Definition of Hyperbolic Functions HYPERBOLIC FUNCTIONS
14 Definition of Hyperbolic Functions HYPERBOLIC FUNCTIONS
14.1. Hyperbolic sine of x
= sinh x =
14.2. Hyperbolic cosine of x
= cosh x =
14.3. Hyperbolic tangent of x
= tanh x =
14.4. Hyperbolic cotangent of x = coth x =
14.5. Hyperbolic secant of x
= sech x =
14.6. Hyperbolic cosecant of x
= csch x =
Relationships Among Hyperbolic Functions
14.7. tanh x = cosh x
sinh x
14.8. coth x = tanh = x sinh x
1 cosh x
14.9. sech x 1 = cosh x
14.10. csch x 1 = sinh x
14.11. cosh 2 x − sinh 2 x = 1
14.12. sech 2 x + tanh 2 x = 1
14.13. coth 2 x − csc h 2 x = 1
Functions of Negative Arguments
14.14. sinh (–x) = – sinh x
14.15. cosh (–x) = cosh x
14.16. tanh (–x) = – tanh x
14.17. csch (–x) = – csch x
14.18. sech (–x) = sech x
14.19. coth (–x) = – coth x
HYPERBOLIC FUNCTIONS
Addition Formulas
14.20. sinh( x ± y ) sinh cosh = x y ± cosh sinh x y
14.21. cosh( x ± y ) cosh cosh = x y ± sinh sinh x y tanh x tanh y
14.22. tanh( x ± y )
1 ± tanh tanh x y coth coth x y 1
14.23. coth( x
± y ) = coth y ± coth x
Double Angle Formulas
14.24. sinh 2 x = 2 sinh cosh x x
14.25. cosh 2 x = cos h 2 x + sin h 2 x = 2 cos h 2 x −=+ 112 sin h 2 x
2 tan h x
14.26. tanh 2 x =
1 + tanh 2 x
Half Angle Formulas
x cosh x sinh
2 2 + if x >− 0 , if x < 0 ]
x cosh x 1
14.28. cosh
cosh x − 1
14.29. tanh =±
[ + if x , if x 0
2 cosh x + 1 >− 0 < ]
sinh x
cosh x − 1
= ccosh x
= sinh x
Multiple Angle Formulas
14.30. sinh 3 x = 3 sinh x + 4 sinh 3 x
14.31. cosh 3 x = 4 cosh 3 x − 3 cosh x
3 tanh tanh 3
14.32. tanh 3 x +
13 + tanh 2 x
14.33. sinh 4 x = 8 sinh 3 x cosh x + 4 sinh cosh x x
14.34. cosh 4 x = 8 cosh 4 x − 8 cosh 2 x + 1
14.35. tanh 4 x x + =
4 tanh x 4 tanh 3
16 + tanh 2 x + tanh 4 x
HYPERBOLIC FUNCTIONS
Powers of Hyperbolic Functions
14.36. sinh 2 x = 1 2 cosh 2 x − 1 2
14.37. cosh 2 x = 1 2 cosh 2 x + 1 2
14.38. sinh 3 x = 1 4 sinh 3 x − 4 3 sinh x
14.39. cosh 3 x = 1 cosh 3 x + 4 3 4 cosh x
14.40. sinh 4 x =− 3 8 1 2 cosh 2 x + 1 8 cosh 4 x
14.41. cosh 4 x =+ 3 8 1 2 cosh 2 x + 1 8 cosh 4 x
Sum, Difference, and Product of Hyperbolic Functions
14.42. sinh x + sinh y = 2 sinh ( 1 2 x + y ) cosh ( 1 2 x − y )
14.43. sinh x − sinh y = 2 cosh ( 1 2 1 x + y )sinh ( 2 x − y )
14.44. cosh x + cosh y = 2 cosh ( 1 2 x + y ) cosh ( 1 2 x − y )
14.45. cosh x − cosh y = 2 sinh ( 1 2 x + y )sinh ( 1 2 x − y )
14.46. sinh sinh x y = 1 2 {cosh ( x + y ) cosh ( − x − y )}
14.47. cosh cosh x y = 1 2 {cosh ( x ++ y ) cosh ( x − y )}
14.48. sinh cosh x y = 1 2 {sinh ( x ++ y ) sinh ( x − y )}
Expression of Hyperbolic Functions in Terms of Others In the following we assume x > 0. If x < 0, use the appropriate sign as indicated by formulas 14.14 to
sech x =u csch x =u sinh x
sinh x =u
cosh x =u
tanh x =u
coth x =u
u u 2 − 1 u /1 − u 2 1 /u 2 − 1 1 −uu 2 / 1/u cosh x
1 +uu −u 2 / tanh x
coth x u 2 +/ 1 u
uu / 2 − 1 1/u
11 / −u 2 1 +u 2
sech x
csch x 1/u
1 /u 2 − 1 1 −uu 2 /
u 2 − 1 u /1 − u 2 u
HYPERBOLIC FUNCTIONS
Graphs of Hyperbolic Functions
14.49. y = sinh x
14.50. y = cosh x 14.51. y = tanh x
Fig. 14-1
Fig. 14-2
Fig. 14-3
14.52. y = coth x
14.53. y = sech x 14.54. y = csch x
Fig. 14-4
Fig. 14-5
Fig. 14-6
Inverse Hyperbolic Functions If x = sinh y, then y = sinh –1 x is called the inverse hyperbolic sine of x. Similarly we define the other inverse
hyperbolic functions. The inverse hyperbolic functions are multiple-valued and as in the case of inverse trigo- nometric functions [see page 49] we restrict ourselves to principal values for which they can be considered as single-valued.
The following list shows the principal values (unless otherwise indicated) of the inverse hyperbolic func- tions expressed in terms of logarithmic functions which are taken as real valued.
14.55. sinh − 1 x = ln ( x + x 2 + 1 )
−<< ⬁ x ⬁
x ⭌ 1 (cosh − 1 x > 0 is prinncipal value) − 1 1 tanh 1 14.57. + x x = ln
14.56. cosh − 1 x = ln ( x + x 2 − 1 )
−<< 1 x 2 1
14.58. coth − 1 = ln x + x 1
1 or
x <−
14.59. sech − 1 ln 1 1 x = + − x
2 1 < x ⬉ 1 ( sech − 1 x > 0 is pr iincipal value)
csch − 1 ln 1 1 1 14.60. x =
HYPERBOLIC FUNCTIONS
Relations Between Inverse Hyperbolic Functions
14.61. csch − 1 x = sinh ( / ) − 1 1 x
14.62. sech − 1 x = cosh ( / ) − 1 1 x
14.63. coth − 1 x = tanh ( / ) − 1 1 x
14.64. sinh ( ) − 1 −=− x sinh − 1 x
14.65. tanh ( ) − 1 −=− x tanh − 1 x
14.66. coth ( ) − 1 −=− x coth − 1 x
14.67. csch − 1 () −=− x csch − 1 x
Graphs of Inverse Hyperbolic Functions
14.68. y = sinh − 1 x
14.69. y = cosh − 1 x 14.70. y = tanh − 1 x
Fig. 14-7
Fig. 14-8
Fig. 14-9
14.71. y = coth − 1 x 14.72. y = sech − 1 x 14.73. y = csch − 1 x
Fig. 14-10
Fig. 14-11
Fig. 14-12
HYPERBOLIC FUNCTIONS
Relationship Between Hyperbolic and Trigonometric Functions
14.74. sin ( ) ix = i sinh x 14.75. cos ( ) cosh ix = x 14.76. tan ( ) ix = i tanh x
14.77. csc ( ) ix = − csch 14.78. i x sec ( ) ix = sech 14.79. x cot ( ) ix =− i coth x
14.80. sinh ( ) ix = i sin x 14.81. cosh ( ) cos ix = x 14.82. tanh ( ) ix = i tan x
14.83. csch ( ) ix =− i csc x 14.84. sech ( ) sec ix = x 14.85. coth ( ) ix =− i cot x
Periodicity of Hyperbolic Functions In the following k is any integer.
14.86. sinh ( x + 2 ki π ) sinh = x 14.87. cosh ( x + 2 ki π ) cosh = x 14.88. tanh ( x + ki π ) tanh = x
14.89. csch ( x + 2 ki π ) = csch x 14.90. sech ( x + 2 ki π ) = sech x 14.91. coth ( x + ki π ) coth = x
Relationship Between Inverse Hyperbolic and Inverse Trigonometric Functions
14.92. sin ( ) − 1 ix = i sin − 1 x 14.93. sinh ( ) − 1 ix = i sin − 1 x
14.94. cos − 1 x =± i cosh − 1 x 14.95. cosh − 1 x =± i cos − 1 x
14.96. tan ( ) − 1 ix = i tanh − 1 x 14.97. tanh ( ) − 1 ix = i tan − 1 x
14.98. cot ( ) − 1 ix = i coth − 1 x 14.99. coth ( ) − 1 ix =− i cot − 1 x
14.100. sec − 1 x =± i sech − 1 x 14.101. sech − 1 x =± i sec − 1 x
14.102. csc ( ) − 1 ix =− i csch − 1 x 14.103. csch − 1 () ix =− i csc − 1 x
Section IV: Calculus DERIVATIVES
Definition of a Derivative Suppose y = f(x). The derivative of y or f(x) is defined as
dy
− fx 15.1. () = lim
where h = ∆x. The derivative is also denoted by y′, df/dx or f′(x). The process of taking a derivative is called differentiation .
General Rules of Differentiation In the following, u, , w are functions of x; a, b, c, n are constants (restricted if indicated); e = 2.71828 … is the
natural base of logarithms; ln u is the natural logarithm of u (i.e., the logarithm to the base e) where it is assumed that u > 0 and all angles are in radians.
15.2. () c dx
15.3. () cx c dx
15.4. ( cx n ) = ncx n −1 dx
15.5. ( u ±±± w ⋯ ) =
15.6. () cu c dx
= dx
d du
15.7. () u = u + dx
15.8. ( uw ) = u
15.10. () u n nu n dx
(Chain rule) du dx
DERIVATIVES
du
15.12. = dx dx du /
dy dy du /
15.13. = dx dx du /
Derivatives of Trigonometric and Inverse Trigonometric Functions
15.14. sin u = cos u dx
d du
dx
15.15. cos u =− sin u dx
d du
dx
15.16. tan u = sec 2 u
15.17. cot u =− csc 2 u
15.18. sec u = sec tan u u dx
d du
dx
15.19. csc u csc cot u u dx
15.20. sin − 1 u =
15.21. cos − 1 u
15.22. tan − 1 u
15.23. cot − 1 u −
15.24. sec − 1 u
15.25. csc u
|| u u 2 − 1 dx uu 2 − 1 dx + if − π / 2 < csc − 1 u < 0
Derivatives of Exponential and Logarithmic Functions
d log a e du
dx = u dx
15.26. log a u
d d 1 du
15.27. ln u = log u dx
e = u dx
dx
15.28. a u = a u ln a dx
d du
dx
DERIVATIVES
15.29. e u e u dx = dx
Derivatives of Hyperbolic and Inverse Hyperbolic Functions
15.31. sinh u cosh u dx
d du
dx
15.32. cosh u = sinh u dx
d du
dx
dx = sech dx
15.33. tanh u
d 2 du
dx = −csch u dx
15.34. coth u
d du
15.35. sech u sech u tanh u dx
15.36. d csch du u =− csch u coth u dx
dx
d 1 du
15.37. sinh − 1 u dx
u 2 + 1 dx
d − 1 ± 1 du
15.38. cosh u =
2 + if cosh − 1 u > 0 , dx u > 1 u − 1 dx − if cosh − 1 u < 0 , u > 1
15.39. d tanh − 1 1 u du =− 1 u 2 [–1 < u dx < 1] dx
d − 1 15.40. 1 du
coth u =−
2 − if sech 1 u > 00 , << u 1 dx u 1 − u dx if sech − 1 + u < 00 , << u 1
d ∓ 1 du
15.41. sech − 1 u
15.42. csch u =
Higher Derivatives The second, third, and higher derivatives are defined as follows.
15.43. Second derivative d dy
2 dy
2 = ′′ f () x = ′′ dx y dx = dx
d dy 2 dy 15.44. 3 Third derivative =
d d n − 1 dy 15.45. n nth derivative = y
dx dx n − 1 = n =
f () n () x y () n
dx
DERIVATIVES
Leibniz’s Rule for Higher Derivatives of Products
du Let D p p stand for the operator
so that D u p
= dx p = the pth derivative of u. Then
dx
15.46. Du n () n uD n
where n n , ,… are the binomial coefficients (see 3.5). 1 2
As special cases we have
du d
du 2
15.47. 2 () u = u 2 + 2 +
15.48. 3 () u = u 3 + 3 2 + 3 +
Differentials Let y = f(x) and ∆ y = fx ( + ∆ x ) − fx ( ). Then
where ⑀ → 0 as ∆x → 0. Thus,
15.50. ∆ y =′ fx () ∆ x + ⑀ ∆ x
If we call ∆x = dx the differential of x, then we define the differential of y to be
15.51. dy = ′( ) f x dx
Rules for Differentials The rules for differentials are exactly analogous to those for derivatives. As examples we observe that
15.52. du ( ±±± w ⋯ ) = du ± d ± dw ± ⋯
15.53. du () = ud + du u du − ud
15.54. d =
15.55. du () n = n nu −1 du
15.56. d (sin ) cos u = u du
15.57. d (cos ) u =− sin u du
DERIVATIVES 66
DERIVATIVES
Partial Derivatives Let z = f (x, y) be a function of the two variables x and y. Then we define the partial derivative of z or f(x, y) with
respect to x, keeping y constant, to be
This partial derivative is also denoted by ∂∂ zxf /,, x or z x . Similarly the partial derivative of z = f (x, y) with respect to y, keeping x constant, is defined to be
∂ f fxy 15.59. (, lim + ∆ y ) − fxy (,)
∂= y ∆ y → 0 ∆ y
This partial derivative is also denoted by ∂∂ zyf /,, y or z y . Partial derivatives of higher order can be defined as follows:
15.61. ∂∂= xy ∂ x ∂ y ∂∂= yx ∂ y ∂ x The results in 15.61 will be equal if the function and its partial derivatives are continuous; that is, in such
cases, the order of differentiation makes no difference. Extensions to functions of more than two variables are exactly analogous.
Multivariable Differentials The differential of z = f(x, y) is defined as
15.62. dz df ∂
= = dx + dy ∂ x
where dx = ∆x and dy = ∆y. Note that dz is a function of four variables, namely x, y, dx, dy, and is linear in the variables dx and dy.
Extensions to functions of more than two variables are exactly analogous.
EXAMPLE: Let z =x 2 + 5xy + 2y 3 . Then z x = 2x + 5y and z y = 5x + 6y 2
and hence
dz = (2x + 5y) dx + (5x + 6y 2 ) dy Suppose we want to find dz for dx = 2, dy = 3 and at the point P (4, 1), i.e., when x = 4 and y = 1. Substitution
yields
dz = (8 + 5)2 + (20 + 6)3 = 26 + 78 = 104