hyperbolic identities

hyperbolic identities

There are many formulas involving

hyperbolic functions

, many of which are to formulas for

trigonometric functions

. Below is a list of some of these formulas (usually for real

arguments

).

1.

Hyperbolic version of

Pythagorean identities

–

cosh

2

⁡

x

-

sinh

2

⁡

x

=

1

–

1

-

tanh

2

⁡

x

=

sech

2

⁡

x

–

coth

2

⁡

x

-

1

=

csch

2

⁡

x

2.

Fractional

identities

–

tanh

⁡

x

=

sinh

⁡

x

cosh

⁡

x

–

coth

⁡

x

=

cosh

⁡

x

sinh

⁡

x

–

coth

⁡

x

=

1

tanh

⁡

x

–

tanh

⁡

x

=

1

coth

⁡

x

–

csch

⁡

x

=

1

sinh

⁡

x

–

sech

⁡

x

=

1

cosh

⁡

x

3.

Hyperbolic functions of a

purely imaginary number

–

sinh

⁡

(

i

⁢

x

)

=

i

⁢

sin

⁡

x

–

cosh

⁡

(

i

⁢

x

)

=

cos

⁡

x

–

tanh

⁡

(

i

⁢

x

)

=

i

⁢

tan

⁡

x

–

cosh

⁡

(

i

⁢

x

)

=

i

⁢

cot

⁡

x

–

csch

⁡

(

i

⁢

x

)

=

i

⁢

csc

⁡

x

–

sech

⁡

(

i

⁢

x

)

=

sec

⁡

x

4.

Addition formulas

and subtraction formulas (

http://planetmath.org/AdditionAndSubtractionFormulasForHyperbolicFunctions

)

–

sinh

⁡

(

x

±

y

)

=

sinh

⁡

x

⁢

cosh

⁡

y

±

cosh

⁡

x

⁢

sinh

⁡

y

–

cosh

⁡

(

x

±

y

)

=

cosh

⁡

x

⁢

cosh

⁡

y

±

sinh

⁡

x

⁢

sinh

⁡

y

–

tanh

⁡

(

x

±

y

)

=

tanh

⁡

x

±

tanh

⁡

y

1

±

tanh

⁡

x

⁢

tanh

⁡

y

5.

Formulas for hyperbolic functions of a

complex number

–

sinh

⁡

(

x

+

i

⁢

y

)

=

sinh

⁡

x

⁢

cos

⁡

y

+

i

⁢

cosh

⁡

x

⁢

sin

⁡

y

–

cosh

⁡

(

x

+

i

⁢

y

)

=

cosh

⁡

x

⁢

cos

⁡

y

+

i

⁢

sinh

⁡

x

⁢

sin

⁡

y

–

tanh

⁡

(

x

+

i

⁢

y

)

=

tanh

⁡

x

+

i

⁢

tan

⁡

y

1

+

i

⁢

tanh

⁡

x

⁢

tan

⁡

y

6.

Opposite formulas

–

sinh

⁡

(

-

x

)

=

-

sinh

⁡

x

–

cosh

⁡

(

-

x

)

=

cosh

⁡

x

–

tanh

⁡

(

-

x

)

=

-

tanh

⁡

x

7.

Double argument formulas

–

sinh

⁡

(

2

⁢

x

)

=

2

⁢

sinh

⁡

x

⁢

cosh

⁡

x

–

cosh

⁡

(

2

⁢

x

)

=

cosh

2

⁡

x

+

sinh

2

⁡

x

=

2

⁢

cosh

2

⁡

x

-

1

=

1

+

2

⁢

sinh

2

⁡

x

–

tanh

⁡

(

2

⁢

x

)

=

2

⁢

tanh

⁡

x

1

+

tanh

2

⁡

x

8.

Periodicity

(

http://planetmath.org/Periodic

) formulas

–

sinh

⁡

(

z

+

2

⁢

π

⁢

i

)

=

sinh

⁡

z

–

cosh

⁡

(

z

+

2

⁢

π

⁢

i

)

=

cosh

⁡

z

–

tanh

⁡

(

z

+

π

⁢

i

)

=

tanh

⁡

z

Cf (

http://planetmath.org/Cf

). the periodicity of

exponential function

.

9.

Exponential formulas (

http://planetmath.org/ExponentialFunction

)

–

cosh

⁡

x

=

e

x

+

e

-

x

2

–

sinh

⁡

x

=

e

x

-

e

-

x

2

–

tanh

⁡

x

=

e

x

-

e

-

x

e

x

+

e

-

x

–

e

x

=

cosh

⁡

x

+

sinh

⁡

x

–

e

-

x

=

cosh

⁡

x

-

sinh

⁡

x

Note that the first three formulas given in this are definitions.

Title

hyperbolic identities

Canonical name

HyperbolicIdentities

Date of creation

2013-03-22 17:50:42

Last modified on

2013-03-22 17:50:42

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

11

Author

Wkbj79 (1863)

Entry type

Topic

Classification

msc 33B10

Classification

msc 26A09

Synonym

hyperbolic formulas

Synonym

hyperbolic formulae

Related topic

GoniometricFormulae

Related topic

AdditionAndSubtractionFormulasForHyperbolicFunctions

Related topic

ExamplesOfPeriodicFunctions

Related topic

TaylorSeriesOfHyperbolicFunctions

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