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Национальный авиационный университет

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∞

2.1.7. а)

∑

;

1+ x

n=1

2.1.8. а)

∞

∑ (15 − x2 )n ;

n=1

∞

2.1.9. а)

∑

tg2 n x ;

n=1

2.1.10. а)

∞

∑ (8 − x2 )n ;

n=1

∞

2.1.11. а)

∑

;

4 + x

n=1

∞

3n sin2 n x

2.1.12. а)

∑

;

n=1

∞

2.1.13. а)

∑

;

n=1

2.1.14. а)

∞

∑ (n + 1)en x ;

n=1

∞

2.1.15. а)

∑

;

n=1 x

2.1.16. а)

∞

;

∑ ne− n x

n=1

∞

2.1.17. а) ∑ xn tgn

;

n=1

∞

2.1.18. а)

∑

;

(x + e)

n=1 ln

∞

2.1.19. а)

∑

;

3 n4

n=1

∞

n x

2.1.20. а)

∑

;

n x

n=1

∞

2.1.21. а)

∑

sinn

x ;

n=1

∞

∑

;

n+1

n=1 n!(x +

∞

∑ e− n2 x ;

n=1

∞

(−1)n+1

∑

;

+ n

n=1

∞

3n + 2

∑

;

x+1

n=1 x n

∞

∑

x4n

sin

;

n=1

	∞	1			(2x) .
c) ∑		1		tgn
c) ∑		3		tgn
	n=1 n
	∞		(−1)		n−1
c)	∑		(−1)				.
c)	∑	4	n			2n	.
	n=1	4		(x −1)
	∞	x + 1			2n
c)	∑				.
c)	∑				.
	n=1	x −1

∞

c) ∑ ne− nx .

n=1

∞sin nx

c)∑ .

n= 2 n ln3 n

∞

∑

;

∑

(

)(

)

n=1

x −

n=1 ln (x − e)

n + 1

∞

2n + 1

∞

b) ∑

;

c) ∑

(n + 1)

(4n −

n=1

∞

sin(2n − 1)x

∞

∑

;

∑

(

)

n=1 (2n −1)

n=1 n

−

∞

(4x − 1)

∑

27n x3n arctg

;

∑

n=1

(n + 2)3

∞

b) ∑

;

c) ∑

(x +

n=1 ln

n=1 x

∞

cos πnx

∞

∑

;

∑

n ln

(5n −

2 x

n= 2

n=1

∞

3x +

(

) .

∑

sinn

x ;

∑

n=1

∞

− nx

∞

∑ ne

;

∑

+ 3

n=1

∞

4n sin2n x

∑

;

∑

(x −

n=1

∞

(x2 − 2)n

∑

;

∑

4n (n2 + 2)

n=1 xn

n=1

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2.1.22. а)

∑

3 sin

3 n

;

∑ nsin x

;

∑ (x − 1)

3n+1

∞

n=1 27n

n=1

∞

−1

(5 − x2 )

(

3x−)2 .

2.1.23. а)

∑

;

∑

;

∑

n=1 n + x

n=1

∞

2.1.24. а)

∑

(−1)

;

∑

;

∑

1/ 2

(

)(

)

n=1 (n − x)

n=1

x −

n=1 3 + 2

+ 1

∞

2.1.25. а)

∑

;

∑

;

c) ∑

n x

n=1

n=1 e

∞

2 + xn

∞

lnn (x + 1)

∞

8n sin3n x

2.1.26. а)

∑

;

∑

;

∑

2 − x

3 n4

n=1

∞

n + 2

∞

(−1)

n−1

∞

2.1.27. а) ∑

;

∑

;

∑

tg2n

x .

n=1

nx+1

n=1 n 3n (x − 5)n

n=1

∞

sin nx

∞

x2n

2.1.28. а)

∑

;

∑

;

∑

(

)

n=1 n (n + x )

n=1 n

n=1 4

− 1

∞

−1

n+1

(−1)

(

)

2.1.29. а)

∑

;

∑

x ;

∑

2x+1 .

+ n)

n=1

n=1 n

n=1

∞

(−1)n

∞

(x2 − 1)n

2.1.30. а)

∑

;

∑

;

∑

x +

+ 1

n=1

2.2. Investigate the functional series for uniform convergence within the given interval.

∞	(sin x +							3 cos x)n
2.2.1. ∑										, x R.
2.2.1. ∑						4		n		, x R.
n=1						4
∞				n + 1
2.2.3. ∑							, x		R.
2.2.3. ∑			n	4	+ x	4	, x		R.
n=1			n		+ x
∞			− nx
2.2.5. ∑		3			, x [0;				∞).
2.2.5. ∑		2			, x [0;				∞).
n= 2		n			nx2 , x R.
2.2.7. ∑ cos2					nx2 , x R.
∞
n=1 x + n
∞	(sin x −							3 cos x)n
2.2.9. ∑										, x R.
2.2.9. ∑							n			, x R.
n=1						3

∞

2.2.2. ∑ sin

, x R.

n=1

∞

2.2.4. ∑

, x [−1; 1].

+ 1

n=1 n

∞

2.2.6. ∑ 2− n sin

, x R.

n=1

∞

2.2.8. ∑

, x R.

n=1 2

+ x

∞

− nx

2.2.10. ∑

x [0; ∞).

n= 2 n

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3sin nx

∞

(3x)

2.2.11. ∑

−

;

n=1 n(1+ n)

∞

2.2.13. ∑

, x [0; ∞).

n=1

1+ xn

∞

[

]

2.2.15.

∑

−1; 1 .

n= 2 n ln n

∞

(sin x − cos x)n

2.2.17. ∑

, x R.

n=1

∞

n + 1

2.2.19. ∑

, x [0; ∞).

(2x + n)

n=1

∞

2.2.21. ∑

2− n cos

, x R.

n=1

∞

x |

2.2.23. ∑

, x R.

3+ | x | n

n=1

∞

n + 2

2.2.25. ∑

x R.

+ | x |

n=1

∞

2.2.27. ∑

x [0; ∞).

(2x + n)

n=1

∞

, x

2.2.29. ∑

−

;

n=1 2 + 3 n7

∞

2.2.12. ∑

, x R.

4 + x

n=1

∞

2.2.14. ∑

, x [0; ∞).

(x + n)

n=1

∞

− nx

2.2.16. ∑

x [0; ∞).

+ 3

n= 2 n

∞

2.2.18. ∑

sin

, x R.

n=1

∞

[

2.2.20.

∑

−1; 1 .

n= 2 n ln

∞

− nx

2.2.22. ∑

x [0;∞).

n= 2

∞

2.2.24. ∑

, x [−1; 1].

n=1 n

+ n

∞

cos nx

2.2.26. ∑

x R.

+ n

n=1 x

∞

(sin x + 1)n

2.2.28. ∑

, x R.

n=1

∞

2.2.30. ∑ , x R.

n=1 x4 + n3

2.3. Find the domain of convergence of power series.

∞

(x −

2.3.1. ∑

(3n + 1) 2

n=1

∞

(3n − 2)(x − 3)n

2.3.3. ∑

(n + 1)

n+1

n=1

∞

(x −

2.3.5. ∑

(−1)

n=1

(n + 1) 5

∞

(x − 5)n

2.3.7. ∑

(n + 2)ln(n +

n=1

∞

(−1)

n−1

2.3.9. ∑

(x + 1)

n=1

+ 1

∞						1
2.3.2. ∑	(x +		5)n tg			1		.
2.3.2. ∑	(x +		5)n tg				n	.
n=1					3
∞	(x + 2)			n
2.3.4. ∑	(x + 2)						.
2.3.4. ∑	(2n + 1) 3				n		.
n=1	(2n + 1) 3
∞			n
2.3.6. ∑ sin			n	(x − 2)n .
2.3.6. ∑ sin		n + 1		(x − 2)n .
n=1		n + 1
∞	n2 (x − 3)n
2.3.8. ∑						.
2.3.8. ∑	(n	4	+ 1)	2		.
n=1	(n		+ 1)

∞

2.3.10. ∑ (−1)n (2n + 1)2 xn .

n=1

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∞

2.3.11. ∑ (n + 1)!(x − 1)n .

n=1

∞

2.3.13. ∑

n=1 n + 1

∞

2.3.15. ∑

(x − 4)

n=1

n 2

∞

xn .

2.3.17. ∑ 3n2

n=1

∞

2.3.19. ∑

(x − 3)

n=1

n 5

∞

(x − 2)n

2.3.21. ∑

+ n2

n−1

n=1

∞

(2n − 1)n (x + 1)n

2.3.23. ∑

n−1

n=1

∞

2.3.25. ∑

(−1)n−1

(x − 5)

n=1

n 3

∞

(−1)n+1

(x −1)

2.3.27. ∑

(3n − 2)

n=1

∞

(−1)n

n + 2

2.3.29. ∑

(x − 2)n .

n=1

n + 1

∞

2n−1

2.3.12. ∑

n=1

2n + 1

∞

(x + 3)n−1

2.3.14. ∑

n= 2

3 ln n

∞

(x + 2)n

2.3.16. ∑

n=1

∞

2.3.18. ∑

+ 1)n .

(n + 1)

n=1

∞

2.3.20. ∑ nn (x + 3)n+1.

n=1

∞

n! (x + 5)n

2.3.22. ∑

n=1

∞

(3n − 2)(x − 6)n

2.3.24. ∑

(n +

n+1

n=1

∞

(x + 3)n

2.3.26. ∑

n=1

∞

(x + 1)

2.3.28. ∑

(

)

n=1

+ 1 ln (n + 1)

∞

(−1)n

(x − 3)

2.3.30. ∑

(4n + 1)

n + 1

n=1

2.4. Expand the given functions into Taylor’s series in powers of x − a and find its domain of convergence.

2.4.1.	1			,	a = 3.	2.4.2. ln(x2 + 4x + 5), a = −2.

	x2 − 6x + 5
2.4.3.	x	,	a = 2.			2.4.4. sin	2 x, a =	π .
	x + 3
								4
2.4.5.	x + 2	,	a = 1.			2.4.6. ex ,	a = 1.
	x − 5

2.4.7. ln(x2 − 6x + 10), a = 3.			2.4.8. ln(6x + 19), a = −3.
2.4.9.	1	, a = −1.	2.4.10.	1	, a = − 4.
	x2 + 2x + 3			x2 + 8x + 17

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2.4.11. ex2 − 2 x+1, a = 1.

2.4.13. ln(3x + 7),				a = −2.
2.4.15. ex2 − 4 x+1 , a = 2.
2.4.17. cos x, a =				π .
		1		2
2.4.19.				, a = 3.
	x2 + 3x − 4
2.4.21. ln(2x − 5),				a = 3.
2.4.23.	3x		, a = −1.
	x +	2

2.4.25.		1		, a = −2.
	x2 + 4x + 6
2.4.27. ln(4x − 5),				a = 2.

2.4.12. cos2 x, a = π4 .

2.4.14. cos π6x , a = 3.

2.4.16. sin πx , a = 2.
4
2.4.18. e− x , a = 2.
1
2.4.20. x2 − 9x + 20	, a = 3.

2.4.22.2x + 1, a = 2.

x− 3

2.4.24.	1	,	a = 1.
	x2 + 3x + 2
2.4.26.	2x + 4	,	a = 4.
	x2 + 4x + 3

2.4.28.2x + 3 , a = 1.

x+ 1

2.4.29.	1	, a = 1.	2.4.30.	3	, a = 2.
	x2 + x − 6			2 − x − x2

2.5. Expand the given function into Maclaurin’s series and find its domain of convergence.

2.5.1.	sin2 x			.
		x

2.5.4.	1				.
	4 16 − x
2.5.7.	1		.
	x2 − 1
2.5.10.		1− cos x				.

			x
2.5.13.				1			.
		x2 + 3x + 2

2.5.16. cos3 x .

2.5.2.	x − 2	.
	x + 1

2.5.5.	x2	.
	x − 1

2.5.8. .

327 + x3

2.5.11.ch 3x −1 .

2.5.3. sin3		x .
2.5.6.	x3		.
	x +	1

2.5.9.	1			.
	x2 + 1

2.5.12. cos2 x .

2.5.14. ln(1− 5x + 6x2 ) . 2.5.15. xe− x .

2.5.17. sh 2x .

2.5.18. 4 − x2 .

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2.5.19.x − ln(1+ x) .

2.5.22. ln(1+ x − 2x2 ) .

2.5.25. xx+−13 .

2.5.28. 2x sin2 2x − x .

2.5.20.		1		.		2.5.21.		1− cos 2x					.
	x2 −		4						x
2.5.23. (1+ x)ex .						2.5.24.	1					.
2.5.23. (1+ x)ex .						2.5.24.			9 − x2			.
									9 − x2
2.5.26.	x2		.			2.5.27.			2 − x	.
2.5.26.	x	− 2	.			2.5.27.			5 − x	.
	x	− 2							5 − x
2.5.29.		x			.	2.5.30.			x2		.
2.5.29.					.	2.5.30.			2 + x		.
		4 − x							2 + x

2.6. Evaluate the values of functions with accuracy ε.

2.6.1. 3 30 ,					ε = 0, 001.			2.6.2. ln1,1,				ε = 0, 001.
2.6.3. sin10° ,							ε = 0, 0001.	2.6.4. e−0,5 ,				ε = 0, 001.
2.6.5. cos 9° ,							ε = 0, 0001.	2.6.6.	4 85 , ε = 0, 001.
2.6.7.		105 , ε = 0, 0001.						2.6.8. sin2 42°, ε = 0, 0001.
2.6.9.	1		,	ε = 0, 001.				2.6.10.		1		, ε = 0, 001.
									5 40
	4 e
2.6.11.		6 60 , ε = 0, 0001.						2.6.12. cos2 66° , ε = 0, 0001.
2.6.13. ln1, 05 , ε = 0, 001.								2.6.14.		4 266 , ε = 0, 0001.
2.6.15. arctg(0,1) , ε = 0, 0001.								2.6.16.		3 130 , ε = 0, 001.
2.6.17.		1				,	ε = 0, 001.	2.6.18.		6 1, 2 , ε = 0, 0001.
	4 20

2.6.19. sin15° , ε = 0, 0001.								2.6.20. ln			1, 08 , ε = 0, 001.
2.6.21. sin 20° , ε = 0, 0001.								2.6.22. cos 80° , ε = 0, 0001.
2.6.23.		1			,		ε = 0, 001.	2.6.24.		3 8, 4 , ε = 0, 0001..
	5 36

2.6.25. cos 96° , ε = 0, 0001.								2.6.26. ln			1, 04 , ε = 0, 001.
2.6.27.	4 1,1 ,						ε = 0, 0001.	2.6.28.	1		,	ε = 0, 001.
										3 e

2.6.29. cos2 85° , ε = 0, 0001.								2.6.30. arctg(0, 2) , ε = 0, 0001.

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2.7. Evaluate the integrals with accuracy ε.

0,5			dx
2.7.1.	∫		dx		,	ε = 10−3 .
2.7.1.	∫	4			,	ε = 10−3 .
	0 1+ x
0,25			dx
2.7.3.	∫		dx			, ε = 10−4 .
2.7.3.	∫	1+ x		3		, ε = 10−4 .
	0	1+ x
1/ 3
2.7.5.	∫ cos x2 dx ,							ε = 10−4 .
	0
0.5
2.7.7.	∫ cos			xdx ,				ε = 10−4 .
	0
1
2.7.9. ∫ e− x2 dx ,						ε = 10−3 .
0
	0,5		dx
2.7.11.	∫		dx			, ε = 10−4 .
2.7.11.	∫			5		, ε = 10−4 .
	0	1+ x
	0,5	1− cos x
2.7.13.	∫						dx , ε = 10−4 .
2.7.13.	∫		x	2			dx , ε = 10−4 .
	0		x
	0,5	2			x
2.7.15.	∫		sin		x	dx , ε = 10−4 .
2.7.15.	∫		2			dx , ε = 10−4 .
	0		x
	0,5		dx
2.7.17.	∫		dx				,	ε = 10−3 .
2.7.17.	∫		1+ x4				,	ε = 10−3 .
	0		1+ x4

	1
2.7.19. ∫ x4 cos x2 dx , ε = 10−4 .
	0
	0,5	sin	2	x
2.7.21.	∫	sin		x		dx , ε = 10−4 .
2.7.21.	∫	x				dx , ε = 10−4 .
	0	x
	0
	0,25
2.7.23.	∫	ln(1+					x)dx , ε = 10−3 .
	0
	0,5	dx
2.7.25.	∫	dx			,		ε = 10−4 .
2.7.25.	∫	1+ x		6	,		ε = 10−4 .
	0	1+ x

0,5

2.7.2. ∫

0,5

2.7.4. ∫

1/ 3

2.7.6. ∫

dx , ε = 10−3 . 1− x3

sinx x dx , ε = 10−3 .

sin x2 dx , ε = 10−3 .

	0
1/ 4
2.7.8.	∫	sin				xdx , ε = 10−4 .
	0
	1
2.7.10. ∫				xe−		x dx ,			ε = 10−3 .
	0
	0,5
2.7.12. ∫ ln(1+ x4 )dx ,										ε = 10−4 .
	0
	0,2		ln(1+ x)
2.7.14. ∫			ln(1+ x)					dx ,		ε = 10−4 .
2.7.14. ∫								dx ,		ε = 10−4 .
	0				x
	0
	0,5			e	− x	−1
2.7.16.	∫			e		−1	dx ,		ε = 10−4 .
2.7.16.	∫				x		dx ,		ε = 10−4 .
	0				x
	0
	1
2.7.18. ∫ xe− x4 dx , ε = 10−4 .
	0
	0,5			e	− x2	− 1
2.7.20.	∫			e		− 1		dx ,		ε = 10−4 .
2.7.20.	∫							dx ,		ε = 10−4 .
	0				x
	0
	0,5
2.7.22.	∫			3 1+ x2 dx ,					ε = 10−3 .
	0
	0,5
2.7.24.	∫				1− x3 dx ,				ε = 10−3 .
	0
	0,5
2.7.26. ∫			4 1+ x3 dx ,						ε = 10−3 .
	0

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	0,5				0,5
2.7.27.	∫ cos x3 dx , ε = 10−4 .				2.7.28.	∫ sin x3dx , ε = 10−4 .
	0					0
	0,5	dx			1		dx
2.7.29.	∫	dx		, ε = 10−3 .	2.7.30. ∫		dx		, ε = 10−3 .
2.7.29.	∫	8 + x	3	, ε = 10−3 .	2.7.30. ∫	16 + x		4	, ε = 10−3 .
	0	8 + x			0	16 + x

2.8. Find the approximate solution of the Cauchy’s problem using the first four nonzero members of expansion of this solution in power series.

2.8.1.y′ = x3 + y3 , y(0) = 1 .

2.8.2.y′ = xy2 + y4 , y(0) = 2 .

2.8.3.y′ = x2 y + y4 , y(0) = −1 .

2.8.4. y′′ − xy′ + y + ex = 0 , y(0) = 1, y′(0) = −1 .

2.8.5.y′′ + 2xy′ + y2 − x3 = 0 , y(0) = 0, y′(0) = 2 .

2.8.6.y′′ − yy′ + xy + x2 = 0 , y(0) = 1, y′(0) = 3 .

2.8.7.y′′ + yy′ + y + x3 = 0 , y(0) = 2, y′(0) = 1 .

2.8.8. y′ + x2 y2 = xy,	y(2) = 1.
2.8.9. xy′ = y2 − ex ,	y(0) = −2 .
2.8.10. yy′ + y3 = x2 ,	y(1) = 1.

2.8.11.x2 y′ = y2 − 2x2 , y(0) = 1 .

2.8.12.xy′ + y3 = ex x , y(0) = 1 .


2.8.13.	y′′ + yy′ − xy + x2 = 0 , y(0) = 1, y′(0) = −1 .
2.8.14.	y′′ + xy′ + xy2 = 0 , y(0) = −1,				y′(0) = 2 .
2.8.15.	y′ = 4 + y / x + ( y / x)2 ,			y(1) = 2 .
2.8.16.	y′′ + xy2 = y ,	y(0) = 2,		y′(0) = 3 .
2.8.17.	y′′ + 4xy′ − 2 y2	= 0 , y(0) = 2,			y′(0) = −2 .
2.8.18.	y′′ + 2 y′ + y4 = x , y(0) = −1,				y′(0) = −2 .
2.8.19.	y′′ + 4yy′ − 3x2	= 0 ,	y(0) = 1,		y′(0) = −1 .
2.8.20.	y′′ + ( y − 2x) y′ − x2		= 0 ,	y(0) = 1, y′(0) = 2 .

2.8.21.xy′ − y( y + x) = x4 , y(1) = −1 .

2.8.22.( y − x) y′ = y2 − 3xy , y(1) = 1.

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2.8.23.	yy′ = xy + x , y(1) = 1 .
2.8.24.	y′′ + yex + ey = 0 ,	y(0) = 1,	y′(0) = 1.
2.8.25.	y′ + 3x2 y = x5 y3 ,	y(0) = 3 .
2.8.26.	y′(x + 2 y) = x3 y ,	y(0) = 2 .
2.8.27.	y′′ + x3 + 3y2 = 0 ,	y(0) = 2,	y′(0) = 1.

2.8.28.x2 y′ − y2 = x3 , y(1) = 0 .

2.8.29.y′(x + y) = ye2x , y(0) = −1 .

2.8.30.yy′ + xey = ex , y(0) = −2 .

Micromodule 3

BASIC THEORETICAL INFORMATION.

FOURIER SERIES

Trigonometric Fourier series. Fourier coefficients. Dirichlet’s theorem.

Fourier series for odd and even functions. Fourier series for 2π-and 2l-periodic functions. Fourier series for functions defined on a segment [0; l] or on arbitrary segment [a; b]. Complex form of Fourier series.

Key words: Fourier series — ряд Фур’є, trigonometric series — триго-

нометричний ряд, orthogonal functions — ортогональні функції, piecewise monotone — кусково монотонний, Weierstrass’ test — ознака Вейєрштрасса, Dirichlet’s theorem — теорема Діріхлє, to extend — продовжувати, in even way — парним чином, in odd way — непарним чином, to expand — розкладати, Fourier coefficients — коефіцієнти Фур’є, integrable function —

інтегровна функція.

Literature: [3, chapter 5, section 5.6], [9, chapter 9, §3], [14, chapter 3, §3], [15, chapter 13, section 13.4], [16, chapter 17, §1— 6], [17, chapter 6, §20—21].

3.1. Trigonometric Fourier Series. Fourier Coefficients

Definition. A functional series

a20 + a1 cos x + b1 sin x + a2 cos x + b2 sin x + ... + an cos nx +

		a0	∞
+bn	sin nx + ... =	a0	+ ∑ (an cosnx+bn sin nx)	(3.1)
+bn	sin nx + ... =		+ ∑ (an cosnx+bn sin nx)	(3.1)
	2		n=1
			n=1
				69

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is called a trigonometric series. The constants a0 , an , bn ( n N ) are called

the coefficients of trigonometric series.

If series (3.1) converges then its sum is a periodic function with period 2π because cos nx and sin nx are periodic functions with period 2π. ( f (x) = f (x+2π)).

Suppose a periodic function				f (x)	with period	2π may be represented as a
trigonometric series convergent to the given function within interval							(−π; π) .
That is				∞
		f (x) =	a0		(a cos nx + b sin nx).		(3.2)
				+ ∑

2				n =	n	n
					1
We assume that the series
	a0	+ \| a1 \| + \| b1 \| + \| a2 \| + \| b2 \| +…+ \| an \| + \| bn \|… is convergent.

	2

Then the series (3.1) is absolutely and uniformly convergent by Weierstrass’ test. We can integrate this series term-by-term.

For calculation of coefficients of series (3.2) we’ll use the following formulas. Suppose m and n are natural numbers. Then

∫

−π

∫π	cos nxdx = 0, if n ≠ 0,
− π	2π, if n = 0;

∫sin nxdx = 0;

−π

	0,	if	m ≠ n,
cos mx cos nxdx =		if	m = n;
	π,
π	0,		if	m ≠ n,
∫
	sin mxsin nxdx =		if	m = n;
−π	π,

∫ sin mx cos nxdx = 0.

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

−π

For calculation of integrals (3.5) — (3.7) we should use formulas of transformation of product of trigonometric functions to the sum.

Note. A sequence of functions ϕ1 (x), ϕ2 (x), …, ϕn (x), …is called orthogonal on a segment [a; b] if

∫ ϕi (x)ϕ j (x)dx = 0 ( i ≠ j ),

∫ ϕi2 (x)dx = λi > 0 ( i = j ).

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