cosh or ch — hyperbolic cosine function

1. Definition

Hyperbolic cosine is defined as

coshx ≡ (e^x + e^−x) /2

Hyperbolic cosine is symmetric function defined everywhere on real axis. Its catenary graph is depicted below — fig. 1.

Fig. 1. Graph of the hyperbolic cosine function y = coshx.

Function codomain is range [1, +∞).

Base:

cosh²x − sinh²x = 1

sinhx + coshx = e^x
coshx − sinhx = e^−x

By definition:

coshx ≡ 1 /sechx

Property of symmetry:

cosh−x = coshx

Half-argument:

cosh(x/2) = √[(coshx + 1) /2]
coshx = [1 + tanh²(x/2)] /[1 − tanh²(x/2)]

Doulbe argument:

cosh(2x) = sinh²x + cosh²x
cosh(2x) = 2 cosh²x − 1
cosh(2x) = 2 sinh²x + 1

Triple argument:

cosh(3x) = 4 cosh³x − 3 coshx

Quadruple argument:

cosh(4x) = 8 cosh⁴x − 8 cosh²x + 4 = 8 cosh²x sinh²x + 1 = 8 sinh⁴x + 8 sinh²x + 1

Power reduction:

cosh²x = (cosh(2x) + 1) /2
cosh³x = (cosh(3x) + 3 coshx) /4
cosh⁴x = (cosh(4x) + 4 cosh(2x) + 3) /8
cosh⁵x = (cosh(5x) + 5 cosh(3x) + 10 coshx) /16

Sum and difference of arguments:

cosh(x + y) = coshx coshy + sinhx sinhy
cosh(x − y) = coshx coshy − sinhx sinhy

Product-to-sum:

coshx coshy = [cosh(x + y) + cosh(x − y)] /2
sinhx coshy = [sinh(x + y) + sinh(x − y)] /2

Sum-to-product:

coshx + coshy = 2 cosh[(x + y) /2] cosh[(x − y) /2]
coshx − coshy = 2 sinh[(x + y) /2] sinh[(x − y) /2]

Hyperbolic cosine derivative:

cosh′x = sinhx

Indefinite integral of the hyperbolic cosine:

∫ coshx dx = sinhx + C

where C is an arbitrary constant.

To calculate hyperbolic cosine of the number:

cosh(−1);

To get hyperbolic cosine of the complex number:

cosh(−1−i);

To get hyperbolic cosine of the current result:

cosh(rslt);

To get hyperbolic cosine of the number z in calculator memory:

cosh(mem[z]);

Hyperbolic cosine of the real argument is supported in free version of the Librow calculator.

Hyperbolic cosine of the complex argument is supported in professional version of the Librow calculator.