Editing Hyperbola (section)

==Arc length==
The arc length of a hyperbola does not have an [[elementary function|elementary expression]]. The upper half of a hyperbola can be parameterized as

<math display="block">y = b\sqrt{\frac{x^{2}}{a^{2}}-1}.</math>

Then the integral giving the arc length <math>s</math> from <math>x_{1}</math> to <math>x_{2}</math> can be computed as:

<math display="block">s = b\int_{\operatorname{arcosh}\frac{x_{1}}{a}}^{\operatorname{arcosh}\frac{x_{2}}{a}} \sqrt{1+\left(1+\frac{a^{2}}{b^{2}}\right) \sinh ^{2}v} \, \mathrm dv.</math>

After using the substitution <math>z = iv</math>, this can also be represented using the [[elliptic integral#Incomplete elliptic integral of the second kind|incomplete elliptic integral of the second kind]] <math>E</math> with parameter <math>m = k^2</math>:

<math display="block">s = ib \Biggr[E\left(iv \, \Biggr| \, 1 + \frac{a^2}{b^2}\right)\Biggr]^{\operatorname{arcosh}\frac{x_1}{a}}_{\operatorname{arcosh}\frac{x_2}{a}}.</math>

Using only real numbers, this becomes<ref>{{dlmf |last=Carlson |first=B. C. |id=19.7.E7 |title=Elliptic Integrals}}</ref>

<math display="block">s=b\left[F\left(\operatorname{gd}v\,\Biggr|-\frac{a^2}{b^2}\right) - E\left(\operatorname{gd}v\,\Biggr|-\frac{a^2}{b^2}\right) + \sqrt{1+\frac{a^2}{b^2}\tanh^2 v}\,\sinh v\right]_{\operatorname{arcosh}\tfrac{x_1}{a}}^{\operatorname{arcosh}\tfrac{x_2}{a}}</math>

where <math>F</math> is the [[elliptic integral#Incomplete elliptic integral of the first kind|incomplete elliptic integral of the first kind]] with parameter <math>m = k^2</math> and <math>\operatorname{gd}v=\arctan\sinh v</math> is the [[Gudermannian function]].