Hyperbolic functions

A2.1

APPENDIX 2 Hjrperbolic functions Hyperbolic functions are combinations of positive and negative exponentials. They resemble goniometric functions and derive their names from the fact that they describe the coordinates of points on rectangular hyperbolas. They are often encountered in diffuse double layer theory . The definitions and some important properties are summarized below. a. Defining

equations

sinh X = i(e^ - e-^)

(A2.1J

c o s h x = i{e^ + e-^)

IA2.21

, sinh X e^ - e""*^ t a n h x = cosh— X = -e-' + e— -'

, *« «, (A2.31

cosh X e^ + e - ' c o t h x = —-—— sinh X = — e' - e - '

[A2.4I

1

2 cosech X = — ; — = sinhx e'^-e"''

[A2.51

1

2 = -~ [A2.61 cosh X e + e" The cosh and sech are even functions (cosh x = cosh (-x), etc.) and always positive: all others are uneven (sinh x = -sinh (-x), etc.), and may be positive or negative. Trends are sketched in fig. A2.1. sech X =

b. Series

expansions

x^ x^ sinh x = x + -— + —- + ... 3! 5!

[A2.7]

x^ x"^ cosh X = 1 + —- + — + ... 2! 4! tanhx = x - - ~ - - 4 - ^ ^ - . . . 3 15

[A2.81 {x^
[A2.9]

A2.2

(a)

sinh X

(b)

sinh X

Figure A2.1. Hyperbolic functions, (a) even: (b) uneven functions. Limits are indicated.

c. Inverse

functions

Note: sinli"^ x is also written as arc sinh x, etc. = cosh"^ J x ^ + 1

[A2.10]

cosh-^ X = ln(x + V^2 - l ) = [ 7 = - = sinh-^ ^ ^ ^ - 1

[A2.111

sinh'^ X = Inf A: + - J x ^ + 1 1 = I ,

(lxl
[A2.121

A2.3 {x>l o r x < l )

[A2.141

sechr^ X = In cosech"* X = In fi

rr

r

dx

{A2.15]

x"V3?"^

2 3

2.4

5

2.4.6

7

sinh-x = l n 2 x a - ^ - M 1 ^13^ 1 2 2x2 2.4 4x^ 2.4.6 ex^ cosh' X = ln2x

d.

11

1.3

2 2x2

2.4 4x^

X^

1

[A2.131

X^

1

X^

1.3.5

1

(x^ < 1)

[A2.161

(x^ >l)

[A2.171

(x2 >1)

(A2.181

(x^ < 1)

IA2.191

2.4.6 6x^

Integrals slnh X d X = cosh x

[A2.201

cosh X d x = sinh x

[A2.211

tanh X d x = In cosh x

1A2.221

coth X d x = In sinh x

[A2.231

sech X d x = tan"^ (sinh x)

IA2.241

cosech X d x = ln(tanh (x/2))

(A2.251

e. Relations between the functions sinh X =

tanh X

[A2.261

^ 1 - tanh 2 X cosh X

1 V l - tanh^ X

(A2.271

A2.4 cosh^ X - sinh^ x = 1

1A2.281

tanh x = ± ^l^-sech^x

[ A2.291

coth X = ^cosech^ x + 1

IA2.301

sech X = ^l-tanh^x

IA2.311

cosech X = ±^jcoth^ x - 1

(A2.32J

/ . Relations between Junctions involving x/2 2tanh(x/2) sinh X = , , _ , 2 ... /o. l - t a n h ^ (x/2)

IA2.331

sinh X = 2 sinh (x / 2) cosh (x / 2)

1A2.341

, l + tanh2(x/2) ^os^ ^ = ;—I—^2 l - t a n h ^ ( x / .2J)

r.«o^, A2.35

cosh X = cosh ^ (x / 2) + sinh ^ (x / 2)

(A2.361

cosh X = 2cosh^ ( x / 2 ) - 1

IA2.371

cosh X = 1 + 2sinh2(x/2)

[A2.38]

^ , 2tanh(x/2) tanh X = ■;—-—^2 r /o^ l + tanh2(x/2)

,.^ ^ , A2.39

sinh (X / 2) = -^i (cosh x - 1)

[A2.401

cosh (x / 2) = ^1 (cosh x + 1)

| A2.41)