Page 698 - Handbook Of Integral Equations

1.2. Hyperbolic Functions

Definitions
x
x
x
x
e – e –x e + e –x e – e –x e + e –x
sinh x = , cosh x = , tanh x = , coth x = .
x
x
2 2 e + e –x e – e –x
Simplest relations
2
2
cosh x – sinh x =1, tanh x coth x =1,
sinh(–x)= – sinh x, cosh(–x) = cosh x,
sinh x cosh x
tanh x = , coth x = ,
cosh x sinh x
tanh(–x)= – tanh x, coth(–x)= – coth x,
1 2 1
2
1 – tanh x = 2 , coth x – 1= 2 .
cosh x sinh x
Relations between hyperbolic functions of single argument (x ≥ 0)
tanh x 1
2
sinh x = cosh x – 1= √ = √ ,
2
2
1 – tanh x coth x – 1
1 coth x

2
cosh x = sinh x +1 = √ = √ ,
2
2
1 – tanh x coth x – 1
√
2
sinh x cosh x – 1 1
tanh x = √ 2 = = ,
sinh x +1 cosh x coth x
√
2
sinh x +1 cosh x 1
coth x = = √ = .
sinh x cosh x – 1 tanh x
2
Addition formulas
sinh(x ± y) = sinh x cosh y ± sinh y cosh x, cosh(x ± y) = cosh x cosh y ± sinh x sinh y,
tanh x ± tanh y coth x coth y ± 1
tanh(x ± y)= , coth(x ± y)= .
1 ± tanh x tanh y coth y ± coth x
Addition and subtraction of hyperbolic functions
x ± y x ∓ y

sinh x ± sinh y = 2 sinh cosh ,
2 2
x + y x – y

cosh x + cosh y = 2 cosh cosh ,
2 2
x + y x – y

cosh x – cosh y = 2 sinh sinh ,
2 2
2
2
2
2
sinh x – sinh y = cosh x – cosh y = sinh(x + y) sinh(x – y),
2
2
sinh x + cosh y = cosh(x + y) cosh(x – y),
sinh(x ± y) sinh(x ± y)
tanh x ± tanh y = , coth x ± coth y = ± .
cosh x cosh y sinh x sinh y
Products of hyperbolic functions
1
sinh x sinh y = [cosh(x + y) – cosh(x – y)],
2
1
cosh x cosh y = [cosh(x + y) + cosh(x – y)],
2
1
sinh x cosh y = [sinh(x + y) + sinh(x – y)].
2
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 682