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