1 Elementary Functions

Elementary Functions. Power of Binomials. Power series.0 + q =+q + qq + +! qq...q + + =! q If q is neither a natural number nor zero, the series converges absolutely for < and diverges for >. For =, the series converges for q> and diverges for q. For =,theseries converges absolutely for q>0. For =, it converges absolutely for q>0 and diverges for q<0. If q = n is a natural number, the series.0 is reduced to the finite sum.. FI II 5 n n. a + n = a n.. + = + 3 + =. + = +3 3 + = 3. + / =+ + 3 6 3 3 5 6 8 +.... + / = + 3 3 5 6 3 +....3.. = < ] see also. q + + = q + q qq 3 qq q 5 ] 3 + + +...!! 3! <, q is a real number ] AD 635. 5

6 The Eponential Function.. + q q q q... q ] + + =+ +! q q 3... q ] +q + q + +! <, q isarealnumber ] AD635.. Series of rational fractions... = + = = + < ] AD 6350.3 > ] AD 6350.3. The Eponential Function. Series representation.. e =!. a ln a =! 3. e =!. e = lim + n n. e + =.3. e e = e.5 n +! e = + + +! B! + 53 3!. e sin =+ +! 3!. e cos = e + 5! 85 5!! + 36 +...! 6! +... <π] FI II 50 AD 660.3 36 + 567 +... AD 660. 6! 7! AD 660.5

.3 Series of eponentials 7 3. e tan =+ +! + 33 + 9 + 375 +... AD 660.6 3!! 5!.6. e arcsin =+ +! + 3 + 5 +... AD 660.7 3!!. e arctan =+ +! 3 3! 7 +... AD 660.8!.7. π eπ + e π e π e π =. π e π e π = = = + cf.. 3 AD 6707. + cf.. 3 AD 6707.. Functional relations.. a = e ln a. a log a = a log a =.. e =cosh +sinh. e i =cos + i sin.3 ] e a e b =a bep a + b + a b ] π MO 6.3 Series of eponentials.3 a = a a>and<0or0<a<and>0].3. tanh =+ e >0]. sech = e + 3. cosech =. sin =ep e + n= ] cos n n >0] >0] 0 π]

8 Trigonometric and Hyperbolic Functions.3.3. Trigonometric and Hyperbolic Functions.30 Introduction The trigonometric and hyperbolic sines are related by the identities sinh = i sini, sin = i sinhi. The trigonometric and hyperbolic cosines are related by the identities cosh =cosi, cos =coshi. Because of this duality, every relation involving trigonometric functions has its formal counterpart involving the corresponding hyperbolic functions, and vice versa. In many though not all cases, both pairs of relationships are meaningful. The idea of matching the relationships is carried out in the list of formulas given below. However, not all the meaningful pairs are included in the list..3 The basic functional relations.3. sin = e i e i i = i sinhi. sinh = e e = i sini 3. cos = e i + e i =coshi. cosh = e + e =cosi 5. tan = sin cos = i tanhi 6. tanh = sinh cosh = i tani 7. cot = cos sin = tan = i cothi 8. coth = cosh sinh = = i cot i tanh.3. cos +sin =

.3 The basic functional relations 9. cosh sinh =.33. sin ± y =sincos y ± sin y cos. sinh ± y =sinhcosh y ± sinh y cosh 3. sin ± iy =sincosh y ± i sinh y cos. sinh ± iy =sinhcos y ± i sin y cosh 5. cos ± y = cos cos y sin sin y 6. cosh ± y =coshcosh y ± sinh sinh y 7. cos ± iy = cos cosh y i sin sinh y 8. cosh ± iy =coshcos y ± i sinh sin y 9. tan ± tan y tan ± y = tan tan y 0. tanh ± tanh y tanh ± y = ± tanh tanh y. tan ± i tanh y tan ± iy = i tan tanh y. tanh ± i tan y tanh ± iy = ± i tanh tan y.3. sin ± sin y =sin ± ycos y. sinh ± sinh y =sinh ± ycosh y 3. cos +cosy = cos + ycos y. cosh +coshy =cosh + ycosh y 5. cos cos y =sin + ysin y 6. cosh cosh y =sinh + ysinh y 7. sin ± y tan ± tan y = cos cos y sinh ± y 8. tanh ± tanh y = cosh cosh y 9. sin ± cos y = ±sin + y ± π ] sin y ± π ] = ±cos + y π ] cos y π ] =sin ± y ± π ] cos y π ]

30 Trigonometric and Hyperbolic Functions.35 0. a sin ± b cos = a + ] b b sin ± arctan a a. ±a sin + b cos = b + a a ] cos arctan b b a 0]. a sin ± b cos y = q + r sin ± y + arctan q q =a + bcos b 0] ] r q ] y, r =a bsin s s ] 3. a cos + b cos y = t + cos t y + arctan t 0] t ] t t = s + cos y arctan s 0] s s s =a bsin ± y.35. sin sin y =sin + ysin y = cos y cos. sinh sinh y =sinh + ysinh y =cosh cosh y 3. cos sin y =cos + ycos y = cos y sin. sinh +cosh y =cosh + ycosh y =cosh +sinh y.36. cos + i sin n =cosn + i sin n n is an integer]. cosh +sinh n =sinhn +coshn n is an integer].37. sin = ± cos ] y ], t =a + bcos q 0] ] ± y. sinh = ± cosh 3. cos = ± + cos. cosh = cosh + 5. tan = cos = sin sin + cos

.3 Trigonometric and hyperbolic functions: epansion in multiple angles 3 6. tanh = cosh sinh = sinh cosh + The signs in front of the radical in formulas.37,.37, and.37 3aretaensoastoagree with the signs of the left-hand members. The sign of the left hand members depends in turn on the value of..3 The representation of powers of trigonometric and hyperbolic functions in terms of functions of multiples of the argument angle.30. sin n = n { n. sinh n = n n n n { n n n 3. sin n = n n. sinh n = n n { 5. cos n = n n { 6. cosh n = n n 7. cos n = n 8. cosh n = n n n+ cos n + n n n+ n n n n cos n + n n cosh n + cosh n + } n KR 56 0, n } n n sinn KR 56 0, sinhn } n KR 56 0, n } n n cosn KR 56 0, 3 coshn Special cases.3. sin = cos +. sin 3 = sin 3 +3sin 3. sin = cos cos +3 8. sin 5 = sin 5 5sin3 +0sin 6

3 Trigonometric and Hyperbolic Functions.3 5. sin 6 = cos 6 +6cos 5 cos + 0 3 6. sin 7 = sin 7 +7sin5 sin 3 +35sin 6.3. sinh = cosh. sinh 3 = sinh 3 3sinh 3. sinh = cosh cosh +3 8. sinh 5 = sinh 5 5sinh3 +0sinh 6 5. sinh 6 = cosh 6 6cosh +5cosh + 0 3 6. sinh 7 = sinh 7 7sinh5 +sinh3 +35sinh 6.33. cos = cos +. cos 3 = cos 3 + 3 cos 3. cos = cos +cos +3 8. cos 5 = cos 5 +5cos3 +0cos 6 5. cos 6 = cos 6 +6cos +5cos + 0 3 6. cos 7 = cos 7 +7cos5 +cos3 +35cos 6.3. cosh = cosh +. cosh 3 = cosh 3 +3cosh 3. cosh = cosh + cosh +3 8. cosh 5 = cosh 5 + 5 cosh 3 +0cosh 6 5. cosh 6 = cosh 6 + 6 cosh +5cosh + 0 3 6. cosh 7 = cosh 7 + 7 cosh 5 +cosh3 +35cosh 6

.33 Trigonometric and hyperbolic functions: epansion in powers 33.33 The representation of trigonometric and hyperbolic functions of multiples of the argument angle in terms of powers of these functions.33 n n. 7 sin n = n cos n sin cos n 3 sin 3 + cos n 5 sin 5...; 3 5 n =sin n cos n n 3 cos n 3 n 3 n + n 5 cos n 5 n 7 cos n 7 +... 3. sinh n = n+/] =sinh 3. cos n =cos n. 3 cosh n = n /] n sinh cosh n + n n cos n sin + = n cos n n n 3 cos n + n n n n 7 cos n 6 +... 3 n/] n sinh cosh n n cosh n n cos n sin...; n 3 n 5 cos n n/] = n cosh n + n n n cosh n.33 {. sin n =ncos sin n n sin 3 n } + sin 5... 3! 5! { = n cos n sin n n n 3 sin n 3! n 3n + n 5 sin n 5! } n n 5n 6 n 7 sin n 7 +... 3! AD 3.75 AD 3.75 AD 3.7 AD 3.73

3 Trigonometric and Hyperbolic Functions.333. sinn =n {sin n sin 3 3! n ] n 3 ] } + sin 5... 5! { = n n sin n n n sin n 3! n n + n 6 sin n 5! } n n 5n 6 n 8 sin n 7 +... 3! 3. cos n = n! sin + n n! = n { n sin n n! n 3 sin n sin n n n 6! sin 6 +... AD 3.7 AD 3.7 AD 3.7 + nn 3 n 5 sin n! } nn n 5 n 7 sin n 6 +... 3! AD 3.73a. cosn =cos { n sin! n ] n 3 ] } + sin...! { = n cos n sin n n 3 n sin n! n n 5 + n 6 sin n 6! } n 5n 6n 7 n 8 sin n 8 +... 3! AD 3.7 AD 3.7 By using the formulas and values of.30, we can write formulas for sinh n, sinhn, coshn, and coshn that are analogous to those of.33, just as was done in the formulas in.33. Special cases.333. sin =sincos. sin 3 =3sin sin 3 3. sin =cos sin 8sin 3. sin 5 =5sin 0 sin 3 +6sin 5 5. sin 6 =cos 6sin 3 sin 3 +3sin 5

.337 Trigonometric and hyperbolic functions: epansion in powers 35 6. sin 7 =7sin 56 sin 3 + sin 5 6 sin 7.33. sinh =sinhcosh. sinh 3 =3sinh +sinh 3 3. sinh =cosh sinh +8sinh 3. sinh 5 =5sinh +0sinh 3 +6sinh 5 5. sinh 6 =cosh 6sinh +3sinh 3 +3sinh 5 6. sinh 7 =7sinh +56sinh 3 + sinh 5 +6sinh 7.335. cos = cos. cos 3 = cos 3 3cos 3. cos = 8 cos 8cos +. cos 5 =6cos 5 0 cos 3 + 5 cos 5. cos 6 =3cos 6 8 cos +8cos 6. cos 7 =6cos 7 cos 5 +56cos 3 7cos.336. cosh =cosh. cosh 3 =cosh 3 3cosh 3. cosh =8cosh 8cosh +. cosh 5 =6cosh 5 0 cosh 3 +5cosh 5. cosh 6 =3cosh 6 8 cosh +8cosh 6. cosh 7 =6cosh 7 cosh 5 +56cosh 3 7cosh.337. cos 3 cos 3 = 3tan. cos cos = 6tan +tan 3. cos 5 cos 5 = 0 tan +5tan. cos 6 cos 6 = 5 tan +5tan tan 6 5. sin 3 cos 3 =3tan tan3 6. sin cos =tan tan3

36 Trigonometric and Hyperbolic Functions.3 7. sin 5 cos 5 =5tan 0 tan3 +tan 5 8. sin 6 cos 6 =6tan 0 tan3 +6tan 5 9. cos 3 sin 3 =cot3 3cot 0. cos sin =cot 6cot +. cos 5 sin 5 =cot5 0 cot 3 + 5 cot. cos 6 sin 6 =cot6 5 cot +5cot 3. sin 3 sin 3 = 3 cot. sin sin = cot3 cot 5. sin 5 sin 5 = 5 cot 0 cot + 6. sin 6 sin 6 = 6 cot5 0 cot 3 + 6 cot.3 Certain sums of trigonometric and hyperbolic functions.3.. 3.. 5. 6. n n n n n sin + y =sin sinh + y =sinh cos + y = cos cosh + y =cosh + n y + n y + n y sin ny cosec y + n y sinh ny sinh y sin ny cosec y sinh ny sinh y cos + y =sin + n y sin ny sec y n sin + y =sin + n y + π sin ny + π sec y AD 36.8 AD 36.9 JO 0 AD 0a

.35 Sums of powers of trigonometric functions of multiple angles 37 Special cases.3. n. 0 n 3...33.. 3. sin =sin n + sin n cosec cos =cos n + sin n cosec + =cos n n + sin cosec = + sin n + sin AD 36. AD 36. n sin =sin n cosec AD 36.7 n cos = sin n cosec JO 07 n cos = + n cos n+ cos n + n+ sin n sin = cos n cos 3 + n sin =sinn cos n +sinn cos +sin cosec AD 36. AD 36.0 JO 08.3.. 3. n sin π n =cot π n n sin π n n = + cos nπ sin nπ n cos π n n = + cos nπ +sinnπ AD 36.9 AD 36.8 AD 36.7.35 Sums of powers of trigonometric functions of multiple angles.35. n sin = n +sin sinn +]cosec = n cosn +sin n sin AD 36.3

38 Trigonometric and Hyperbolic Functions.35. 3.. 5. 6. n cos = n + cos n sinn +cosec = n cosn +sin n + sin n sin 3 = 3 n cos 3 = 3 n + sin sin n cosec 3n + sin sin 3n cosec 3 n + cos sin n cosec + 3n + cos sin 3n cosec 3 AD 36.a JO 0 JO a n sin = 3n cosn +sin n cosec + cos n +sin n cosec ] JO 8 n cos = 3n +cosn +sin n cosec + cos n +sin n cosec ] JO 3 8.35. n sin n sin = sin. n n cos n sin cos = n sin n cos n sin sin AD 36.5 AD 36.6.353.. 3.. n p sin = p sin pn sin n + p n+ sinn p cos + p AD 36.a n p sinh = p sinh pn sinh n + p n+ sinhn p cosh + p n p cos = p cos pn cos n + p n+ cosn p cos + p AD 36.3a n p cosh = p cosh pn cosh n + p n+ coshn p cosh + p JO 396.36 Sums of products of trigonometric functions of multiple angles.36.. n sin sin + = n +sin sin n +]cosec JO n sin sin + = n cos cosn +3sin n cosec JO 6

.38 Sums leading to hyperbolic tangents and cotangents 39 3. n sin cos y =sin ny + n + sin ny n + n +y sin cosec +y ny sin cosec y JO 7.36.. n n sin = n sin n sin AD 36.5 sec =cosec n cosec n AD 36..37 Sums of tangents of multiple angles.37.. n n tan = n cot cot AD 36.6 n tan = n+ 3 n + cot n cot n AD 36.0.38 Sums leading to hyperbolic tangents and cotangents.38.. n n tanh + n sin + n tanh + tan n π tanh n sin π n + tanh π tan n π =tanhn JO 0a =cothn tanh +coth JO 03 n

0 Trigonometric and Hyperbolic Functions.38 3.. n tanh n +sin + n + π + tanh + tan n + π tanh n n +sin π n + tanh + π tan n + =tanhn + tanh n + =cothn + coth n + JO 0 JO 05.38. n = n tanh n JO 06 + sin n π + sinh tanh. n =ncoth n coth JO 07 π sin n + sinh tanh 3.. n n + =n +tanh + sin n + π + sinh tanh tanh n n + =n +coth coth π sin n + + sinh tanh JO 08 JO 09

.395 Representing sines and cosines as finite products.39 The representation of cosines and sines of multiples of the angle as finite products.39. sin n = n sin cos. cos n = n 3. sin n = n sin. cos n =cos n n n sin π sin n sin sin π n sin sin π n is even] JO 568 n sin n is even] JO 569 π sin n.39 n. sin n = n sin + π n. cos n = n n.393. n n..39 n cos + n π sin + n π = sin + n π n is odd] JO 570 n is odd] JO 57a = cos n n odd] n = n ] n cos n n even] n n sin n n odd] JO 58 JO 59 JO 53 = n n cos n n even] JO 5 { y cos α + π } + y = n n y n cos nα + y n JO 573 n.395 n {. cos n cos ny = n cos cos y + π } n JO 573

Trigonometric and Hyperbolic Functions.396 n. cosh n cos ny = n { cosh cos y + π } n JO 538.396.. 3.. n n n n cos πn + = n π cos n + + = n+ π +cos n + + = n+ + KR 58 8. KR 58 8. KR 58 8.3 +π cos + = n + KR 58 8. n. The epansion of trigonometric and hyperbolic functions in power series.. sin =. sinh = 3. cos = + +! + +!!. cosh =! ] 5. tan = B < π! 6. tanh = 3 3 + 5 5 7 35 7 + = B! ] < π 7. cot = FI II 53 B <π ] FI II 53a! 8. coth = + 3 3 5 + 5 95 = + B! <π ] FI II 5a

. Trigonometric and hyperbolic functions: power series epansion 3 9. sec = E! 0. sech = + 5 66 70 + =+. cosec = + B! E. cosech = 6 + 73 360 35 50 + = ] < π! ] < π CE 330a CE 330 <π ] CE 39a B! <π ] JO 8.. sin = +!. cos = 3. sin 3 =. cos 3 = +! JO 5a JO 3 + 3+ 3 +! + JO 5aa 3 +3! JO 3a.3. sinh =cosec +!. cosh =sec +sec 3. sinh =sec! /]!. cosh =cosec /]! JO 508 JO 507 JO 50 JO 509.. cos n ln + ] + = n +0 n +... n + ] + +! < ] AD 656.

Trigonometric and Hyperbolic Functions.. sin n ln + ] + = n n n + + n +3... n + ] + +! < ] AD 656. Power series for ln sin, ln cos, and lntan see.58.. Epansion in series of simple fractions.. tan π = π. 0 tanh π = π 3. cot π = π + π. coth π = π + π 5. tan π =.. sec π = π BR* 9, AD 695. + = π + π = 0 + cf..7 AD 695., JO 50a 3... ] JO 50 + AD 695.3a. sec π = π { } + + 3. cosec π = π + π. cosec π = π 5. + cosec 6. cosec π = π.3 = = = JO 5a see also.7 AD 695.a = π + π + π + JO 6 JO 9 JO 50b π π m cosec m + π m cot π m = m JO 77

.39 Representation in the form of an infinite product 5.3 Representation in the form of an infinite product.3. sin =. sinh = 3. cos =. cosh = π + π + π + + π EU EU EU EU.3. cos cos y = y sin y. cosh cos y = +.33 cos π sin π = y sin y + π + y + + ] π + y π y π y AD 653. AD 653. BR* 89.3 cos = π ] π + + MO 6 π sin π + a.35 = + a + MO 6 sin πa a a + a.36 sin π ] sin πa = MO 6 a = sin 3.37 sin = ] MO 6 + π = cosh cos a ].38 = + MO 6 cos a π + a.39. sin =. sin = cos < ] AD 65, MO 6 = 3 sin 3 ] MO 6

6 Trigonometric and Hyperbolic Functions...5 Trigonometric Fourier series... 3.. sin cos = π 0 <<π] FI III 539 = ln cos ] 0 <<π] FI III 530a, AD 68 sin cos = =ln cos π <<π] FI III 5 π <<π] FI III 550... 3. sin cos = π sign π <<π] FI III 5 = ln cot 0 <<π] sin = π ln tan + π ] <<π. 0 cos = π = π π ] <<π ] π <<3π BR* 68, JO 66, GI III95 BR* 68, JO 68a BR* 68, JO 69.3. 8 cos π n = n n πn n! n = n π n B n n! n B n ρ 0, ρ = ] CE 30, GE 7. sin π n+ = n n πn+ n +! π n+ n+ = n n +! B n+ n + B n + ρ 0 <<; ρ = ] CE 30

.5 Trigonometric Fourier series 7 3.. 5. 6. 7.... 3.. 5. cos cos sin 3 cos sin 5 = π 6 π + sin + + cos + + = π = π 6 π + 3 = π 90 π + π3 8 = π 90 π 3 36 + π 8 5 0 0 π] FI III 57 π π] FI III 5 0 π] =sin π sin sin cos ln sin π =cos sin +sin ln sin 0 π] AD 667 0 π] AD 688 0 π] BR* 68, GI III 90 0 π] BR* 68 sin + =sin cos + + cos sin ln MO 3 cos + =cos cos + sin + cos ln MO 3 sin + + = π π π ] = π π π 3 ] π 6. 6 7..5.. cos = π π cos + = π 0 sin π ] sin + α = π sinh απ sinh απ MO 3 π π] FI III 56 JO 59 0 <<π] BR* 57, JO cos + α = π cosh απ α sinh απ α 0 π] BR* 57, JO 0

8 Trigonometric and Hyperbolic Functions.6 3.. 5. 6. 7. cos + α sin + α = π sinh α sinh απ = π cosh α α sinh απ α π π] FI III 56 sin sin {αm +π ]} = π α sinαπ π <<π] FI III, 56 if =mπ, then ] =0 mπ < < m +π, α not an integer] MO 3 cos α = α π cos α {m +π }] α sin απ mπ m +π, α not an integer] MO 3 sin sinαmπ ] = π if =m +π, then ] =0, α sinαπ m π <<m +π, α not an integer] FI III 55a 8. cos α = α π cosαmπ ] α sin απ m π m +π, α not an integer] FI III 55a 9. n= e inα n β + γ = π e iβα π sinhγα+e iβα sinh γπ α] γ coshπγ cosπβ 0 α π].6.7.. + cos + + +3 = π 8 cos 3 cos π π ] BR* 56, GI III 89 p p sin sin = p cos + p p < ] FI II 559 p p cos cos = p cos + p p < ] FI II 559 3. + p p cos = p cos + p p < ] FI II 559a, MO 3

.9 Trigonometric Fourier series 9.8. p sin =arctan p sin p cos 0 <<π, p ] FI II 559. 3. p cos p sin = ln p cos + p 0 <<π, p ] FI II 559 = arctan p sin p 0 <<π, p ] JO 59. p cos = ln +p cos + p p cos + p 0 <<π, p ] JO 59 5. p sin = ln +p sin + p p sin + p 0 <<π, p ] JO 6 6. p cos = arctan p cos p 0 <<π, p ] JO 597.9.. p sin! p cos! = e p cos sin p sin = e p cos cos p sin p ] JO 86 p ] JO 85 Let S = cos + and C = sin. 3. n n a Sn =π Ca cotπasa] 0 <<π, a 0, ±, ±,...] n=. n= 5. n= n a Cn = a π Sa cotπaca] a 0 π, a 0, ±, ±,...] n n n a Sn =π cosecπasa π <<π, a 0, ±, ±,...]

50 Trigonometric and Hyperbolic Functions.5 6. n= 7. n= 8. n= 9. n= 0. n= n n Cn = a a + π cosecπaca π <<π, a 0, ±, ±,...] a n πa ] n a Sn =π Ca+tan Sa n a Cn = π Sa tan a n n a Sn = π πa a sec Sa n n πa n a Cn =π sec Ca πa Ca 0 <<π, a 0, ±, ±,...] ] 0 π, a 0, ±, ±,...] π π ], a 0, ±, ±,... π π, a 0, ±, ±,... ] Fourier epansions of hyperbolic functions.5 +0 +... + ]. sinh =cos sin + JO 50 +! + +3... + ]. cosh =cos +cos sin JO 503!.5. sinh cos θ=sec sin θ. cosh cos θ=sec sin θ 3. sinh cos θ=cosec sin θ. cosh cos θ=cosec sin θ + cos +θ +! cos θ! sin θ! + sin +θ +! < ] JO 39 < ] JO 390 <, sin θ 0 ] JO 393 <, sin θ 0 ] JO 39

.80 Lobachevsiy s Angle of Parallelism 5.6 Series of products of eponential and trigonometric functions.6. e t sin = sin cosh t cos. +.6 9.63 e t sinh t cos = cosh t cos sin sin y. e cos ϕ cos sin ϕ =. e cos ϕ sin sin ϕ = e t = ln n=0 n= n cos nϕ n! n sin nϕ n! sin + y sin y t >0] MO 3 t >0] MO 3 +sinh t MO +sinh t < ] AD 676. < ] AD 676..7 Series of hyperbolic functions.7.. 3..7.. sinh! cosh! = e cosh sinh sinh. JO 395 = e cosh cosh sinh. JO 39 m +π + 3 tanh + tanh ] m +π = π3 6 p p sinh sinh = p p cosh + p < ] JO 396 p cosh = p cosh p cosh + p p < ] JO 397a.8 Lobachevsiy s Angle of Parallelism Π.80 Definition.. Π = arccot e = arctan e 0] LO III 97, LOI 0

5 Trigonometric and Hyperbolic Functions.8. Π =π Π <0] LO III 83, LOI 93.8 Functional relations. sin Π = LO III 97 cosh. cos Π =tanh LO III 97 3. tan Π = LO III 97 sinh. cot Π =sinh LO III 97 5. sin Π + y = 6. cos Π + y = sin ΠsinΠy + cos ΠcosΠy cos Π + cos Πy + cos ΠcosΠy.8 Connection with the Gudermannian. gd =Π π Definite integral of the angle of parallelism: cf..58 and.56..9 The hyperbolic amplitude the Gudermannian gd LO III 97 LO III 83.90 Definition. dt. gd = 0 cosh t = arctan e π gd dt gd. = cos t =lntan + π 0 JA JA.9 Functional relations.. cosh = secgd AD 33., JA. sinh = tangd AD 33., JA π 3. e = secgd +tangd =tan + gd = +singd AD 33.5, JA cosgd. tanh = singd AD 33.3, JA 5. tanh =tan gd AD 33., JA 6. arctan tanh = gd AD 33.6a.9 If γ =gd, theni =gdiγ JA.93 Series epansion.. gd = + tanh+ JA

.53 Series representation 53. = + tan+ gd 3. gd = 3 6 + 5 67 500. =gd + gd 3 6.5 The Logarithm + gd 5.5 Series representation + JA + 6gd 7 500.5 ln + = + 3 3 + =.5 +.... ln = + 3 3 =. ln = 3. ln = + + 3 + 3 + + 5. ɛ ln = lim ɛ 0 ɛ + 3 + gd < π ] < ] + 5 +...] + 3 + = 0 < ] = 0 <] ] + JA JA AD 6.6.53. ln + = < ] FI II. ln + = > ] AD 6.9 3. ln. ln 5. = or>] JO 88a = ln = + <] JO 88b <] JO 0

5 The Logarithm.5 6. 7. ln = 3 ln n= n = 3 +.5 ln cos ϕ + =.55. ln + + cos ϕ ; see.63,.6,.6,.66 < ] JO 88e <] AD 65. ln + + =arcsinh, cos ϕ ] MO 98, FI II 85 + + =ln+ 3 + 3 5 6 6 6...! =ln! ] JO 9. ln + + =ln + 3 3 + 3 5 5... =ln + +!!! + + ] AD 6. 3.. + ln + + =!! + +! ] JO 93 ln + + + =! +! + ] JO 9.56.. {ln ± } = 6 {ln + }3 = + + + + + 3. ln + ln =. + n= n= n= { + ln + } +ln n n + n+ n n m= = + m < ] JO 86, JO 85 < ] AD 6. < ] JO 87 + 0 <<] AD 65.

.5 Series of logarithms cf..3 55.57. 6. 3..58 { ln + arctan } = + arctan ln = arctan ln + = n= n n + + + n= + + n 0 < ] AD 65.3 < ] BR* 63 ] AD 655.3. ln sin =ln 6 80 6 835... B =ln +! 0 <<π] AD 63.a. 3 ln cos = 6 5 78 50... B = =! sin ] < π 3. ln tan =ln + 3 + 7 90 + 6 835 6 + 7 8, 900 8 +... =ln + + B! 0 << π ] FI II 5 AD 63.3a.5 Series of logarithms cf..3.5.. ln π =lncos ln π =lnsin ln π ] <<π 0 <<π]

56 The Inverse Trigonometric and Hyperbolic Functions.6.6 The Inverse Trigonometric and Hyperbolic Functions.6 The domain of definition The principal values of the inverse trigonometric functions are defined by the inequalities:. π arcsin π ; 0 arccos π ] FI II 553. π < arctan <π ; 0 < arccot <π <<+ ] FI II 55.6.63 Functional relations.6 The relationship between the inverse and the direct trigonometric functions.. arcsin sin = nπ = +n +π nπ π nπ + π ] n +π π n +π + π ]. arccos cos = nπ nπ n +π] = +n +π n +π n +π] 3. arctan tan = nπ nπ π <<nπ+ π ]. arccot cot = nπ nπ < < n +π].6 The relationship between the inverse trigonometric functions, the inverse hyperbolic functions, and the logarithm.. arcsin z = i ln iz + z = i arcsinhiz. arccos z = z i ln + z = i arccosh z 3. arctan z = +iz ln i iz = i arctanhiz. arccot z = iz ln i iz + = i arccothiz 5. arcsinh z =ln z + z + = i arcsiniz 6. arccosh z =ln z + z = i arccos z 7. arctanh z = 8. arccoth z = ln +z z = i arctaniz ln z + z = i arccot iz

.6 Functional relations 57 Relations between different inverse trigonometric functions.63. arcsin + arccos = π. arctan + arccot = π NV 3 NV 3.6. arcsin = arccos 0 ] NV 7 5 = arccos 0] NV 6. arcsin =arctan < ] 3. arcsin = arccot 0 < ] = arccot π <0] NV 9 0. arccos =arcsin 0 ] = π arcsin 0] NV 8 6 5. arccos =arctan = π +arctan 6. arccos = arccot 7. arctan =arcsin + 8. arctan = arccos + = arccos + 0 < ] <0] NV 8 8 <] NV 6 NV 6 3 0] 0] NV 8 7 9. arctan = arccot >0] = arccot π <0] NV 9 9 0. arccot =arcsin + = π arcsin +. arccot = arccos + >0] <0] NV 9 NV 6

58 The Inverse Trigonometric and Hyperbolic Functions.65. arccot =arctan = π +arctan >0] <0] NV 9.65. arcsin +arcsiny =arcsin y + y y 0or + y ] = π arcsin y + y >0, y > 0and + y > ] = π arcsin y + y <0, y < 0and + y > ]. arcsin +arcsiny = arccos y y = arccos y y 0, y 0] NV 5, GI I 880 <0, y < 0] NV 55 3. arcsin +arcsiny =arctan y + y y y y 0or + y < ] =arctan y + y y y + π >0, y > 0and + y > ] =arctan y + y y y π <0, y < 0and + y > ]. arcsin arcsin y =arcsin y y y 0or + y ] = π arcsin y y >0, y < 0and + y > ] = π arcsin y y <0, y > 0and + y > ] 5. arcsin arcsin y = arccos y + y = arccos y + y y > y] NV 56 NV 55 <y] NV 56 6. arccos + arccos y = arccos y y =π arccos y y 7. arccos arccos y = arccos y + y = arccos y + y + y 0] + y<0] NV 57 3 y] <y] NV 57

.67 Functional relations 59 8. arctan +arctany =arctan + y y = π +arctan + y y = π +arctan + y y 9. arctan arctan y =arctan y +y = π +arctan y +y = π +arctan y +y y < ] >0, y > ] <0, y > ] y > ] >0, y < ] <0, y < ] NV 595, GI I 879 NV 596.66. arcsin =arcsin ] = π arcsin ] < = π arcsin < ] NV 6 7. arccos = arccos 0 ] =π arccos <0] NV 6 8 3. arctan =arctan < ] =arctan + π >] =arctan π < ] NV 6 9.67. arctan +arctan = π = π >0] <0] GI I 878. arctan +arctan + = π > ] = 3 π < ] NV 6, GI I 88

60 The Inverse Trigonometric and Hyperbolic Functions.68.68. arcsin = π arctan + ] = arctan ] = π arctan ] NV 65. arccos + = arctan 0] = arctan 0] NV 66.69 π arctan tan π = E GI 886.63 Relations between the inverse hyperbolic functions.. arcsinh = arccosh +=arctanh + JA. arccosh =arcsinh = arctanh 3. arctanh =arcsinh = arccosh = arccoth. arcsinh ± arcsinh y =arcsinh +y ± y + 5. arccosh ± arccosh y = arccosh y ± y 6. arctanh ± arctanh y =arctanh ± y ± y JA JA JA JA JA.6 Series representations.6. arcsin = π arccos = + 3 3 + 3 5 5 + 3 5 6 7 7 +...! =! + + = F, ; 3 ; ] FI II 79. arcsinh = 3 3 + 3 5 5...; =!! + + = F, ; 3 ; ] FI II 80

.65 Series representations 6.6. arcsinh =ln + 3 +... +! =ln +!. arccosh =ln.63!!. arctan = 3 3 + 5 5 7 7 +... + = +. arctanh = + 3 3 + 5 5 + = + + ] AD 680.a ] AD 680.3a ] FI II 79 < ] AD 680..6. arctan = =! +! + F +, ; 3 ; + + < ] AD 6.3. arctan = π + 3 3 5 5 + 7 7 = π + + AD 6..65. arcsec = π 3 3 3 5 5 = π! +! + = π F, ; 3 ; > ]. arcsin =! + +! + 3. arcsin 3 = 3 + 3! 5! 3 + 3 5 + 3! 7! 3 5 AD 6.5 ] AD 6., GI III 5a + 3 + 5 7 +... ] BR* 88, AD 6., GI III 53a

6 The Inverse Trigonometric and Hyperbolic Functions.66.66. arcsinh = arcosech =!! + ] AD 680.5. arccosh =arcsech =ln!! 0 < ] AD 680.6 3. arcsinh = arcosech =ln + +!! 0 < ] AD 680.7a. arctanh = arccoth = + + > ] AD 680.8.67.. tanh π/ n+3 = πn+3 n j j n j+ Bj B n j+3 j!n j +! j= + n n+ B n+ n +!] sech π/ n+ = πn+ n n+3 j= n =0,,,..., j B j B n j j!n j! + B n n! + n Bn n]! n =,,..., The summation term on the right is to be omitted for n =. See page iii for the definition of B r.