1 Elementary Functions - Elsevier.com · 2013. 12. 20. · 3. cos2 x−sin2 y =cos(x+y)cos(x−y) = cos2 y −sin2 x 4. sinh2 x+cosh2 y =cosh(x+y)cosh(x−y)=cosh2 x+sinh2 y 1.316

1 Elementary Functions

1.1 Power of Binomials

1.11 Power series

1.110 (1 + x)q = 1 + qx +q(q − 1)

2!x2 + · · · + q(q − 1) . . . (q − k + 1)

k!xk + · · · =

∞∑

k=0

( q

k

)xk

If q is neither a natural number nor zero, the series converges absolutely for |x| < 1 and diverges for|x| > 1. For x = 1, the series converges for q > −1 and diverges for q ≤ −1. For x = 1, the seriesconverges absolutely for q > 0. For x = −1, it converges absolutely for q > 0 and diverges for q < 0. Ifq = n is a natural number, the series 1.110 is reduced to the finite sum 1.111. FI II 425

1.111 (a + x)n =n∑

k=0

(n

k

)xkan−k

1.112

1. (1 + x)−1 = 1 − x + x2 − x3 + · · · =∞∑

k=1

(−1)k−1xk−1

(see also 1.121 2)

2. (1 + x)−2 = 1 − 2x + 3x2 − 4x3 + · · · =∞∑

k=1

(−1)k−1kxk−1

3.11 (1 + x)1/2 = 1 +12x − 1 · 1

2 · 4x2 +1 · 1 · 32 · 4 · 6x3 − 1 · 1 · 3 · 5

2 · 4 · 6 · 8x4 + . . .

4. (1 + x)−1/2 = 1 − 12x +

1 · 32 · 4x2 − 1 · 3 · 5

2 · 4 · 6x3 + . . .

1.113x

(1 − x)2=

∞∑

k=1

kxk[x2 < 1

]

1.114

1.(1 +

√1 + x

)q= 2q

[1 +

q

1!

(x

4

)+

q(q − 3)2!

(x

4

)2

+q(q − 4)(q − 5)

3!

(x

4

)3

+ . . .

]

[x2 < 1, q is a real number

]

AD (6351.1)

25

26 The Exponential Function 1.121

2.(x +

√1 + x2

)q

= 1 +∞∑

k=0

q2(q2 − 22

) (q2 − 42

). . .

[q2 − (2k)2

]x2k+2

(2k + 2)!

+qx + q∞∑

k=1

(q2 − 12

) (q2 − 32

). . .

[q2 − (2k − 1)2

]

(2k + 1)!x2k+1

[x2 < 1, q is a real number

]AD(6351.2)

1.12 Series of rational fractions

1.121

1.x

1 − x=

∞∑

k=1

2k−1x2k−1

1 + x2k−1 =∞∑

k=1

x2k−1

1 − x2k

[x2 < 1

]AD (6350.3)

2.1

x − 1=

∞∑

k=1

2k−1

x2k−1 + 1

[x2 > 1

]AD (6350.3)

1.2 The Exponential Function

1.21 Series representation

1.211

1.11 ex =∞∑

k=0

xk

k!

2. ax =∞∑

k=0

(x ln a)k

k!

3. e−x2=

∞∑

k=0

(−1)k x2k

k!

4.∗ ex = limn→∞

(1 +

x

n

)n

1.212 ex(1 + x) =∞∑

k=0

xk(k + 1)k!

1.213x

ex − 1= 1 − x

2+

∞∑

k=1

B2kx2k

(2k)![x < 2π] FI II 520

1.214 eex

= e

(1 + x +

2x2

2!+

5x3

3!+

15x4

4!+ . . .

)AD (6460.3)

1.215

1. esin x = 1 + x +x2

2!− 3x4

4!− 8x5

5!− 3x6

6!+

56x7

7!+ . . . AD (6460.4)

2. ecos x = e

(1 − x2

2!+

4x4

4!− 31x6

6!+ . . .

)AD (6460.5)

1.232 Series of exponentials 27

3. etan x = 1 + x +x2

2!+

3x3

3!+

9x4

4!+

37x5

5!+ . . . AD (6460.6)

1.216

1. earcsin x = 1 + x +x2

2!+

2x3

3!+

5x4

4!+ . . . AD (6460.7)

2. earctan x = 1 + x +x2

2!− x3

3!− 7x4

4!+ . . . AD (6460.8)

1.217

1. πeπx + e−πx

eπx − e−πx= x

∞∑

k=−∞

1x2 + k2

(cf. 1.421 3) AD (6707.1)

2.2π

eπx − e−πx= x

∞∑

k=−∞

(−1)k

x2 + k2(cf. 1.422 3) AD (6707.2)

1.22 Functional relations

1.2211. ax = ex ln a

2. aloga x = a1

logx a = x

1.222

1. ex = cosh x + sinh x

2. eix = cos x + i sin x

1.223 eax − ebx = (a − b)x exp[12(a + b)x

] ∞∏

k=1

[1 +

(a − b)2x2

2k2π2

]MO 216

1.23 Series of exponentials

1.231∞∑

k=0

akx =1

1 − ax[a > 1 and x < 0 or 0 < a < 1 and x > 0]

1.232

1. tanhx = 1 + 2∞∑

k=1

(−1)ke−2kx [x > 0]

2. sech x = 2∞∑

k=0

(−1)ke−(2k+1)x [x > 0]

3. cosechx = 2∞∑

k=0

e−(2k+1)x [x > 0]

4.∗ sin x = exp

[

−∞∑

n=1

cos2n x

2n

]

[0 ≤ x ≤ π]

28 Trigonometric and Hyperbolic Functions 1.311

1.3–1.4 Trigonometric and Hyperbolic Functions

1.30 Introduction

The trigonometric and hyperbolic sines are related by the identities

sinh x =1i

sin(ix), sinx =1i

sinh(ix).

The trigonometric and hyperbolic cosines are related by the identities

cosh x = cos(ix), cos x = cosh(ix).

Because of this duality, every relation involving trigonometric functions has its formal counterpart involv-ing the corresponding hyperbolic functions, and vice versa. In many (though not all) cases, both pairs ofrelationships are meaningful.

The idea of matching the relationships is carried out in the list of formulas given below. However, notall the meaningful “pairs” are included in the list.

1.31 The basic functional relations

1.311

1. sin x=12i

(eix − e−ix

)

= −i sinh(ix)

2. sinh x=12

(ex − e−x

)

= −i sin(ix)

3. cosx=12

(eix + e−ix

)

= cosh(ix)

4. cosh x=12

(ex + e−x

)

= cos(ix)

5. tanx =sin x

cos x=

1i

tanh(ix)

6. tanhx =sinh x

cosh x=

1i

tan(ix)

7. cotx =cos x

sin x=

1tanx

= i coth(ix)

8. coth x =cosh x

sinh x=

1tanh x

= i cot (ix)

1.312

1. cos2 x + sin2 x = 1

1.314 The basic functional relations 29

2. cosh2 x − sinh2 x = 1

1.3131. sin (x ± y) = sin x cos y ± sin y cos x

2. sinh (x ± y) = sinhx cosh y ± sinh y cosh x

3. sin (x ± iy) = sin x cosh y ± i sinh y cos x

4. sinh (x ± iy) = sinh x cos y ± i sin y cosh x

5. cos (x ± y) = cosx cos y ∓ sinx sin y

6. cosh (x ± y) = cosh x cosh y ± sinh x sinh y

7. cos (x ± iy) = cos x cosh y ∓ i sin x sinh y

8. cosh (x ± iy) = cosh x cos y ± i sinh x sin y

9. tan (x ± y) =tan x ± tan y

1 ∓ tan x tan y

10. tanh (x ± y) =tanh x ± tanh y

1 ± tanh x tanh y

11. tan (x ± iy) =tanx ± i tanh y

1 ∓ i tan x tanh y

12. tanh (x ± iy) =tanh x ± i tan y

1 ± i tanh x tan y

1.314

1. sin x ± sin y = 2 sin12

(x ± y) cos12

(x ∓ y)

2. sinh x ± sinh y = 2 sinh12

(x ± y) cosh12

(x ∓ y)

3. cosx + cos y = 2 cos12(x + y) cos

12(x − y)

4. cosh x + cosh y = 2 cosh12(x + y) cosh

12(x − y)

5. cosx − cos y = 2 sin12(x + y) sin

12(y − x)

6. cosh x − cosh y = 2 sinh12(x + y) sinh

12(x − y)

7. tanx ± tan y =sin (x ± y)cos x cos y

8. tanhx ± tanh y =sinh (x ± y)cosh x cosh y

9.∗ sin x ± cos y = ±2 sin[12(x + y) ± π

4

]sin

[12(x − y) ± π

4

]

= ±2 cos[12(x + y) ∓ π

4

]cos

[12(x − y) ∓ π

4

]

= 2 sin[12(x ± y) ± π

4

]cos

[12(x ∓ y) ∓ π

4

]


10.∗ a sin x ± b cos x = a

√

1 +(

b

a

)2

sin[x ± arctan

(b

a

)]

[a �= 0]

11.∗ ±a sin x + b cos x = b

√

1 +(a

b

)2

cos[x ∓ arctan

(a

b

)]

[b �= 0]

12.∗ a sin x ± b cos y = q

√

1 +(

r

q

)2

sin[12(x ± y) + arctan

(r

q

)]

q = (a + b) cos[12(x ∓ y)

], r = (a − b) sin

[12(x ∓ y)

][q �= 0]

13.∗ a cos x + b cos y = t

√

1 +(s

t

)2

cos[12(x ∓ y) + arctan

(s

t

)][t �= 0]

= −s

√

1 +(

t

s

)2

cos[12(x ∓ y) − arctan

(t

s

)][s �= 0]

s = (a − b) sin[12(x ± y)

], t = (a + b) cos

[12(x ± y)

]

1.315

1. sin2 x − sin2 y = sin(x + y) sin(x − y) = cos2 y − cos2 x

2. sinh2 x − sinh2 y = sinh(x + y) sinh(x − y) = cosh2 x − cosh2 y

3. cos2 x − sin2 y = cos(x + y) cos(x − y) = cos2 y − sin2 x

4. sinh2 x + cosh2 y = cosh(x + y) cosh(x − y) = cosh2 x + sinh2 y

1.316

1. (cos x + i sin x)n = cos nx + i sin nx [n is an integer]

2. (cosh x + sinh x)n = sinh nx + cosh nx [n is an integer]

1.317

1. sinx

2= ±

√12

(1 − cos x)

2. sinhx

2= ±

√12

(cosh x − 1)

3. cosx

2= ±

√12

(1 + cosx)

4. coshx

2=

√12

(cosh x + 1)

5. tanx

2=

1 − cos x

sinx=

sin x

1 + cosx

1.321 Trigonometric and hyperbolic functions: expansion in multiple angles 31

6. tanhx

2=

cosh x − 1sinh x

=sinh x

cosh x + 1

The signs in front of the radical in formulas 1.317 1, 1.317 2, and 1.317 3 are taken so as to agreewith the signs of the left-hand members. The sign of the left hand members depends in turn on the valueof x.

1.32 The representation of powers of trigonometric and hyperbolic functions in termsof functions of multiples of the argument (angle)

1.320

1. sin2n x =1

22n

{n−1∑

k=0

(−1)n−k2(

2n

k

)cos 2(n − k)x +

(2n

n

)}

KR 56 (10, 2)

2. sinh2n x =(−1)n

22n

{n−1∑

k=0

(−1)n−k2(

2n

k

)cosh 2(n − k)x +

(2n

n

)}

3. sin2n−1 x =1

22n−2

n−1∑

k=0

(−1)n+k−1

(2n − 1

k

)sin(2n − 2k − 1)x KR 56 (10, 4)

4. sinh2n−1 x =(−1)n−1

22n−2

n−1∑

k=0

(−1)n+k−1

(2n − 1

k

)sinh(2n − 2k − 1)x

5. cos2n x =1

22n

{n−1∑

k=0

2(

2n

k

)cos 2(n − k)x +

(2n

n

)}

KR 56 (10, 1)

6. cosh2n x =1

22n

{n−1∑

k=0

2(

2n

k

)cosh 2(n − k)x +

(2n

n

)}

7. cos2n−1 x =1

22n−2

n−1∑

k=0

(2n − 1

k

)cos(2n − 2k − 1)x KR 56 (10, 3)

8. cosh2n−1 x =1

22n−2

n−1∑

k=0

(2n − 1

k

)cosh(2n − 2k − 1)x

Special cases

1.321

1. sin2 x =12

(− cos 2x + 1)

2. sin3 x =14

(− sin 3x + 3 sinx)

3. sin4 x =18

(cos 4x − 4 cos 2x + 3)

4. sin5 x =116

(sin 5x − 5 sin 3x + 10 sinx)


5. sin6 x =132

(− cos 6x + 6 cos 4x − 15 cos 2x + 10)

6. sin7 x =164

(− sin 7x + 7 sin 5x − 21 sin 3x + 35 sinx)

1.322

1. sinh2 x =12

(cosh 2x − 1)

2. sinh3 x =14

(sinh 3x − 3 sinh x)

3. sinh4 x =18

(cosh 4x − 4 cosh 2x + 3)

4. sinh5 x =116

(sinh 5x − 5 sinh 3x + 10 sinhx)

5. sinh6 x =132

(cosh 6x − 6 cosh 4x + 15 cosh 2x + 10)

6. sinh7 x =164

(sinh 7x − 7 sinh 5x + 21 sinh 3x + 35 sinhx)

1.323

1. cos2 x =12

(cos 2x + 1)

2. cos3 x =14

(cos 3x + 3 cosx)

3. cos4 x =18

(cos 4x + 4 cos 2x + 3)

4. cos5 x =116

(cos 5x + 5 cos 3x + 10 cos x)

5. cos6 x =132

(cos 6x + 6 cos 4x + 15 cos 2x + 10)

6. cos7 x =164

(cos 7x + 7 cos 5x + 21 cos 3x + 35 cos x)

1.324

1. cosh2 x =12

(cosh 2x + 1)

2. cosh3 x =14

(cosh 3x + 3 cosh x)

3. cosh4 x =18

(cosh 4x + 4 cosh 2x + 3)

4. cosh5 x =116

(cosh 5x + 5 cosh 3x + 10 coshx)

5. cosh6 x =132

(cosh 6x + 6 cosh 4x + 15 cosh 2x + 10)

6. cosh7 x =164

(cosh 7x + 7 cosh 5x + 21 cosh 3x + 35 coshx)

1.332 Trigonometric and hyperbolic functions: expansion in powers 33

1.33 The representation of trigonometric and hyperbolic functions of multiples ofthe argument (angle) in terms of powers of these functions

1.331

1.7 sin nx= n cosn−1 x sin x −(n

3

)cosn−3 x sin3 x +

(n

5

)cosn−5 x sin5 x − . . . ;

= sin x

2n−1 cosn−1 x −

(n − 2

1

)2n−3 cosn−3 x

+(

n − 32

)2n−5 cosn−5 x −

(n − 4

3

)2n−7 cosn−7 x + . . .

AD (3.175)

2. sinh nx= x

[(n+1)/2]∑

k=1

(n

2k − 1

)sinh2k−2 x coshn−2k+1 x

= sinh x

[(n−1)/2]∑

k=0

(−1)k

(n − k − 1

k

)2n−2k−1 coshn−2k−1 x

3. cosnx= cosn x −(n

2

)cosn−2 x sin2 x +

(n

4

)cosn−4 x sin4 x − . . . ;

= 2n−1 cosn x − n

12n−3 cosn−2 x +

n

2

(n − 3

1

)2n−5 cosn−4 x

−n

3

(n − 4

2

)2n−7 cosn−6 x + . . .

AD (3.175)

4.3 cosh nx=[n/2]∑

k=0

( n

2k

)sinh2k x coshn−2k x

= 2n−1 coshn x + n

[n/2]∑

k=1

(−1)k 1k

(n − k − 1

k − 1

)2n−2k−1 coshn−2k x

1.332

1. sin 2nx= 2n cos x

{

sin x − 4n2 − 22

3!sin3 x +

(4n2 − 22

) (4n2 − 42

)

5!sin5 x − . . .

}

AD (3.171)

= (−1)n−1 cos x

{22n−1 sin2n−1 x − 2n − 2

1!22n−3 sin2n−3 x

+(2n − 3)(2n − 4)

2!22n−5 sin2n−5 x

− (2n − 4)(2n − 5)(2n − 6)3!

22n−7 sin2n−7 x + . . .

}AD (3.173)


2. sin(2n − 1)x= (2n − 1){

sin x − (2n − 1)2 − 12

3!sin3 x

+

[(2n − 1)2 − 12

] [(2n − 1)2 − 32

]

5!sin5 x − . . .

}

AD (3.172)

= (−1)n−1

{22n−2 sin2n−1 x − 2n − 1

1!22n−4 sin2n−3 x

+(2n − 1)(2n − 4)

2!22n−6 sin2n−5 x

− (2n − 1)(2n − 5)(2n − 6)3!

22n−8 sin2n−7 x + . . .

}AD (3.174)

3. cos 2nx= 1 − 4n2

2!sin2 x +

4n2(4n2 − 22

)

4!sin4 x −

4n2(4n2 − 2

) (4n2 − 42

)

6!sin6 x + . . .

AD (3.171)

= (−1)n

{22n−1 sin2n x − 2n

1!22n−3 sin2n−2 x

+2n(2n − 3)

2!22n−5 sin2n−4 x − 2n(2n − 4)(2n − 5)

3!22n−7 sin2n−6 x + . . .

}

AD (3.173)a

4. cos(2n − 1)x= cos x

{1 − (2n − 1)2 − 12

2!sin2 x

+

[(2n − 1)2 − 12

] [(2n − 1)2 − 32

]

4!sin4 x − . . .

}

AD (3.172)

= (−1)n−1 cos x

{22n−2 sin2n−2 x − 2n − 3

1!22n−4 sin2n−4 x

+(2n − 4)(2n − 5)

2!22n−6 sin2n−6 x

− (2n − 5)(2n − 6)(2n − 7)3!

22n−8 sin2n−8 x + . . .

}AD (3.174)

By using the formulas and values of 1.30, we can write formulas for sinh 2nx, sinh(2n−1)x, cosh 2nx,and cosh(2n − 1)x that are analogous to those of 1.332, just as was done in the formulas in 1.331.

Special cases

1.333

1. sin 2x = 2 sin x cos x

2. sin 3x = 3 sin x − 4 sin3 x

3. sin 4x = cos x(4 sin x − 8 sin3 x

)

4. sin 5x = 5 sin x − 20 sin3 x + 16 sin5 x

5. sin 6x = cos x(6 sin x − 32 sin3 x + 32 sin5 x

)

1.337 Trigonometric and hyperbolic functions: expansion in powers 35

6. sin 7x = 7 sin x − 56 sin3 x + 112 sin5 x − 64 sin7 x

1.334

1. sinh 2x = 2 sinh x cosh x

2. sinh 3x = 3 sinh x + 4 sinh3 x

3.11 sinh 4x = cosh x(4 sinh x + 8 sinh3 x

)

4. sinh 5x = 5 sinh x + 20 sinh3 x + 16 sinh5 x

5.11 sinh 6x = cosh x(6 sinh x + 32 sinh3 x + 32 sinh5 x

)

6. sinh 7x = 7 sinh x + 56 sinh3 x + 112 sinh5 x + 64 sinh7 x

1.335

1. cos 2x = 2 cos2 x − 1

2. cos 3x = 4 cos3 x − 3 cos x

3. cos 4x = 8 cos4 x − 8 cos2 x + 1

4. cos 5x = 16 cos5 x − 20 cos3 x + 5 cosx

5. cos 6x = 32 cos6 x − 48 cos4 x + 18 cos2 x − 1

6. cos 7x = 64 cos7 x − 112 cos5 x + 56 cos3 x − 7 cos x

1.336

1. cosh 2x = 2 cosh2 x − 1

2. cosh 3x = 4 cosh3 x − 3 coshx

3. cosh 4x = 8 cosh4 x − 8 cosh2 x + 1

4. cosh 5x = 16 cosh5 x − 20 cosh3 x + 5 cosh x

5. cosh 6x = 32 cosh6 x − 48 cosh4 x + 18 cosh2 x − 1

6. cosh 7x = 64 cosh7 x − 112 cosh5 x + 56 cosh3 x − 7 cosh x

1.337

1.∗cos 3x

cos3 x= 1 − 3 tan2 x

2.∗cos 4x

cos4 x= 1 − 6 tan2 x + tan4 x

3.∗cos 5x

cos5 x= 1 − 10 tan2 x + 5 tan4 x

4.∗cos 6x

cos6 x= 1 − 15 tan2 x + 15 tan4 x − tan6 x

5.∗sin 3x

cos3 x= 3 tanx − tan3 x

6.∗sin 4x

cos4 x= 4 tanx − 4 tan3 x


7.∗sin 5x

cos5 x= 5 tanx − 10 tan3 x + tan5 x

8.∗sin 6x

cos6 x= 6 tanx − 20 tan3 x + 6 tan5 x

9.∗cos 3x

sin3 x= cot3 x − 3 cot x

10.∗cos 4x

sin4 x= cot4 x − 6 cot2 x + 1

11.∗cos 5x

sin5 x= cot5 x − 10 cot3 x + 5 cotx

12.∗cos 6x

sin6 x= cot6 x − 15 cot4 x + 15 cot2 x − 1

13.∗sin 3x

sin3 x= 3 cot2 x − 1

14.∗sin 4x

sin4 x= 4 cot3 x − 4 cotx

15.∗sin 5x

sin5 x= 5 cot4 x − 10 cot2 x + 1

16.∗sin 6x

sin6 x= 6 cot5 x − 20 cot3 x + 6 cotx

1.34 Certain sums of trigonometric and hyperbolic functions

1.341

1.n−1∑

k=0

sin(x + ky) = sin(

x +n − 1

2y

)sin

ny

2cosec

y

2AD (361.8)

2.n−1∑

k=0

sinh(x + ky) = sinh(

x +n − 1

2y

)sinh

ny

21

sinhy

2

3.n−1∑

k=0

cos(x + ky) = cos(

x +n − 1

2y

)sin

ny

2cosec

y

2AD (361.9)

4.n−1∑

k=0

cosh(x + ky) = cosh(

x +n − 1

2y

)sinh

ny

21

sinhy

2

5.2n−1∑

k=0

(−1)k cos(x + ky) = sin(

x +2n − 1

2y

)sin ny sec

y

2JO (202)

6.n−1∑

k=0

(−1)k sin(x + ky) = sin(

x +n − 1

2(y + π)

)sin

n(y + π)2

secy

2AD (202a)

1.351 Sums of powers of trigonometric functions of multiple angles 37

Special cases

1.342

1.n∑

k=1

sin kx = sinn + 1

2x sin

nx

2cosec

x

2AD (361.1)

2.10n∑

k=0

cos kx= cosn + 1

2x sin

nx

2cosec

x

2+ 1

= cosnx

2sin

n + 12

x cosecx

2=

12

(

1 +sin

(n + 1

2

)x

sin x2

)

AD (361.2)

3.n∑

k=1

sin(2k − 1)x = sin2 nx cosec x AD (361.7)

4.n∑

k=1

cos(2k − 1)x =12

sin 2nx cosec x JO (207)

1.343

1.n∑

k=1

(−1)k cos kx = −12

+(−1)n cos

(2n+1

2 x)

2 cosx

2

AD (361.11)

2.n∑

k=1

(−1)k+1 sin(2k − 1)x = (−1)n+1 sin 2nx

2 cos xAD (361.10)

3.n∑

k=1

cos(4k − 3)x +n∑

k=1

sin(4k − 1)x = sin 2nx (cos 2nx + sin 2nx) (cosx + sin x) cosec 2x

JO (208)

1.344

1.n−1∑

k=1

sinπk

n= cot

π

2nAD (361.19)

2.n−1∑

k=1

sin2πk2

n=

√n

2

(1 + cos

nπ

2− sin

nπ

2

)AD (361.18)

3.n−1∑

k=0

cos2πk2

n=

√n

2

(1 + cos

nπ

2+ sin

nπ

2

)AD (361.17)

1.35 Sums of powers of trigonometric functions of multiple angles

1.351

1.n∑

k=1

sin2 kx=14

[(2n + 1) sin x − sin(2n + 1)x] cosec x

=n

2− cos(n + 1)x sin nx

2 sin xAD (361.3)


2.n∑

k=1

cos2 kx=n − 1

2+

12

cos nx sin(n + 1)x cosec x

=n

2+

cos(n + 1)x sin nx

2 sinxAD (361.4)a

3.n∑

k=1

sin3 kx =34

sinn + 1

2x sin

nx

2cosec

x

2− 1

4sin

3(n + 1)x2

sin3nx

2cosec

3x

2JO (210)

4.n∑

k=1

cos3 kx =34

cosn + 1

2x sin

nx

2cosec

x

2+

14

cos3(n + 1)

2x sin

3nx

2cosec

3x

2JO (211)a

5.n∑

k=1

sin4 kx =18

[3n − 4 cos(n + 1)x sin nx cosec x + cos 2(n + 1)x sin 2nx cosec 2x] JO (212)

6.n∑

k=1

cos4 kx =18

[3n + 4 cos(n + 1)x sin nx cosec x + cos 2(n + 1)x sin 2nx cosec 2x] JO (213)

1.352

1.11n−1∑

k=1

k sin kx =sinnx

4 sin2 x2

−n cos

(2n−1

2 x)

2 sin x2

AD (361.5)

2.11n−1∑

k=1

k cos kx =n sin

(2n−1

2 x)

2 sin x2

− 1 − cos nx

4 sin2 x2

AD (361.6)

1.353

1.n−1∑

k=1

pk sin kx =p sin x − pn sin nx + pn+1 sin(n − 1)x

1 − 2p cos x + p2AD (361.12)a

2.n−1∑

k=1

pk sinh kx =p sinh x − pn sinh nx + pn+1 sinh(n − 1)x

1 − 2p cosh x + p2

3.n−1∑

k=0

pk cos kx =1 − p cos x − pn cos nx + pn+1 cos(n − 1)x

1 − 2p cos x + p2AD (361.13)a¡

4.n−1∑

k=0

pk cosh kx =1 − p cosh x − pn cosh nx + pn+1 cosh(n − 1)x

1 − 2p cosh x + p2JO (396)

1.36 Sums of products of trigonometric functions of multiple angles

1.361

1.n∑

k=1

sin kx sin(k + 1)x =14

[(n + 1) sin 2x − sin 2(n + 1)x] cosec x JO (214)

2.n∑

k=1

sin kx sin(k + 2)x =n

2cos 2x − 1

2cos(n + 3)x sin nx cosec x JO (216)

1.381 Sums leading to hyperbolic tangents and cotangents 39

3. 2n∑

k=1

sin kx cos(2k − 1)y = sin(

ny +n + 1

2x

)sin

n(x + 2y)2

cosecx + 2y

2

− sin(

ny − n + 12

x

)sin

n(2y − x)2

cosec2y − x

2JO (217)

1.362

1.n∑

k=1

(2k sin2 x

2k

)2

=(2n sin

x

2n

)2

− sin2 x AD (361.15)

2.n∑

k=1

(12k

secx

2k

)2

= cosec2 x −(

12n

cosecx

2n

)2

AD (361.14)

1.37 Sums of tangents of multiple angles

1.371

1.n∑

k=0

12k

tanx

2k=

12n

cotx

2n− 2 cot 2x AD (361.16)

2.n∑

k=0

122k

tan2 x

2k=

22n+2 − 13 · 22n−1

+ 4 cot2 2x − 122n

cot2x

2nAD (361.20)

1.38 Sums leading to hyperbolic tangents and cotangents

1.381

1.n−1∑

k=0

tanh

x

1

n sin2

(2k + 1

4nπ

)

1 +tanh2 x

tan2

(2k + 1

4nπ

)= tanh (2nx) JO (402)a

2.n−1∑

k=1

tanh

x

1

n sin2

(kπ

2n

)

1 +tanh2 x

tan2

(kπ

2n

)= coth (2nx) − 1

2n(tanhx + cothx) JO (403)


3.n−1∑

k=0

tanh

x

2

(2n + 1) sin2

(2k + 1

2(2n + 1)π

)

1 +tanh2 x

tan2

(2k + 1

2(2n + 1)π

)= tanh (2n + 1) x − tanh x

2n + 1JO (404)

4.n∑

k=1

tanh

x

2

(2n + 1) sin2

(kπ

2(2n + 1)

)

1 +tanh2 x

tan2

(kπ

(2n + 1)

)= coth (2n + 1)x − coth x

2n + 1JO (405)

1.382

1.n−1∑

k=0

1

sin2

(2k + 1

4nπ

)

sinh x+

12

tanh(x

2

)

= 2n tanh (nx) JO (406)

2.n−1∑

k=1

1

sin2

(kπ

2n

)

sinh x+

12

tanh(x

2

)

= 2n coth (nx) − 2 cothx JO (407)

3.n−1∑

k=0

1

sin2

(2k + 1

2(2n + 1)π

)

sinh x+

12

tanh(x

2

)

= (2n + 1) tanh(

(2n + 1)x2

)− tanh

x

2JO (408)

4.n∑

k=1

1

sin2

(kπ

2n + 1

)

sinh x+

12

tanh(x

2

)

= (2n + 1) coth(

(2n + 1)x2

)− coth

x

2JO (409)

1.395 Representing sines and cosines as finite products 41

1.39 The representation of cosines and sines of multiples of the angle as finiteproducts

1.391

1. sin nx = n sin x cos x

n−22∏

k=1

1 − sin2 x

sin2 kπ

n

[n is even] JO (568)

2. cosnx =

n2∏

k=1

1 − sin2 x

sin2 (2k − 1)π2n

[n is even] JO (569)

3. sin nx = n sin x

n−12∏

k=1

1 − sin2 x

sin2 kπ

n

[n is odd] JO (570)

4. cosnx = cos x

n−12∏

k=1

1 − sin2 x

sin2 (2k − 1)π2n

[n is odd] JO (571)a

1.392

1. sin nx = 2n−1n−1∏

k=0

sin(

x +kπ

n

)JO (548)

2. cosnx = 2n−1n∏

k=1

sin(

x +2k − 1

2nπ

)JO (549)

1.393

1.n−1∏

k=0

cos(

x +2k

nπ

)=

12n−1

cos nx [n odd]

=1

2n−1

[(−1)

n2 − cos nx

][n even]

JO (543)

2.11n−1∏

k=0

sin(

x +2k

nπ

)=

(−1)n−1

2

2n−1sin nx [n odd]

=(−1)

n2

2n−1(1 − cos nx) [n even]

JO (544)

1.394n−1∏

k=0

{x2 − 2xy cos

(α +

2kπ

n

)+ y2

}= x2n − 2xnyn cos nα + y2n JO (573)

1.395

1. cosnx − cos ny = 2n−1n−1∏

k=0

{cos x − cos

(y +

2kπ

n

)}JO (573)


2. cosh nx − cos ny = 2n−1n−1∏

k=0

{cosh x − cos

(y +

2kπ

n

)}JO (538)

1.396

1.n−1∏

k=1

(x2 − 2x cos

kπ

n+ 1

)=

x2n − 1x2 − 1

KR 58 (28.1)

2.n∏

k=1

(x2 − 2x cos

2kπ

2n + 1+ 1

)=

x2n+1 − 1x − 1

KR 58 (28.2)

3.n∏

k=1

(x2 + 2x cos

2kπ

2n + 1+ 1

)=

x2n+1 − 1x + 1

KR 58 (28.3)

4.n−1∏

k=0

(x2 − 2x cos

(2k + 1)π2n

+ 1)

= x2n + 1 KR 58 (28.4)

1.41 The expansion of trigonometric and hyperbolic functions in power series

1.411

1. sin x =∞∑

k=0

(−1)k x2k+1

(2k + 1)!

2. sinh x =∞∑

k=0

x2k+1

(2k + 1)!

3. cosx =∞∑

k=0

(−1)k x2k

(2k)!

4. cosh x =∞∑

k=0

x2k

(2k)!

5. tanx =∞∑

k=1

22k(22k − 1

)

(2k)!|B2k|x2k−1

[x2 <

π2

4

]FI II 523

6.11 tanh x = x − x3

3+

2x5

15− 17

315x7 + · · · =

∞∑

k=1

22k(22k − 1

)

(2k)!B2kx2k−1

[x2 <

π2

4

]

7. cotx =1x−

∞∑

k=1

22k|B2k|(2k)!

x2k−1[x2 < π2

]FI II 523a

8. coth x =1x

+x

3− x3

45+

2x5

945− · · · =

1x

+∞∑

k=1

22kB2k

(2k)!x2k−1

[x2 < π2

]FI II 522a

1.414 Trigonometric and hyperbolic functions: power series expansion 43

9. sec x =∞∑

k=0

|E2k|(2k)!

x2k

[x2 <

π2

4

]CE 330a

10. sech x = 1 − x2

2+

5x4

24− 61x6

720+ · · · = 1 +

∞∑

k=1

E2k

(2k)!x2k

[x2 <

π2

4

]CE 330

11. cosec x =1x

+∞∑

k=1

2(22k−1 − 1

)|B2k|x2k−1

(2k)![x2 < π2

]CE 329a

12. cosechx =1x− 1

6x +

7x3

360− 31x5

15120+ · · · =

1x−

∞∑

k=1

2(22k−1 − 1

)B2k

(2k)!x2k−1

[x2 < π2

]JO (418)

1.412

1. sin2 x =∞∑

k=1

(−1)k+1 22k−1x2k

(2k)!JO (452)a

2. cos2 x = 1 −∞∑

k=1

(−1)k+1 22k−1x2k

(2k)!JO (443)

3. sin3 x =14

∞∑

k=1

(−1)k+1 32k+1 − 3(2k + 1)!

x2k+1 JO (452a)a

4. cos3 x =14

∞∑

k=0

(−1)k

(32k + 3

)x2k

(2k)!JO (443a)

1.413

1. sinh x = cosec x∞∑

k=1

(−1)k+1 22k−1x4k−2

(4k − 1)!JO (508)

2. cosh x = sec x + sec x

∞∑

k=1

(−1)k 22kx4k

(4k)!JO (507)

3. sinh x = sec x

∞∑

k=1

(−1)[k/2] 2k−1x2k−1

(2k − 1)!JO (510)

4. cosh x = cosec x

∞∑

k=1

(−1)[(k−1)/2] 2k−1x2k−1

(2k − 1)!JO (509)

1.414

1. cos[n ln

(x +

√1 + x2

)]= 1 −

∞∑

k=0

(−1)k

(n2 + 02

) (n2 + 22

). . .

[n2 + (2k)2

]

(2k + 2)!x2k+2

[x2 < 1

]AD (6456.1)


2. sin[n ln

(x +

√1 + x2

)]= nx − n

∞∑

k=1

(−1)k+1

(n2 + 12

) (n2 + 32

). . .

[n2 + (2k − 1)2

]x2k+1

(2k + 1)![x2 < 1

]AD (6456.2)

Power series for ln sinx, ln cosx, and ln tanx see 1.518.

1.42 Expansion in series of simple fractions

1.421

1. tanπx

2=

4x

π

∞∑

k=1

1(2k − 1)2 − x2

BR* (191), AD (6495.1)

2.10 tanhπx

2=

4x

π

∞∑

k=1

1(2k − 1)2 + x2

3. cotπx =1

πx+

2x

π

∞∑

k=1

1x2 − k2

=1

πx+

x

π

∞∑

k=−∞k �=0

1k(x − k)

AD (6495.2), JO (450a)

4. coth πx =1

πx+

2x

π

∞∑

k=1

1x2 + k2

(cf. 1.217 1)

5. tan2 πx

2= x2

∞∑

k=1

2(2k − 1)2 − x2

(12 − x2)2 (32 − x2)2 . . . [(2k − 1)2 − x2]2JO (450)

1.422

1. secπx

2=

4π

∞∑

k=1

(−1)k+1 2k − 1(2k − 1)2 − x2

AD (6495.3)a

2. sec2 πx

2=

4π2

∞∑

k=1

{1

(2k − 1 − x)2+

1(2k − 1 + x)2

}JO (451)a

3. cosec πx =1

πx+

2x

π

∞∑

k=1

(−1)k

x2 − k2(see also 1.217 2) AD (6495.4)a

4. cosec2 πx =1π2

∞∑

k=−∞

1(x − k)2

=1

π2x2+

2π2

∞∑

k=1

x2 + k2

(x2 − k2)2JO (446)

5.1 + x cosec x

2x2=

1x2

−∞∑

k=1

(−1)k+1

(x2 − k2π2)JO (449)

6. cosec πx =2π

∞∑

k=−∞

(−1)k

x2 − k2JO (450b)

1.423π2

4m2cosec2 π

m+

π

4mcot

π

m− 1

2=

∞∑

k=1

1(1 − k2m2)2

JO (477)

1.439 Representation in the form of an infinite product 45

1.43 Representation in the form of an infinite product

1.431

1. sin x = x

∞∏

k=1

(1 − x2

k2π2

)EU

2. sinh x = x

∞∏

k=1

(1 +

x2

k2π2

)EU

3. cosx =∞∏

k=0

(1 − 4x2

(2k + 1)2π2

)EU

4. cosh x =∞∏

k=0

(1 +

4x2

(2k + 1)2π2

)EU

1.432

1.11 cos x − cos y = 2(

1 − x2

y2

)sin2 y

2

∞∏

k=1

(

1 − x2

(2kπ + y)2

) (1 − x2

(2kπ − y)2

)AD (653.2)

2. cosh x − cos y = 2(

1 +x2

y2

)sin2 y

2

∞∏

k=1

(1 +

x2

(2kπ + y)2

) (1 +

x2

(2kπ − y)2

)AD (653.1)

1.433 cosπx

4− sin

πx

4=

∞∏

k=1

[1 +

(−1)kx

2k − 1

]BR* 189

1.434 cos2 x =14(π + 2x)2

∞∏

k=1

[

1 −(

π + 2x

2kπ

)2]2

MO 216

1.435sin π(x + a)

sin πa=

x + a

a

∞∏

k=1

(1 − x

k − a

)(1 +

x

k + a

)MO 216

1.436 1 − sin2 πx

sin2 πa=

∞∏

k=−∞

[

1 −(

x

k − a

)2]

MO 216

1.437sin 3x

sin x= −

∞∏

k=−∞

[

1 −(

2x

x + kπ

)2]

MO 216

1.438cosh x − cos a

1 − cos a=

∞∏

k=−∞

[

1 +(

x

2kπ + a

)2]

MO 216

1.439

1. sin x = x

∞∏

k=1

cosx

2k[|x| < 1] AD (615), MO 216

2.sin x

x=

∞∏

k=1

[1 − 4

3sin2

( x

3k

)]MO 216


1.44–1.45 Trigonometric (Fourier) series

1.441

1.∞∑

k=1

sin kx

k=

π − x

2[0 < x < 2π] FI III 539

2.∞∑

k=1

cos kx

k= −1

2ln [2 (1 − cos x)] [0 < x < 2π] FI III 530a, AD (6814)

3.∞∑

k=1

(−1)k−1 sin kx

k=

x

2[−π < x < π] FI III 542

4.∞∑

k=1

(−1)k−1 cos kx

k= ln

(2 cos

x

2

)[−π < x < π] FI III 550

1.442

1.11∞∑

k=1

sin(2k − 1)x2k − 1

=π

4sign x [−π < x < π] FI III 541

2.∞∑

k=1

cos(2k − 1)x2k − 1

=12

ln cotx

2[0 < x < π]

BR* 168, JO (266), GI III(195)

3.∞∑

k=1

(−1)k−1 sin(2k − 1)x2k − 1

=12

ln tan(π

4+

x

2

) [−π

2< x <

π

2

]BR* 168, JO (268)a

4.10∞∑

k=1

(−1)k−1 cos(2k − 1)x2k − 1

=π

4

[−π

2< x <

π

2

]

= −π

4

[π

2< x <

3π

2

]

BR* 168, JO (269)

1.443

1.8∞∑

k=1

cos kπx

k2n= (−1)n−122n−1 π2n

(2n)!

2n∑

k=0

(2n

k

)B2n−kρk

= (−1)n−1 12

(2π)2n

(2n)!B2n

(x

2

)

[0 ≤ x ≤ 2, ρ =

x

2−

⌊x

2

⌋]CE 340, GE 71

2.∞∑

k=1

sin kπx

k2n+1= (−1)n−122n π2n+1

(2n + 1)!

2n+1∑

k=0

(2n + 1

k

)B2n−k+1ρ

k

= (−1)n−1 12

(2π)2n+1

(2n + 1)!B2n+1

(x

2

)

[0 < x < 1; ρ =

x

2−

⌊x

2

⌋]CE 340

1.445 Trigonometric (Fourier) series 47

3.∞∑

k=1

cos kx

k2=

π2

6− πx

2+

x2

4[0 ≤ x ≤ 2π] FI III 547

4.∞∑

k=1

(−1)k−1 cos kx

k2=

π2

12− x2

4[−π ≤ x ≤ π] FI III 544

5.∞∑

k=1

sin kx

k3=

π2x

6− πx2

4+

x3

12[0 ≤ x ≤ 2π]

6.∞∑

k=1

cos kx

k4=

π4

90− π2x2

12+

πx3

12− x4

48[0 ≤ x ≤ 2π] AD (6617)

7.∞∑

k=1

sin kx

k5=

π4x

90− π2x3

36+

πx4

48− x5

240[0 ≤ x ≤ 2π] AD (6818)

1.444

1.∞∑

k=1

sin 2(k + 1)xk(k + 1)

= sin 2x − (π − 2x) sin2 x − sin x cos x ln(4 sin2 x

)

[0 ≤ x ≤ π] BR* 168, GI III (190)

2.∞∑

k=1

cos 2(k + 1)xk(k + 1)

= cos 2x −(π

2− x

)sin 2x + sin2 x ln

(4 sin2 x

)

[0 ≤ x ≤ π] BR* 168

3.∞∑

k=1

(−1)k sin(k + 1)xk(k + 1)

= sin x − x

2(1 + cos x) − sin x ln

∣∣∣2 cosx

2

∣∣∣ MO 213

4.∞∑

k=1

(−1)k cos(k + 1)xk(k + 1)

= cos x − x

2sin x − (1 + cosx) ln

∣∣∣2 cosx

2

∣∣∣ MO 213

5.∞∑

k=0

(−1)k sin(2k + 1)x(2k + 1)2

=π

4x

[−π

2≤ x ≤ π

2

]

=π

4(π − x)

[π

2≤ x ≤ 3

2π

]

MO 213

6.6∞∑

k=1

cos(2k − 1)x(2k − 1)2

=π

4

(π

2− |x|

)[−π ≤ x ≤ π] FI III 546

7.∞∑

k=1

cos 2kx

(2k − 1)(2k + 1)=

12− π

4sinx

[0 ≤ x ≤ π

2

]JO (591)

1.445

1.∞∑

k=1

k sin kx

k2 + α2=

π

2sinh α(π − x)

sinh απ[0 < x < 2π] BR* 157, JO (411)

2.∞∑

k=1

cos kx

k2 + α2=

π

2α

cosh α(π − x)sinh απ

− 12α2

[0 ≤ x ≤ 2π] BR* 257, JO (410)


3.∞∑

k=1

(−1)k cos kx

k2 + α2=

π

2α

cosh αx

sinh απ− 1

2α2[−π ≤ x ≤ π] FI III 546

4.∞∑

k=1

(−1)k−1 k sin kx

k2 + α2=

π

2sinh αx

sinh απ[−π < x < π] FI III, 546

5.∞∑

k=1

k sin kx

k2 − α2= π

sin {α[(2m + 1)π − x]}2 sinαπ

[if x = 2mπ, then

∑· · · = 0

]

[2mπ < x < (2m + 2)π, α not an integer] MO 213

6.∞∑

k=1

cos kx

k2 − α2=

12α2

− π

2cos [α {(2m + 1)π − x}]

α sin απ

[2mπ ≤ x ≤ (2m + 2)π, α not an integer] MO 213

7.∞∑

k=1

(−1)k k sin kx

k2 − α2= π

sin[α(2mπ − x)]2 sin απ

[if x = (2m + 1)π, then

∑· · · = 0

],

[(2m − 1)π < x < (2m + 1)π, α not an integer] FI III 545a

8.∞∑

k=1

(−1)k cos kx

k2 − α2=

12α2

− π

2cos[α(2mπ − x)]

α sin απ

[(2m − 1)π ≤ x ≤ (2m + 1)π, α not an integer] FI III 545a

9.∗∞∑

n=−∞

einα

(n − β)2 + γ2=

π

γ

eiβ(α−2π) sinh(γα) + eiβα sinh [γ(2π − α)]cosh(2πγ) − cos(2πβ)

[0 ≤ α ≤ 2π]

1.446∞∑

k=1

(−1)k+1 cos(2k + 1)x(2k − 1)(2k + 1)(2k + 3)

=π

8cos2 x − 1

3cos x

[−π

2≤ x ≤ π

2

]BR* 256, GI III (189)

1.447

1.∞∑

k=1

pk sin kx=p sin x

1 − 2p cos x + p2

[|p| < 1] FI II 559

2.∞∑

k=0

pk cos kx=1 − p cos x

1 − 2p cos x + p2

[|p| < 1] FI II 559

3. 1 + 2∞∑

k=1

pk cos kx=1 − p2

1 − 2p cos x + p2

[|p| < 1] FI II 559a, MO 213

1.449 Trigonometric (Fourier) series 49

1.448

1.∞∑

k=1

pk sin kx

k= arctan

p sin x

1 − p cos x[0 < x < 2π, p2 ≤ 1

]FI II 559

2.∞∑

k=1

pk cos kx

k= −1

2ln

(1 − 2p cos x + p2

)

[0 < x < 2π, p2 ≤ 1

]FI II 559

3.∞∑

k=1

p2k−1 sin(2k − 1)x2k − 1

=12

arctan2p sin x

1 − p2

[0 < x < 2π, p2 ≤ 1

]JO (594)

4.∞∑

k=1

p2k−1 cos(2k − 1)x2k − 1

=14

ln1 + 2p cos x + p2

1 − 2p cos x + p2

[0 < x < 2π, p2 ≤ 1

]JO (259)

5.∞∑

k=1

(−1)k−1p2k−1 sin(2k − 1)x2k − 1

=14

ln1 + 2p sin x + p2

1 − 2p sin x + p2

[0 < x < π, p2 ≤ 1

]JO (261)

6.∞∑

k=1

(−1)k−1p2k−1 cos(2k − 1)x2k − 1

=12

arctan2p cos x

1 − p2

[0 < x < π, p2 ≤ 1

]JO (597)

1.449

1.∞∑

k=1

pk sin kx

k!= ep cos x sin (p sin x)

[p2 ≤ 1

]JO (486)

2.∞∑

k=0

pk cos kx

k!= ep cos x cos (p sin x)

[p2 ≤ 1

]JO (485)

Let S(x) = − 1x cos x + 1

x and C(x) = 1x sin x.

3.∗∞∑

n=1

n

n2 − a2S(nx) =

π

2[C(ax) − cot(πa)S(ax)] [0 < x < 2π, a �= 0,±1,±2, . . .]

4.∗∞∑

n=1

1n2 − a2

C(nx) =1

2a2− π

2a[S(ax) − cot(πa)C(ax)]

[0 ≤ x ≤ 2π, a �= 0,±1,±2, . . .]

5.∗∞∑

n=1

(−1)n−1n

n2 − a2S(nx) =

π

2cosec(πa)S(ax) [−π < x < π, a �= 0,±1,±2, . . .]


6.∗∞∑

n=1

(−1)n−1

n2 − a2C(nx) = − 1

2a2+

π

2acosec(πa)C(ax) [−π < x < π, a �= 0,±1,±2, . . .]

7.∗∞∑

n=1

2n − 1(2n − 1)2 − a2

S(nx) =π

4

[C(ax) + tan

(πa

2

)S(ax)

]

[0 < x < π, a �= 0,±1,±2, . . .]

8.∗∞∑

n=1

1(2n − 1)2 − a2

C(nx) = − π

4a

[S(ax) − tan

(πa

2

)C(ax)

]

[0 ≤ x ≤ π, a �= 0,±1,±2, . . .]

9.∗∞∑

n=1

(−1)n−1

(2n − 1)2 − a2S(nx) =

π

4asec

(πa

2

)S(ax)

[−π

2≤ x ≤ π

2, a �= 0,±1,±2, . . .

]

10.∗∞∑

n=1

(−1)n−1(2n − 1)(2n − 1)2 − a2

C(nx) =π

4sec

(πa

2

)C(ax)

[−π

2≤ x ≤ π

2, a �= 0,±1,±2, . . .

]

Fourier expansions of hyperbolic functions

1.451

1. sinh x = cos x

∞∑

k=0

(12 + 02

) (12 + 22

). . .

[12 + (2k)2

]

(2k + 1)!sin2k+1 x JO (504)

2. cosh x = cos x + cos x

∞∑

k=1

(12 + 12

) (12 + 32

). . .

[12 + (2k − 1)2

]

(2k)!sin2k x JO (503)

1.452

1. sinh (x cos θ) = sec (x sin θ)∞∑

k=0

x2k+1 cos(2k + 1)θ(2k + 1)!

[x2 < 1

]JO (391)

2. cosh (x cos θ) = sec (x sin θ)∞∑

k=0

x2k cos 2kθ

(2k)![x2 < 1

]JO (390)

3. sinh (x cos θ) = cosec (x sin θ)∞∑

k=1

x2k sin 2kθ

(2k)![x2 < 1, x sin θ �= 0

]JO (393)

4. cosh (x cos θ) = cosec (x sin θ)∞∑

k=0

x2k+1 sin(2k + 1)θ(2k + 1)!

[x2 < 1, x sin θ �= 0

]JO (392)

1.480 Lobachevskiy’s “Angle of Parallelism” 51

1.46 Series of products of exponential and trigonometric functions

1.461

1.∞∑

k=0

e−kt sin kx =12

sinx

cosh t − cos x[t > 0] MO 213

2. 1 + 2∞∑

k=1

e−kt cos kx =sinh t

cosh t − cos x[t > 0] MO 213

1.4629

∞∑

k=1

sin kx sin ky

ke−2k|t| =

14

ln

sin2 x + y

2+ sinh2 t

sin2 x − y

2+ sinh2 t

MO 214

1.463

1. ex cos ϕ cos (x sin ϕ) =∞∑

n=0

xn cos nϕ

n![x2 < 1

]AD (6476.1)

2. ex cos ϕ sin (x sin ϕ) =∞∑

n=1

xn sin nϕ

n![x2 < 1

]AD (6476.2)

1.47 Series of hyperbolic functions

1.471

1.∞∑

k=1

sinh kx

k!= ecosh x sinh (sinh x) . JO (395)

2.∞∑

k=0

cosh kx

k!= ecosh x cosh (sinh x) . JO (394)

3.∞∑

k=0

1(2k + 1)3

[1x

tanh(2m + 1)πx

2+ x tanh

(2m + 1)π2x

]=

π3

16

1.472

1.∞∑

k=1

pk sinh kx =p sinh x

1 − 2p cosh x + p2

[p2 < 1

]JO (396)

2.∞∑

k=0

pk cosh kx =1 − p cosh x

1 − 2p cosh x + p2

[p2 < 1

]JO (397)a

1.48 Lobachevskiy’s “Angle of Parallelism” Π(x)

1.480 Definition.

1. Π(x) = 2 arccot ex = 2 arctan e−x [x ≥ 0] LO III 297, LOI 120


2. Π(x) = π − Π(−x) [x < 0] LO III 183, LOI 193

1.481 Functional relations

1. sin Π(x) =1

cosh xLO III 297

2. cosΠ(x) = tanhx LO III 297

3. tan Π(x) =1

sinh xLO III 297

4. cotΠ(x) = sinhx LO III 297

5. sin Π(x + y) =sin Π(x) sin Π(y)

1 + cosΠ(x) cos Π(y)LO III 297

6. cosΠ(x + y) =cosΠ(x) + cos Π(y)

1 + cosΠ(x) cos Π(y)LO III 183

1.482 Connection with the Gudermannian.gd(−x) = Π(x) − π

2(Definite) integral of the angle of parallelism: cf. 4.581 and 4.561.

1.49 The hyperbolic amplitude (the Gudermannian) gd x

1.490 Definition.

1. gdx =∫ x

0

dt

cosh t= 2 arctan ex − π

2JA

2. x =∫ gd x

0

dt

cos t= ln tan

(gd x

2+

π

4

)JA

1.491 Functional relations.

1. cosh x = sec(gd x) AD (343.1), JA

2. sinh x = tan(gdx) AD (343.2), JA

3. ex = sec(gdx) + tan(gdx) = tan(

π

4+

gd x

2

)=

1 + sin(gdx)cos(gd x)

AD (343.5), JA

4. tanhx = sin(gdx) AD (343.3), JA

5. tanhx

2= tan

(12

gd x

)AD (343.4), JA

6. arctan (tanh x) =12

gd 2x AD (343.6a)

1.492 If γ = gdx, then ix = gd iγ JA

1.493 Series expansion.

1.gd x

2=

∞∑

k=0

(−1)k

2k + 1tanh2k+1 x

2JA

1.513 Series representation 53

2.x

2=

∞∑

k=0

12k + 1

tan2k+1

(12

gd x

)JA

3. gdx = x − x3

6+

x5

24− 61x7

5040+ · · · JA

4. x = gd x +(gd x)3

6+

(gd x)5

24+

61(gdx)7

5040+ . . .

[gd x <

π

2

]JA

1.5 The Logarithm

1.51 Series representation

1.511 ln(1 + x) = x − 12x2 +

13x3 − 1

4x4 + · · · =

∞∑

k=1

(−1)k+1 xk

k

[−1 < x ≤ 1]1.512

1. lnx = (x − 1) − 12(x − 1)2 +

13(x − 1)3 − · · · =

∞∑

k=1

(−1)k+1 (x − 1)k

k

[0 < x ≤ 2]

2. lnx = 2

[x − 1x + 1

+13

(x − 1x + 1

)3

+15

(x − 1x + 1

)5

+ . . .

]

= 2∞∑

k=1

12k − 1

(x − 1x + 1

)2k−1

[0 < x]

3. lnx =x − 1

x+

12

(x − 1

x

)2

+13

(x − 1

x

)3

+ · · · =∞∑

k=1

1k

(x − 1

x

)k

[x ≥ 1

2

]AD (644.6)

4.∗ lnx = limε→0

(xε − 1

ε

)

1.513

1. ln1 + x

1 − x= 2

∞∑

k=1

12k − 1

x2k−1[x2 < 1

]FI II 421

2. lnx + 1x − 1

= 2∞∑

k=1

1(2k − 1)x2k−1

[x2 > 1

]AD (644.9)

3. lnx

x − 1=

∞∑

k=1

1kxk

[x ≤ −1 or x > 1] JO (88a)

4. ln1

1 − x=

∞∑

k=1

xk

k[−1 ≤ x < 1] JO (88b)

5.1 − x

xln

11 − x

= 1 −∞∑

k=1

xk

k(k + 1)[−1 ≤ x < 1] JO (102)

54 The Logarithm 1.514

6.1

1 − xln

11 − x

=∞∑

k=1

xkk∑

n=1

1n

[x2 < 1

]JO (88e)

7.(1 − x)2

2x3ln

11 − x

=1

2x2− 3

4x+

∞∑

k=1

xk−1

k(k + 1)(k + 2)[−1 ≤ x < 1] AD (6445.1)

1.514 ln(1 − 2x cos ϕ + x2

)= −2

∞∑

k=1

cos kϕ

kxk; ln

(x +

√1 + x2

)= arcsinh x

(see 1.631, 1.641, 1.642, 1.646)[x2 ≤ 1, x cos ϕ �= 1

]MO 98, FI II 485

1.515

1.11 ln(1 +

√1 + x2

)= ln 2 +

1 · 12 · 2x2 − 1 · 1 · 3

2 · 4 · 4x4 +1 · 1 · 3 · 52 · 4 · 6 · 6x6 − . . .

= ln 2 −∞∑

k=1

(−1)k (2k − 1)!22k (k!)2

x2k

[x2 ≤ 1

]JO (91)

2. ln(1 +

√1 + x2

)= lnx +

1x− 1

2 · 3x3+

1 · 32 · 4 · 5x5

− . . .

= lnx +1x

+∞∑

k=1

(−1)k (2k − 1)!22k−1 · k!(k − 1)!(2k + 1)x2k+1

[x2 ≥ 1

]AD (644.4)

3.√

1 + x2 ln(x +

√1 + x2

)= x −

∞∑

k=1

(−1)k 22k−1(k − 1)!k!(2k + 1)!

x2k+1

[x2 ≤ 1

]JO (93)

4.ln

(x +

√1 + x2

)

√1 + x2

=∞∑

k=0

(−1)k 22k (k!)2

(2k + 1)!x2k+1

[x2 ≤ 1

]JO (94)

1.516

1.12{ln (1 ± x)}2 =

∞∑

k=1

(∓1)k+1xk+1

k + 1

k∑

n=1

1n

[x2 < 1

]JO (86), JO (85)

2.16{ln(1 + x)}3 =

∞∑

k=1

(−1)k+1xk+2

k + 2

k∑

n=1

1n + 1

n∑

m=1

1m

[x2 < 1

]AD (644.14)

3. − ln(1 + x) · ln(1 − x) =∞∑

k=1

x2k

k

2k−1∑

n=1

(−1)n+1

n

[x2 < 1

]JO (87)

4.14x

{1 + x√

xln

1 +√

x

1 −√

x+ 2 ln(1 − x)

}=

12x

+∞∑

k=1

xk−1

(2k − 1)2k(2k + 1)

[0 < x < 1] AD (6445.2)

1.521 Series of logarithms (cf. 1.431) 55

1.517

1.612x

{1 − ln(1 + x) − 1 − x√

xarctan

√x

}=

∞∑

k=1

(−1)k+1xk−1

(2k − 1)2k(2k + 1)

[0 < x ≤ 1] AD (6445.3)

2.12

arctanx ln1 + x

1 − x=

∞∑

k=1

x4k−2

2k − 1

2k−1∑

n=1

(−1)n−1

2n − 1[x2 < 1

]BR* 163

3.12

arctanx ln(1 + x2

)=

∞∑

k=1

(−1)k+1x2k+1

2k + 1

2k∑

n=1

1n

[x2 ≥ 1

]AD (6455.3)

1.518

1. ln sinx= lnx − x2

6− x4

180− x6

2835− . . .

= lnx +∞∑

k=1

(−1)k22k−1B2kx2k

k(2k)!

[0 < x < π] AD (643.1)a

2.3 ln cos x= −x2

2− x4

12− x6

45− 17x8

2520− . . .

= −∞∑

k=1

22k−1(22k − 1

)|B2k|

k(2k)!x2k = −1

2

∞∑

k=1

sin2k x

k[x2 <

π2

4

]FI II 524

3. ln tanx= lnx +x2

3+

790

x4 +62

2835x6 +

12718, 900

x8 + . . .

= lnx +∞∑

k=1

(−1)k+1

(22k−1 − 1

)22kB2kx2k

k(2k)![0 < x <

π

2

]AD (643.3)a

1.52 Series of logarithms (cf. 1.431)

1.521

1.∞∑

k=1

ln(

1 − 4x2

(2k − 1)2π2

)= ln cosx

[−π

2< x <

π

2

]

2.∞∑

k=1

ln(

1 − x2

k2π2

)= ln sinx − ln x [0 < x < π]

56 The Inverse Trigonometric and Hyperbolic Functions 1.621

1.6 The Inverse Trigonometric and Hyperbolic Functions

1.61 The domain of definition

The principal values of the inverse trigonometric functions are defined by the inequalities:

1. −π

2≤ arcsinx ≤ π

2; 0 ≤ arccosx ≤ π [−1 ≤ x ≤ 1] FI II 553

2. −π

2< arctanx <

π

2; 0 < arccotx < π [−∞ < x < +∞] FI II 552

1.62–1.63 Functional relations

1.621 The relationship between the inverse and the direct trigonometric functions.

1. arcsin (sinx) = x − 2nπ[2nπ − π

2≤ x ≤ 2nπ +

π

2

]

= −x + (2n + 1)π[(2n + 1)π − π

2≤ x ≤ (2n + 1)π +

π

2

]

2. arccos (cos x) = x − 2nπ [2nπ ≤ x ≤ (2n + 1)π]

= −x + 2(n + 1)π [(2n + 1)π ≤ x ≤ 2(n + 1)π]

3. arctan (tanx) = x − nπ[nπ − π

2< x < nπ +

π

2

]

4. arccot (cotx) = x − nπ [nπ < x < (n + 1)π]

1.622 The relationship between the inverse trigonometric functions, the inverse hyperbolic functions,and the logarithm.

1. arcsin z =1i

ln(iz +

√1 − z2

)=

1i

arcsinh(iz)

2. arccos z =1i

ln(z +

√z2 − 1

)=

1i

arccosh z

3. arctan z =12i

ln1 + iz

1 − iz=

1i

arctanh(iz)

4. arccot z =12i

lniz − 1iz + 1

= i arccoth(iz)

5. arcsinh z = ln(z +

√z2 + 1

)=

1i

arcsin(iz)

6. arccosh z = ln(z +

√z2 − 1

)= i arccos z

7. arctanh z =12

ln1 + z

1 − z=

1i

arctan(iz)

8. arccoth z =12

lnz + 1z − 1

=1i

arccot(−iz)

1.624 Functional relations 57

Relations between different inverse trigonometric functions

1.623

1. arcsinx + arccos x =π

2NV 43

2. arctanx + arccotx =π

2NV 43

1.624

1. arcsinx= arccos√

1 − x2 [0 ≤ x ≤ 1] NV 47 (5)

= − arccos√

1 − x2 [−1 ≤ x ≤ 0] NV 46 (2)

2. arcsinx = arctanx√

1 − x2

[x2 < 1

]

3. arcsinx= arccot√

1 − x2

x[0 < x ≤ 1]

= arccot√

1 − x2

x− π [−1 ≤ x < 0] NV 49 (10)

4. arccos x= arcsin√

1 − x2 [0 ≤ x ≤ 1]

= π − arcsin√

1 − x2 [−1 ≤ x ≤ 0] NV 48 (6)

5. arccos x= arctan√

1 − x2

x[0 < x ≤ 1]

= π + arctan√

1 − x2

x[−1 ≤ x < 0] NV 48 (8)

6. arccos x = arccotx√

1 − x2[−1 ≤ x < 1] NV 46 (4)

7. arctanx = arcsinx√

1 + x2NV 6 (3)

8. arctanx= arccos1√

1 + x2[x ≥ 0]

= − arccos1√

1 + x2[x ≤ 0] NV 48 (7)

9. arctanx= arccot1x

[x > 0]

= − arccot1x− π [x < 0] NV 49 (9)

10.11 arccotx= arcsin1√

1 + x2[x > 0]

= π − arcsin1√

1 + x2[x < 0] NV 49 (11)

11. arccotx = arccosx√

1 + x2NV 46 (4)


12. arccotx= arctan1x

[x > 0]

= π + arctan1x

[x < 0] NV 49 (12)

1.625

1. arcsinx + arcsin y = arcsin(x√

1 − y2 + y√

1 − x2) [

xy ≤ 0 or x2 + y2 ≤ 1]

= π − arcsin(x√

1 − y2 + y√

1 − x2) [

x > 0, y > 0 and x2 + y2 > 1]

= −π − arcsin(x√

1 − y2 + y√

1 − x2) [

x < 0, y < 0 and x2 + y2 > 1]

NV 54(1), GI I (880)

2. arcsinx + arcsin y = arccos(√

1 − x2√

1 − y2 − xy)

[x ≥ 0, y ≥ 0]

= − arccos(√

1 − x2√

1 − y2 − xy)

[x < 0, y < 0] NV 55

3. arcsinx + arcsin y = arctanx√

1 − y2 + y√

1 − x2

√1 − x2

√1 − y2 − xy

[xy ≤ 0 or x2 + y2 < 1

]

= arctanx√

1 − y2 + y√

1 − x2

√1 − x2

√1 − y2 − xy

+ π[x > 0, y > 0 and x2 + y2 > 1

]

= arctanx√

1 − y2 + y√

1 − x2

√1 − x2

√1 − y2 − xy

− π[x < 0, y < 0 and x2 + y2 > 1

]

NV 56

4. arcsinx − arcsin y = arcsin(x√

1 − y2 − y√

1 − x2) [

xy ≥ 0 or x2 + y2 ≤ 1]

= π − arcsin(x√

1 − y2 − y√

1 − x2) [

x > 0, y < 0 and x2 + y2 > 1]

= −π − arcsin(x√

1 − y2 − y√

1 − x2) [

x < 0, y > 0 and x2 + y2 > 1]

NV 55(2)

5. arcsinx − arcsin y = arccos(x√

1 − x2√

1 − y2 + xy)

[xy > y]

= − arccos(√

1 − x2√

1 − y2 + xy)

[x < y] NV 56

6. arccos x + arccos y = arccos(xy −

√1 − x2

√1 − y2

)[x + y ≥ 0]

= 2π − arccos(xy −

√1 − x2

√1 − y2

)[x + y < 0] NV 57 (3)

7.11 arccos x − arccos y = − arccos(xy +

√1 − x2

√1 − y2

)[x ≥ y]

= arccos(xy +

√1 − x2

√1 − y2

)[x < y] NV 57 (4)

1.627 Functional relations 59

8. arctanx + arctan y = arctanx + y

1 − xy[xy < 1]

= π + arctanx + y

1 − xy[x > 0, xy > 1]

= −π + arctanx + y

1 − xy[x < 0, xy > 1]

NV 59(5), GI I (879)

9. arctanx − arctan y = arctanx − y

1 + xy[xy > −1]

= π + arctanx − y

1 + xy[x > 0, xy < −1]

= −π + arctanx − y

1 + xy[x < 0, xy < −1]

NV 59(6)

1.626

1. 2 arcsinx= arcsin(2x

√1 − x2

) [|x| ≤ 1√

2

]

= π − arcsin(2x

√1 − x2

) [1√2

< x ≤ 1]

= −π − arcsin(2x

√1 − x2

) [−1 ≤ x < − 1√

2

]

NV 61 (7)

2. 2 arccos x= arccos(2x2 − 1

)[0 ≤ x ≤ 1]

= 2π − arccos(2x2 − 1

)[−1 ≤ x < 0] NV 61 (8)

3. 2 arctanx= arctan2x

1 − x2[|x| < 1]

= arctan2x

1 − x2+ π [x > 1]

= arctan2x

1 − x2− π [x < −1]

NV 61 (9)

1.627

1. arctanx + arctan1x

=π

2[x > 0]

= −π

2[x < 0] GI I (878)

2. arctanx + arctan1 − x

1 + x=

π

4[x > −1]

= −34π [x < −1] NV 62, GI I (881)


1.628

1. arcsin2x

1 + x2= −π − 2 arctanx [x ≤ −1]

= 2 arctanx [−1 ≤ x ≤ 1]

= π − 2 arctanx [x ≥ 1]

NV 65

2. arccos1 − x2

1 + x2 = 2 arctanx [x ≥ 0]

= −2 arctan x [x ≤ 0] NV 66

1.6292x − 1

2− 1

πarctan

(tan

2x − 12

π

)= E (x) GI (886)

1.631 Relations between the inverse hyperbolic functions.

1. arcsinhx = arccosh√

x2 + 1 = arctanhx√

x2 + 1JA

2. arccosh x = arcsinh√

x2 − 1 = arctanh√

x2 − 1x

JA

3. arctanhx = arcsinhx√

1 − x2= arccosh

1√1 − x2

= arccoth1x

JA

4. arcsinhx ± arcsinh y = arcsinh(x√

1 + y2 ± y√

1 + x2)

JA

5. arccosh x ± arccosh y = arccosh(xy ±

√(x2 − 1) (y2 − 1)

)JA

6. arctanhx ± arctanh y = arctanhx ± y

1 ± xyJA

1.64 Series representations

1.641

1. arcsinx=π

2− arccos x = x +

12 · 3x3 +

1 · 32 · 4 · 5x5 +

1 · 3 · 52 · 4 · 6 · 7x7 + . . .

=∞∑

k=0

(2k)!22k (k!)2 (2k + 1)

x2k+1 = xF(

12,12;32; x2

)

[x2 ≤ 1

]FI II 479

2. arcsinhx= x − 12 · 3x3 +

1 · 32 · 4 · 5x5 − . . . ;

=∞∑

k=0

(−1)k (2k)!22k (k!)2 (2k + 1)

x2k+1

= xF(

12 , 1

2 ; 32 ;−x2

)

[x2 ≤ 1

]FI II 480

1.645 Series representations 61

1.642

1. arcsinhx= ln 2x +12

12x2

− 1 · 32 · 4

14x4

+ . . .

= ln 2x +∞∑

k=1

(−1)k+1 (2k)!x−2k

22k (k!)2 2k[x ≥ 1]

AD (6480.2)a

2. arccosh x = ln 2x −∞∑

k=1

(2k)!x−2k

22k (k!)2 2k[x ≥ 1] AD (6480.3)a

1.643

1. arctanx= x − x3

3+

x5

5− x7

7+ . . .

=∞∑

k=0

(−1)kx2k+1

2k + 1[x2 ≤ 1

]FI II 479

2. arctanhx = x +x3

3+

x5

5+ · · · =

∞∑

k=0

x2k+1

2k + 1[x2 < 1

]AD (6480.4)

1.644

1. arctanx=x√

1 + x2

∞∑

k=0

(2k)!22k (k!)2 (2k + 1)

(x2

1 + x2

)k

=x√

1 + x2F

(12,12;32;

x2

1 + x2

)[x2 < ∞

]

AD (641.3)

2. arctanx =π

2− 1

x+

13x3

− 15x5

+1

7x7− · · · =

π

2−

∞∑

k=0

(−1)k 1(2k + 1)x2k+1

AD (641.4)

1.645

1. arcsec x=π

2− 1

x− 1

2 · 3x3− 1 · 3

2 · 4 · 5x5− · · · =

π

2−

∞∑

k=0

(2k)!x−(2k+1)

(k!)2 22k(2k + 1)

=π

2− 1

xF

(12,12;32;

1x2

) [x2 > 1

]

AD (641.5)

2. (arcsinx)2 =∞∑

k=0

22k (k!)2 x2k+2

(2k + 1)!(k + 1)[x2 ≤ 1

]AD (642.2), GI III (152)a

3. (arcsinx)3 = x3 +3!5!

32

(1 +

132

)x5 +

3!7!

32 · 52

(1 +

132

+152

)x7 + . . .

[x2 ≤ 1

]

BR* 188, AD (642.2), GI III (153)a


1.646

1. arcsinh1x

= arcosechx =∞∑

k=0

(−1)k(2k)!22k (k!)2 (2k + 1)

x−2k−1

[x2 ≥ 1

]AD (6480.5)

2. arccosh1x

= arcsech x = ln2x−

∞∑

k=1

(2k)!22k (k!)2 2k

x2k [0 < x ≤ 1] AD (6480.6)

3. arcsinh1x

= arcosechx = ln2x

+∞∑

k=1

(−1)k+1(2k)!22k (k!)2 2k

x2k

[0 < x ≤ 1] AD (6480.7)a

4. arctanh1x

= arccothx =∞∑

k=0

x−(2k+1)

2k + 1[x2 > 1

]AD (6480.8)

1.647

1.∞∑

k=1

tanh(2k − 1) (π/2)(2k − 1)4n+3

=π4n+3

2

2n∑

j=1

(−1)j−1(22j − 1

) (24n−2j+4 − 1

)B∗

2j−1B∗4n−2j+3

(2j)!(4n − 2j + 4)!

+(−1)n

(22n+2 − 1

)2B∗

2n+12

[(2n + 2)!]2

n = 0, 1, 2, . . . ,

2.∞∑

k=1

(−1)k−1 sech(2k − 1) (π/2)(2k − 1)4n+1

=π4n+1

24n+3

2n−1∑

j=1

(−1)jB∗2jB

∗4n−2j

(2j)!(4n − 2j)!+

2B∗4n

(4n)!+

(−1)nB∗2n

2

[(2n)]!2

,

n = 1, 2, . . .

(The summation term on the right is to be omitted for n = 1.) (See page xxxiii for the definition of B∗r .)

Documents

1 Elementary Functions - Elsevier.com · 2013. 12. 20. · 3. cos2 x−sin2 y =cos(x+y)cos(x−y) = cos2 y −sin2 x 4. sinh2 x+cosh2 y =cosh(x+y)cosh(x−y)=cosh2 x+sinh2 y 1.316