas Uiversity 6

7 Determiatio of a a cos m acos cos m bsi cos md a cos md a cos cos md b si cos md a m b m m a m a a am am Ewha Womas Uiversity 7

8 Determiatio of a f ( )cos f ( )cos md d am a m a f ( )cos md m a f ( )cos d Ewha Womas Uiversity 8 m

9 Determiatio of b Multiply both sides by Itegrate o both sides: f ( ) a a cos b si sim f ( )si m a si m a cos b si si m f ( )si md a si m acos si m bsi si m d Ewha Womas Uiversity 9

10 Determiatio of b a si m acos si m bsi si md a si md a cos si md b si si md a b m m b m b b bm bm m Ewha Womas Uiversity

11 Determiatio of b f ( )si f ( )si md d b m b m b f ( )si md m b f ( )si d Ewha Womas Uiversity m

12 Fourier Series Trigoometric Series f ( ) a acos bsi with coefficiets a f ( ) d Ewha Womas Uiversity a f ( )cos d (,,...) b f ( )si d (,,...)

13 Eample Fid the Fourier coefficiets of the followig periodic fuctio. f( ) k, if k, if f ( ) f ( ) Ewha Womas Uiversity 3

14 Eample: Solutio Fourier coefficiets of a periodic fuctio f() a f ( ) d a f ( )cos d (,,...) b f ( )si d (,,...) Ewha Womas Uiversity 4

15 Eample: Solutio () a f ( ) d ( k) d kd k k k k Ewha Womas Uiversity 5

16 Eample: Solutio () a f ( )cos d ( k)cos d k cos d k k si si Ewha Womas Uiversity 6

17 Eample: Solutio (3) b f ( )si d ( k)si d k si d k k cos cos k k cos cos( ) cos cos k cos Ewha Womas Uiversity 7 4k for odd for eve

18 Eample Fourier coefficiets k, if f( ) k, if () a f ( ) d f ( ) f ( ) () a f ( )cos d 4k for odd (3) b f ( )si d for eve Ewha Womas Uiversity 8

19 Eample Fourier series of f ( ) a a cos b si b si b si f() Ewha Womas Uiversity 9 b 4k si si 3 si k for odd for eve

20 Fourier Series Partial Sum 4k S si 4k S si si 3 3 4k S3 si si 3 si Ewha Womas Uiversity

21 Fourier Series Ewha Womas Uiversity

22 Covergece Theorem. If a periodic fuctio with period is piecewise cotiuous i the iterval ad has a left-had derivative ad right-had derivative at each poit of that iterval, the the Fourier series of f() is coverget. Its sum is f(), ecept at a poit at which f() is discotiuous ad the sum of the series is the average of the left- ad right-had limits of at. f() Ewha Womas Uiversity f()

23 Proof f ( ) a a cos b si () a f ( ) d M f ( ) M Etreme Value Theorem Ewha Womas Uiversity 3

24 Proof uvd uv uvd () a f ( )cos d f ( ) si d f ( ) si f ( ) si d f ( )si d f ( ) cos d Ewha Womas Uiversity 4

25 Proof uvd uv uvd f ( ) cos d f ( ) cos f ( ) cos d f ( )cos d f ( )cos d Ewha Womas Uiversity 5

26 Proof a f ( )cos d a f ( )cos d M f ( )cos d ( ) cos f M Ewha Womas Uiversity 6 Etreme Value Theorem

27 Proof uvd uv uvd (3) b f ( )si d f ( ) cos f ( )cos d Ewha Womas Uiversity 7 f ( )cos d f ( ) si f ( )si d f ( )si d

28 Proof b f ( )si d M f ( )si d ( ) si f M Etreme Value Theorem Ewha Womas Uiversity 8

29 Proof f ( ) a a cos b si a acos bsi a a b si, cos M M M M 4M Ewha Womas Uiversity 9 p series: Coverget p

30 Covergece Theorem. If a periodic fuctio with period is piecewise cotiuous i the iterval ad has a left-had derivative ad right-had derivative at each poit of that iterval, the the Fourier series of f() is coverget. Its sum is f(), ecept at a poit at which f() is discotiuous ad the sum of the series is the average of the left- ad right-had limits of at. f() Ewha Womas Uiversity 3 f()

31 Fourier Series f() Periodic Fuctio with a period where f() a,, a b cos a a b si are real costats. Ewha Womas Uiversity 3

32 Fourier Series Periodic Fuctio with a period L L v L If, the f() v L Thus, we ca rewrite with a period gv () f() as a ew fuctio Ewha Womas Uiversity 3

33 Fourier Series gv () Periodic Fuctio with a period f() where a a,, a b cos a a v b si v g() v dv are real costats. a g( v)cos vdv (,,...) b g( v)si vdv (,,...) gv () Ewha Womas Uiversity 33 v L

34 v v L Fourier Series dv d L, L, L a g() v dv L ( ) g L L L d L L f ( ) d L Ewha Womas Uiversity 34

35 v v L Fourier Series dv d L, L, L a g( v)cos vdv L g ( )cos L L L L d L L L f ( )cos d L Ewha Womas Uiversity 35

36 v v L Fourier Series dv d L, L, L b g( v)si vdv L g ( )si L L L L d L L L f ( )si d L Ewha Womas Uiversity 36

37 Fourier Series gv () Periodic Fuctio with a period f() where a a,, a b cos a a v b si v g() v dv are real costats. a g( v)cos vdv (,,...) b g( v)si vdv (,,...) gv () Ewha Womas Uiversity 37 v L

38 Fourier Series Periodic Fuctio with a period where a f() f() a acos bsi L L a,, are real costats. a b L L L f ( ) d L a f ( )cos d (,,...) L L L L b f ( )si d (,,...) L L L L Ewha Womas Uiversity 38

39 Eample Fid the Fourier series of the followig periodic fuctio. f ( ) k f ( 4) f ( ) period: L 4 Ewha Womas Uiversity 39

40 Eample: Solutio L () a f ( ) d L L d kd d k d 4 k Ewha Womas Uiversity 4

41 Eample: Solutio L () a f ( )cos d L L L cos d k cos d cos d k si for eve k si k for,5,9... k for 3,7,... Ewha Womas Uiversity 4

42 Eample: Solutio L (3) b f ( )si d L L L si d k si d si d k cos Ewha Womas Uiversity 4

43 Eample Fourier coefficiets L k () a f ( ) d L L for eve L () a f ( )cos d k for,5,9... L L L k for 3,7,... L (3) b f ( )si d L L L f ( ) k f ( 4) f ( ) period: L 4 Ewha Womas Uiversity 43

44 Eample: Fourier Series a acos bsi L L a a cos L f() k a cos a3 cos a5 cos a7 cos... k k k 3 k 5 k 7 cos cos cos cos k k cos cos cos cos Ewha Womas Uiversity 44

45 Assigmet Fid the Fourier series of the fuctio f ( ), ( ) which is assumed to have the period. Show the details of your work. Ewha Womas Uiversity 45