The Chemistry Maths Book, Second Edition

88 Chapter 3Transcendental functions

The hyperbolic cosine and sine are defined in terms of the exponential function as

(3.47)

(read as ‘cosh’ and ‘shine’). Their graphs are shown in Figure 3.22.

Both the notation and the properties of the hyperbolic functions parallel those of

the trigonometric (circular) functions; for example,

cosh

2

x 1 − 1 sinh

2

x 1 = 11

sinh(x 1 ± 1 y) 1 = 1 sinh 1 x 1 cosh 1 y 1 ± 1 cosh 1 x 1 sinh 1 y (3.48)

cosh(x 1 ± 1 y) 1 = 1 cosh 1 x 1 cosh 1 y 1 ± 1 sinh 1 x 1 sinh 1 y

The functions are called ‘hyperbolic’ because, ifx 1 = 1 a 1 cosh 1 tandy 1 = 1 asinh 1 tare the

coordinates of a point, where tis a parameter, then by the first equation (3.48),

x

2

1 − 1 y

2

1 = 1 a

2

and this is the equation of a hyperbola.

The inverse hyperbolic functions

The inverse hyperbolic functions are defined in the same way as the inverse

trigonometric functions; for example, ifx 1 = 1 cosh 1 ytheny 1 = 1 cosh

− 1

x. These functions

can be expressed in terms of the logarithmic function:

(3.49)

cosh ln ( )

sinh ln

−













−

=±− ≥

=+

12

xxx 11 x

xxx++

=

−













<













−

1

2

1

1

tanh x ln (| |

x

x ))

coshxeesinhxee

xx xx

=+, =−

−













−













1

2

1

2

e

x

e

−x

O

co shx

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Figure 3.22