1124 FINITESUMS ANDINFINITESERIES31... 1126 FINITESUMS ANDINFINITESERIES27... B., Tables of Integrals and Other Mathematical Data, Macmillan, New York, 1961.. M., Tables of Integrals, Se
Trang 1∞
k=1
(–1)k
k2n+1 sin(kx) =
(–1)n–1(2π)2n+1
2(2n+1)! B2n+1
x + π
2π ,
where –π < x≤π for n =0, 1, ; the B n (x) are Bernoulli polynomials.
17
∞
k=1
1
k! sin(kx) = exp(cos x) sin(sin x), xis any number.
18
∞
k=1
(–1)k
k! sin(kx) = – exp(– cos x) sin(sin x), xis any number.
19
∞
k=0
1
(2k)! sin(kx) = sin
sinx
2
sinh
cosx
2
, xis any number
20
∞
k=0
(–1)k
(2k)! sin(kx) = – sin
cosx
2
sinh
sin x
2
, xis any number
21
∞
k=0
a k
k! sin(kx) = exp(k cos x) sin(k sin x), |a| ≤ 1, x is any number.
22
∞
k=0
a k sin(kx) = a sin x
1–2a cos x + a2, |a|<1, x is any number.
23
∞
k=1
ka k sin(kx) = a(1– a2) sin x
(1–2a cos x + a2)2, |a|<1, x is any number.
24
∞
k=1
1
k sin(kx + a) = 1
2(π – x) cos a – ln
2sin x
2
sin a, 0< x <2π
25
∞
k=1
(–1)k–1
k sin(kx + a) = 1
2x cos a + ln
2cosx2
sin a, –π < x < π.
26
∞
k=1
sin[(2k–1)x]
2k–1 =
π
4, 0< x < π.
27
∞
k=1
(–1)k–1sin[(2k–1)x]
2k–1 =
1
2ln tan
x
2 +
π
4
, –π
2 < x <
π
2.
28
∞
k=1
a2k–1sin[(2k–1)x]
2k–1 =
1
2arctan
2a sin x
1– a2 , 0< x <2π, |a| ≤ 1 29
∞
k=1
(–1)k–1a2k–1sin[(2k–1)x]
2k–1 =
1
4 ln
1+2a sin x + a2
1–2a sin x + a2, 0< x < π, |a| ≤ 1 30
∞
k=1
(–1)k sin[(k +1)x]
k (k +1) = sin x –
1
2x(1+ cos x) – sin x ln
2cos x
2
Trang 2
1124 FINITESUMS ANDINFINITESERIES
31
∞
k=0
a2k+1sin[(2k+1)x] = a(1+ a2) sin x
(1+ a2)2–4a2cos2x, |a|<1, x is any number.
32
∞
k=0
(–1)k a2k+1sin[(2k+1)x] = a(1– a2) sin x
(1+ a2)2–4a2sin2x, |a|<1, x is any number.
33
∞
k=1
sin[2(k +1)x]
k (k +1) = sin(2x ) – (π –2x) sin2x – sin x cos x ln(4sin2x), 0 ≤x≤π 34
∞
k=1
(–1)ksin[(2k+1)x]
(2k+1)2 =
1
4πx if –12π ≤x≤ 1
2π, 1
4π (π – x) if 12π ≤x≤ 3
2π.
T1.2.2-3 Trigonometric series in one variable involving cosine
1
∞
k=1
1
k cos(kx) = – ln
2sin x
2
, 0< x <2π
2
∞
k=1
(–1)k–1
k cos(kx) = ln
2cosx
2
, –π < x < π.
3
∞
k=1
a k
k cos(kx) = ln √ 1
1–2a cos x + a2, 0< x <2π, |a| ≤ 1 4
∞
k=0
1
2k+1 cos(kx) =
π
4 sin
x
2 + cos
x
2 ln
cot2 x
4
, 0< x <2π
5
∞
k=0
(–1)k
2k+1 cos(kx) = –
1
4 sin
x
2 ln
cot2 x + π
4
+ π
4 cos
x
2, –π < x < π.
6
∞
k=1
1
k2 cos(kx) =
1
12(3x2–6πx+2π2), 0 ≤x≤ 2π.
7
∞
k=1
(–1)k
k2 cos(kx) =
1
12(3x2– π2), –π≤x≤π.
8
∞
k=1
1
k (k +1) cos(kx) =
1
2(x – π) sin x –2sin2
x
2 ln
2sin x
2
+1, 0 ≤x≤ 2π
9
∞
k=1
(–1)k
k (k +1) cos(kx) = –
1
2x sin x –2cos2
x
2 ln
2cosx
2
+1, –π ≤x≤π
10
∞
k=1
1
k2+ a2 cos(kx) =
π
2a sinh(πa) cosh[a(π – x)] –
1
2a2, 0 ≤x≤ 2π 11
∞
k=1
1
k2– a2 cos(kx) = –
π
2a sin(πa) cos[a(π – x)] +
1
2a2, 0 ≤x≤ 2π
Trang 3∞
k=2
(–1)k
k2–1 cos(kx) =
1
2 –
1
4cos x –
1
2x sin x, –π ≤x≤π.
13
∞
k=2
k
k2–1 cos(kx) = –
1
2 –
1
4cos x – cos x ln
2sin x
2
, 0< x <2π
14
∞
k=1
1
k2n cos(kx) =
(–1)n–1(2π)2n
2(2n)! B2n
x
2π
, where0 ≤x≤ 2π for n =1, 2, ; the B n (x) are Bernoulli polynomials.
15
∞
k=1
(–1)k
k2n cos(kx) =
(–1)n–1(2π)2n
2(2n)! B2n
x + π
2π
,
where –π≤x≤π for n =1,2, ; the B n (x) are Bernoulli polynomials.
16
∞
k=0
1
k! cos(kx) = exp(cos x) cos(sin x), xis any number.
17
∞
k=0
(–1)k
k! cos(kx) = exp(– cos x) cos(sin x), xis any number.
18
∞
k=0
1
(2k)! cos(kx) = cos
sin x
2
cosh
cos x
2
, xis any number
19
∞
k=0
(–1)k
(2k)! cos(kx) = cos
cos x
2
cosh
sin x
2
, xis any number
20
∞
k=0
a k
k! cos(kx) = exp(a cos x) cos(a sin x), |a| ≤ 1, x is any number.
21
∞
k=0
a k cos(kx) = 1– a cos x
1–2a cos x + a2, |a|<1, x is any number.
22
∞
k=1
ka k cos(kx) = a(1+ a2) cos x –2a2
(1–2a cos x + a2)2 , |a|<1, x is any number.
23
∞
k=1
1
k cos(kx + a) = 1
2(x – π) sin a – ln
2sin x
2
cos a, 0< x <2π
24
∞
k=1
(–1)k–1
k cos(kx + a) = –1
2x sin a + ln
2cosx
2
cos a, –π < x < π.
25
∞
k=1
cos[(2k–1)x]
2k–1 =
1
2 ln cot
x
2, 0< x < π.
26
∞
k=1
(–1)k–1cos[(2k–1)x]
2k–1 =
π
4, 0< x < π.
Trang 41126 FINITESUMS ANDINFINITESERIES
27
∞
k=1
a2k–1cos[(2k–1)x]
2k–1 =
1
4 ln
1+2a cos x + a2
1–2a cos x + a2, 0< x <2π, |a| ≤ 1 28
∞
k=1
(–1)k–1a2k–1cos[(2k–1)x]
2k–1 =
1
2arctan
2a cos x
1– a2 , 0< x < π, |a| ≤ 1 29
∞
k=1
cos[(2k–1)x]
(2k–1)2 =
π
4
π
2 –|x|
, –π ≤x≤π
30
∞
k=1
(–1)k cos[(k +1)x]
k (k +1) = cos x –
1
2x sin x – (1+ cos x) ln
2cos x
2
31
∞
k=0
a2k+1cos[(2k+1)x] = a(1– a2) cos x
(1+ a2)2–4a2cos2x, |a|<1, x is any number.
32
∞
k=0
(–1)k a2k+1cos[(2k+1)x] = a(1+ a2) cos x
(1+ a2)2–4a2sin2x, |a|<1, x is any number.
33
∞
k=1
cos[2(k +1)x]
k (k +1) = cos(2x) –
π
2 – x
sin(2x) + sin2xln(4sin2x), 0 ≤x≤π
T1.2.2-4 Trigonometric series in two variables
1
∞
k=1
1
k sin(kx) sin(ky) = 1
2ln
sin x + y2 cosec x – y
2
, x y≠ 0, 2π, 4π ,
2
∞
k=1
(–1)k
k sin(kx) sin(ky) = 1
2ln
cos x + y2 secx – y
2
, x y≠π, 3π, 5π ,
3
∞
k=1
1
k2 sin(kx) sin(ky) =
1
2x (π – y) if –y ≤x≤y,
1
2y (π – x) if y ≤x≤ 2π – y. Here, 0< y < π.
4
∞
k=1
(–1)k+1
k2 sin(kx) sin(ky) =
1
2xy, |x y| ≤π.
5
∞
k=1
a k
k sin(kx) sin(ky) = 1
4 ln
4asin2[(x + y)/2] + (a –1)2
4asin2[(x – y)/2] + (a –1)2, 0< a <1 6
∞
k=1
1
k2 sin2(kx) sin2(ky) =
1
2πx, 0 ≤x≤y≤
π
2.
7
∞
k=1
1
k cos(kx) cos(ky) = –1
2 ln2(cos x – cos y), x y≠ 0, 2π, 4π ,
8
∞
k=1
(–1)k
k cos(kx) cos(ky) = –1
2 ln2(cos x + cos y), x y≠π, 3π,5π ,
Trang 5∞
k=1
1
k sin(kx) cos(ky) =
⎪
⎪
–12 if 0< x < y,
1
4(π –2y) if x = y,
1
2(π – x) if y < x < π.
Here, 0< y < π.
10
∞
k=1
1
k2 cos(kx) cos(ky) =
1 12
3x2+3(y – π)2– π2
if 0 ≤x≤y,
1 12
3y2+3(x – π)2– π2
if y≤x≤π Here, 0< y < π.
11
∞
k=1
(–1)k
k2 cos(kx) cos(ky) =
1 12
3(x2+ y2) – π2
if –(π – y)≤x≤π – y,
1 12
3(x – π)2+3(y – π)2– π2
if π – y≤x≤π + y.
Here, 0< y < π.
References for Chapter T1
Dwight, H B., Tables of Integrals and Other Mathematical Data, Macmillan, New York, 1961.
Gradshteyn, I S and Ryzhik, I M., Tables of Integrals, Series, and Products, 6th Edition, Academic Press,
New York, 2000.
Hansen, E R., A Table of Series and Products, Printice Hall, Englewood Cliffs, London, 1975.
Mangulis, V., Handbook of Series for Scientists and Engineers, Academic Press, New York, 1965.
Prudnikov, A P., Brychkov, Yu A., and Marichev, O I., Integrals and Series, Vol 1, Elementary Functions,
Gordon & Breach, New York, 1986.
Zwillinger, D., CRC Standard Mathematical Tables and Formulae, 31st Edition, CRC Press, Boca Raton, 2002.
Trang 7T2.1 Indefinite Integrals
Throughout Section T2.1, the integration constant C is omitted for brevity.
T2.1.1 Integrals Involving Rational Functions
T2.1.1-1 Integrals involving a + bx.
1
dx
a + bx =
1
b ln|a + bx| 2
(a + bx) n dx= (a + bx) n+
1
b (n +1) , n≠–1 3
x dx
a + bx =
1
b2 a + bx – a ln|a + bx| 4
x2dx
a + bx =
1
b3
*1
2(a + bx)2–2a (a + bx) + a2ln|a + bx|
+
5
dx
x (a + bx) = –
1
alna + bx
x
6
dx
x2(a + bx) = –
1
ax + b
a2 lna + bx
x
7
x dx
(a + bx)2 =
1
b2
ln|a + bx|+ a
a + bx
8
x2dx
(a + bx)2 =
1
b3
a + bx –2aln|a + bx|– a
2
a + bx
9
dx
x (a + bx)2 =
1
a (a + bx) –
1
a2 lna + bx
x
10
x dx
(a + bx)3 =
1
b2
*
– 1
a + bx +
a
2(a + bx)2
+
T2.1.1-2 Integrals involving a + x and b + x.
1
a + x
b + x dx = x + (a – b) ln|b + x|
1129