Hyperbolic function

The hyperbolic functions are functions whose definitions are based on the exponential function, connected by rational operations and are analogous to trigonometric functions. These are:

Curves of hyperbolic functions sinh, cosh and Soh

Curves of hyperbolic functions csch, sech and coth

The hyperbolic sine

${displaystyle sinh(x)={frac {e^{x}-e^{-x}{2}}{2}}}}}$

The hyperbolic cosine

${displaystyle cosh(x)={frac {e^{x}+e^{-x}{2}}{2}}}}}$

The hyperbolic tangent

${displaystyle tanh(x)={frac {sinh(x)}{cosh(x)}}}}}$

and other lines:

${displaystyle coth(x)={frac {cosh(x)}{sinh(x)}}}}}$

(hyperbolic cotangent)

${displaystyle {mbox{sech}}(x)={frac {1}{cosh(x)}}}}}$

(hyperbolic drying)

${displaystyle {mbox{csch}}(x)={frac {1}{sinh(x)}}}}}}$

(hyperbolic cosecrate)

Relation between hyperbolic functions and circular functions

The trigonometric functions sin(t) and cos(t) can be the Cartesian coordinates (x,y) of a point P on the unit circle centered at the origin, where t is the angle, measured in radians, between the positive semi-axis X, and the segment OP, according to the following equalities:

${displaystyle left{begin{matrix}x(t)=cos ty(t)=sin tend{matrix}}}right. !$

The parameter t can also be interpreted as the length of the arc of unitary circumference between the point (1,0) and the point P, or as twice the area of the circular sector determined by the positive semi-axis X, the segment OP and the unit circle.

Animation of representation of hyperbolic sinus.

Similarly, we can define hyperbolic functions as the Cartesian coordinates (x,y) of a point P of the hyperbola equilateral, centered at the origin, whose equation is

${displaystyle x^{2}-y^{2}=1}$

being t twice the area of the region between the positive semiaxis X, and the segment OP and the hyperbola, according to the following equalities:

${displaystyle left{begin{matrix}x(t)=cosh ty(t)=sinh tend{matrix}}right. !$

However, the following description of the hyperbola can also be shown to be valid:

${displaystyle x(t)={frac {e^{t}+e^{-t}{2}}}}}}$

${displaystyle y(t)={frac {e^{t}-e^{-t}{2}}}}}}}$

given that

${displaystyle left({frac {e^{t}+e^{-t}{2}}}}right)^{2}-left({frac {e^{t}-e^{{-t}}{2}}{2}{2}{2}=1}$

So the hyperbolic cosine and hyperbolic sine admit a representation in terms of exponential functions of real variable:

${displaystyle cosh(t)={frac {e^{t}+e^{-t}}{2}}{2}}}}$

${displaystyle sinh(t)={frac {e^{t}-e^{-t}{2}}{2}}}}}$

Relationships

Fundamental equation

${displaystyle cosh ^{2}(x)-,mathrm {sinh} ^{2}(x)=1,}$

Argument duplication

We have the following formulas very similar to their corresponding trigonometric

${displaystyle cosh(x+y)=cosh(x)cosh(y)+sinh(x)sinh(y)}$

${displaystyle cosh(x-y)=cosh(x)cosh(y)-sinh(x)sinh(y)}$

which leads to the following relation:

${displaystyle cosh(2x)=cosh ^{2}(x)+,mathrm {sinh} ^{2}(x)}$

and on the other hand

${displaystyle mathrm {sinh} (x+y)=mathrm {sinh} (x)cosh(y)+mathrm {sinh} (y)cosh(x)}}$

${displaystyle mathrm {sinh} (x-y)=mathrm {sinh} (x)cosh(y)-mathrm {sinh} (y)cosh(x)}}$

which leads to:

${displaystyle mathrm {sinh} (2x)=2,mathrm {sinh} (x)cosh(x)}$

you have this other relationship

${displaystyle tanh(x+y)={frac {tanh(x)+tanh(y)}{1+tanh(x)tanh(y)}}}}$

that allows to obtain

${displaystyle tanh(2x)={frac {2tanh x}{1+tanh ^{2x}}}}}}$

Derivation and integration

Being rational combinations of derivative functions and^x and^-x are derived from the defined points that have similarities with trigonometric functions.

All are deducted from:

${displaystyle {d over dx}{{{bigl (}e^{u}-e^{-u}{bigr)}}{over 2}}$

Therefore, those derived from trigonometric functions are the following:

${displaystyle {frac {d }{dx}}(,mathrm {senh },(x))=cosh(x)}$

${displaystyle {frac {d }{dx}}(cosh(x))=,mathrm {sinh} ,(x)}$

${displaystyle {frac {d}{dx}}(,tanh(x))=mathrm {sech} ^{2}(x)}$

${displaystyle {frac {d }{dx}}(mathrm {coth}(x))=-mathrm {csch} ^{2}(x)}$

${displaystyle {frac {d }{dx}}(mathrm {sech} (x))=-mathrm {sech} (x)tanh(x)}$

${displaystyle {frac {d }{dx}}(mathrm {csch}(x))}=-mathrm {csch} (x)mathrm {coth} (x)}$

These formulas lead in a similar way to those of integration. In addition, since integration is the inverse operation of the derivation, it is trivial in this case.

The graph of the function cosh(x) is called a catenary.

Next, the integral formulas:

${displaystyle int mathrm {senh} (x),dx=cosh(x)+C}$

${displaystyle int mathrm {cosh} (x),dx=senh(x)+C}$

${displaystyle int mathrm {sech} ^{2}(x),dx=mathrm {tanh} (x)+C}$

${displaystyle int mathrm {csch} ^{2}(x),dx=-mathrm {coth} (x)+C}$

${displaystyle int mathrm {sech} (x),mathrm {tanh} (x),dx=-mathrm {sech} (x)+C}$

${displaystyle int mathrm {csch} (x),mathrm {coth} (x),dx=-mathrm {csch} (x)+C}$

Relation to the exponential function

From the relation of the hyperbolic cosine and sine, the following relations can be derived:

${displaystyle e^{x}=cosh x+sinh x!}$

and

${displaystyle e^{-x}=cosh x-sinh x. !$

These expressions are analogous to those in terms of sines and cosines, based on Euler's formula, as a sum of complex exponentials.

${displaystyle e^{ix}=cos x+isin x. !$

Additionally,

${displaystyle e^{x}={sqrt {frac {1+tanh x{1-tanh x}}}}={frac {1+tanh {frac {x}{x}{2}}}{1-tanh {frac {x}}}}}}}}}}$

Expressions in the form of a Taylor series

It is possible to explicitly express the Taylor series at zero (or the Laurent series, if the function is not defined at zero) of the above functions.

${displaystyle sinh x=x+{frac {x^{3}}{3}{3}}{x^{5}}}{5}}}} +{frac {x^{7}}}{7}}}{7!} +cdots =sum _{n=0}{infty }{frac {x^{2n+1}{x}{x}{7}{x}{x}{x1⁄2n+1⁄2n+1⁄2}}{x1⁄2}}}}}{x1⁄2}}{x1⁄1⁄2}}{x1⁄1⁄2}}{x1⁄1⁄2}{x1⁄2}}{x1⁄1⁄2nx1⁄2}}{x1⁄2}}{x1⁄4}}{x1⁄4}}{x1⁄4}{x1⁄2}}{x1⁄2}}{x1⁄2}{x1⁄2}}}}}{$

This series is convergent for every complex value of x. Since the function sinh x is odd, only the odd exponents of x< /span> appear in this Taylor series.

${displaystyle cosh x=1+{frac {x^{2}}{2}}{2}}{x^{4}}}{4!} +{frac {x^{6}}}{6}}}}} +cdots =sum _{n=0}^{infty }{frac {x^{2n}{x}{2n}{2}{x}{x}{2n}}{$

This series is convergent for every complex value of x. Since the function cosh x is even, only the even exponents of x< /span> appear in this Taylor series.

The sum of the series of sinh and cosh is the Taylor series expression of the exponential function.

The following series are obtained from the description of a subset of its radius of convergence, where the series is convergent and its sum is equal to the function.

$♪♪$

where:

${displaystyle B_{n},}$ It's him. n-Sixty number of Bernoulli

${displaystyle E_{n},}$ It's him. n- That's a big number from Euler.

Inverse hyperbolic functions

Inverse hyperbolic functions are the inverse functions of hyperbolic functions. For a given value of a hyperbolic function, the corresponding inverse hyperbolic function gives the hyperbolic angle.