T1.2. INFINITE SERIES 1123 16. ∞ k=1 (–1) k k 2n+1 sin(kx)= (–1) n–1 (2π) 2n+1 2(2n + 1)! B 2n+1 x + π 2π , where –π < x ≤ π for n = 0, 1, ;theB n (x) are Bernoulli polynomials. 17. ∞ k=1 1 k! sin(kx)=exp(cosx) sin(sin x), x is any number. 18. ∞ k=1 (–1) k k! sin(kx)=–exp(–cosx) sin(sin x), x is any number. 19. ∞ k=0 1 (2k)! sin(kx)=sin sin x 2 sinh cos x 2 , x is any number. 20. ∞ k=0 (–1) k (2k)! sin(kx)=–sin cos x 2 sinh sin x 2 , x is 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 sinx 1 – 2a cos x + a 2 , |a| < 1, x is any number. 23. ∞ k=1 ka k sin(kx)= a(1 – a 2 )sinx (1 – 2a cos x + a 2 ) 2 , |a| < 1, x is any number. 24. ∞ k=1 1 k sin(kx + a)= 1 2 (π – x)cosa –ln 2 sin x 2 sin a, 0 < x < 2π. 25. ∞ k=1 (–1) k–1 k sin(kx + a)= 1 2 x cos a +ln 2 cos x 2 sin a,–π < x < π. 26. ∞ k=1 sin[(2k – 1)x] 2k – 1 = π 4 , 0 < x < π. 27. ∞ k=1 (–1) k–1 sin[(2k – 1)x] 2k – 1 = 1 2 ln tan x 2 + π 4 ,– π 2 < x < π 2 . 28. ∞ k=1 a 2k–1 sin[(2k – 1)x] 2k – 1 = 1 2 arctan 2a sinx 1 – a 2 , 0 < x < 2π, |a| ≤ 1. 29. ∞ k=1 (–1) k–1 a 2k–1 sin[(2k – 1)x] 2k – 1 = 1 4 ln 1 + 2a sin x + a 2 1 – 2a sin x + a 2 , 0 < x < π, |a| ≤ 1. 30. ∞ k=1 (–1) k sin[(k + 1)x] k(k + 1) =sinx – 1 2 x(1 +cosx)–sinx ln 2 cos x 2 . 1124 FINITE SUMS AND INFINITE SERIES 31. ∞ k=0 a 2k+1 sin[(2k + 1)x]= a(1 + a 2 )sinx (1 + a 2 ) 2 – 4a 2 cos 2 x , |a| < 1, x is any number. 32. ∞ k=0 (–1) k a 2k+1 sin[(2k + 1)x]= a(1 – a 2 )sinx (1 + a 2 ) 2 – 4a 2 sin 2 x , |a| < 1, x is any number. 33. ∞ k=1 sin[2(k + 1)x] k(k + 1) =sin(2x)–(π – 2x)sin 2 x –sinxcos x ln(4 sin 2 x), 0 ≤ x ≤ π. 34. ∞ k=1 (–1) k sin[(2k + 1)x] (2k + 1) 2 = 1 4 πx if – 1 2 π ≤ x ≤ 1 2 π, 1 4 π(π – x)if 1 2 π ≤ x ≤ 3 2 π. T1.2.2-3. Trigonometric series in one variable involving cosine. 1. ∞ k=1 1 k cos(kx)=–ln 2 sin x 2 , 0 < x < 2π. 2. ∞ k=1 (–1) k–1 k cos(kx)=ln 2 cos x 2 ,–π < x < π. 3. ∞ k=1 a k k cos(kx)=ln 1 √ 1 – 2a cos x + a 2 , 0 < x < 2π, |a| ≤ 1. 4. ∞ k=0 1 2k + 1 cos(kx)= π 4 sin x 2 +cos x 2 ln cot 2 x 4 , 0 < x < 2π. 5. ∞ k=0 (–1) k 2k + 1 cos(kx)=– 1 4 sin x 2 ln cot 2 x + π 4 + π 4 cos x 2 ,–π < x < π. 6. ∞ k=1 1 k 2 cos(kx)= 1 12 (3x 2 – 6πx + 2π 2 ), 0 ≤ x ≤ 2π. 7. ∞ k=1 (–1) k k 2 cos(kx)= 1 12 (3x 2 – π 2 ), –π ≤ x ≤ π. 8. ∞ k=1 1 k(k + 1) cos(kx)= 1 2 (x – π)sinx – 2 sin 2 x 2 ln 2 sin x 2 + 1, 0 ≤ x ≤ 2π. 9. ∞ k=1 (–1) k k(k + 1) cos(kx)=– 1 2 x sin x – 2 cos 2 x 2 ln 2 cos x 2 + 1,–π ≤ x ≤ π. 10. ∞ k=1 1 k 2 + a 2 cos(kx)= π 2a sinh(πa) cosh[a(π – x)] – 1 2a 2 , 0 ≤ x ≤ 2π. 11. ∞ k=1 1 k 2 – a 2 cos(kx)=– π 2a sin(πa) cos[a(π – x)] + 1 2a 2 , 0 ≤ x ≤ 2π. T1.2. INFINITE SERIES 1125 12. ∞ k=2 (–1) k k 2 – 1 cos(kx)= 1 2 – 1 4 cos x – 1 2 x sin x,–π ≤ x ≤ π. 13. ∞ k=2 k k 2 – 1 cos(kx)=– 1 2 – 1 4 cos x –cosx ln 2 sin x 2 , 0 < x < 2π. 14. ∞ k=1 1 k 2n cos(kx)= (–1) n–1 (2π) 2n 2(2n)! B 2n x 2π , where 0 ≤ x ≤ 2π for n = 1, 2, ;theB n (x) are Bernoulli polynomials. 15. ∞ k=1 (–1) k k 2n cos(kx)= (–1) n–1 (2π) 2n 2(2n)! B 2n x + π 2π , where –π ≤ x ≤ π for n = 1, 2, ;theB n (x) are Bernoulli polynomials. 16. ∞ k=0 1 k! cos(kx)=exp(cosx)cos(sinx), x is any number. 17. ∞ k=0 (–1) k k! cos(kx)=exp(–cosx)cos(sinx), x is any number. 18. ∞ k=0 1 (2k)! cos(kx)=cos sin x 2 cosh cos x 2 , x is any number. 19. ∞ k=0 (–1) k (2k)! cos(kx)=cos cos x 2 cosh sin x 2 , x is 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 + a 2 , |a| < 1, x is any number. 22. ∞ k=1 ka k cos(kx)= a(1 + a 2 )cosx – 2a 2 (1 – 2a cos x + a 2 ) 2 , |a| < 1, x is any number. 23. ∞ k=1 1 k cos(kx + a)= 1 2 (x – π)sina –ln 2 sin x 2 cos a, 0 < x < 2π. 24. ∞ k=1 (–1) k–1 k cos(kx + a)=– 1 2 x sin a +ln 2 cos x 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–1 cos[(2k – 1)x] 2k – 1 = π 4 , 0 < x < π. 1126 FINITE SUMS AND INFINITE SERIES 27. ∞ k=1 a 2k–1 cos[(2k – 1)x] 2k – 1 = 1 4 ln 1 + 2a cos x + a 2 1 – 2a cos x + a 2 , 0 < x < 2π, |a| ≤ 1. 28. ∞ k=1 (–1) k–1 a 2k–1 cos[(2k – 1)x] 2k – 1 = 1 2 arctan 2a cos x 1 – a 2 , 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) =cosx – 1 2 x sin x –(1 +cosx)ln 2 cos x 2 . 31. ∞ k=0 a 2k+1 cos[(2k + 1)x]= a(1 – a 2 )cosx (1 + a 2 ) 2 – 4a 2 cos 2 x , |a| < 1, x is any number. 32. ∞ k=0 (–1) k a 2k+1 cos[(2k + 1)x]= a(1 + a 2 )cosx (1 + a 2 ) 2 – 4a 2 sin 2 x , |a| < 1, x is any number. 33. ∞ k=1 cos[2(k + 1)x] k(k + 1) =cos(2x)– π 2 – x sin(2x)+sin 2 x ln(4 sin 2 x), 0 ≤ x ≤ π. T1.2.2-4. Trigonometric series in two variables. 1. ∞ k=1 1 k sin(kx)sin(ky)= 1 2 ln sin x + y 2 cosec x – y 2 , x y ≠ 0, 2π, 4π, 2. ∞ k=1 (–1) k k sin(kx)sin(ky)= 1 2 ln cos x + y 2 sec x – y 2 , x y ≠ π, 3π, 5π, 3. ∞ k=1 1 k 2 sin(kx)sin(ky)= 1 2 x(π – y)if–y ≤ x ≤ y, 1 2 y(π – x)ify ≤ x ≤ 2π – y. Here, 0 < y < π. 4. ∞ k=1 (–1) k+1 k 2 sin(kx)sin(ky)= 1 2 xy, |x y| ≤ π. 5. ∞ k=1 a k k sin(kx)sin(ky)= 1 4 ln 4a sin 2 [(x + y)/2]+(a – 1) 2 4a sin 2 [(x – y)/2]+(a – 1) 2 , 0 < a < 1. 6. ∞ k=1 1 k 2 sin 2 (kx)sin 2 (ky)= 1 2 πx, 0 ≤ x ≤ y ≤ π 2 . 7. ∞ k=1 1 k cos(kx)cos(ky)=– 1 2 ln 2(cos x –cosy) , x y ≠ 0, 2π, 4π, 8. ∞ k=1 (–1) k k cos(kx)cos(ky)=– 1 2 ln 2(cos x +cosy) , x y ≠ π, 3π, 5π, REFERENCES FOR CHAPTER T1 1127 9. ∞ k=1 1 k sin(kx)cos(ky)= ⎧ ⎪ ⎨ ⎪ ⎩ – 1 2 if 0 < x < y, 1 4 (π – 2y)ifx = y, 1 2 (π – x)ify < x < π. Here, 0 < y < π. 10. ∞ k=1 1 k 2 cos(kx)cos(ky)= 1 12 3x 2 + 3(y – π) 2 – π 2 if 0 ≤ x ≤ y, 1 12 3y 2 + 3(x – π) 2 – π 2 if y ≤ x ≤ π. Here, 0 < y < π. 11. ∞ k=1 (–1) k k 2 cos(kx)cos(ky)= 1 12 3(x 2 +y 2 )–π 2 if –(π –y) ≤ x ≤ π–y, 1 12 3(x–π) 2 +3(y –π) 2 –π 2 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.andRyzhik,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. Chapter T2 Integrals T2.1. 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. xdx a + bx = 1 b 2 a + bx – a ln |a + bx| . 4. x 2 dx a + bx = 1 b 3 1 2 (a + bx) 2 – 2a(a + bx)+a 2 ln |a + bx| . 5. dx x(a + bx) =– 1 a ln a + bx x . 6. dx x 2 (a + bx) =– 1 ax + b a 2 ln a + bx x . 7. xdx (a + bx) 2 = 1 b 2 ln |a + bx| + a a + bx . 8. x 2 dx (a + bx) 2 = 1 b 3 a + bx – 2a ln |a + bx| – a 2 a + bx . 9. dx x(a + bx) 2 = 1 a(a + bx) – 1 a 2 ln a + bx x . 10. xdx (a + bx) 3 = 1 b 2 – 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 . 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,. π. References for Chapter T1 Dwight, H. B., Tables of Integrals and Other Mathematical Data, Macmillan, New York, 1961. Gradshteyn,I.S.andRyzhik,I.M.,Tables of Integrals, Series, and Products,. 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