21 Circular Functions of Gudermanian.

Article 21
Circular Functions of Gudermanian.

The six hyperbolic functions of $u$ are expressible in terms of the six circular functions of its gudermanian; for since

\frac{x_{1}}{a_{1}} = cosh u, \frac{x_{2}}{a_{2}} = cosh v,

(see Arts. 6, 7)

in which $u, v$ are the measures of corresponding hyperbolic and elliptic sectors, hence

\begin{aligned} cosh u & = sec v, [eq. (44)] \\ sinh u & = \sqrt{{sec}^{2} v - 1} = tan v, \\ tanh u & = \frac{tan v}{sec v} = sin v, \\ coth u & = csc v, \\ sech u & = cos v, \\ csch u & = cot v . \end{aligned}\}

(46)

The gudermanian is sometimes useful in computation; for instance, if $sinh u$ be given, $v$ can be found from a table of natural tangents, and the other circular functions of $v$ will give the remaining hyperbolic functions of $u$ . Other uses of this function are given in Arts. 22–26, 32–36.

Prob. 49.: Prove that $\begin{array}{l} gd u & = {sec}^{- 1} (cosh u) = {tan}^{- 1} (sinh u) \\ = {cos}^{- 1} (sech u) = {sin}^{- 1} (tanh u) . \end{array}$
Prob. 50.: Prove $\begin{array}{l} {gd}^{- 1} v & = {cosh}^{- 1} (sec v) = {sinh}^{- 1} (tan v) \\ = {sech}^{- 1} (cos v) = {tanh}^{- 1} (sin v) . \end{array}$
Prob. 51.: Prove $\begin{array}{l} gd 0 & = 0, gd \infty = \frac{1}{2} π, gd (- \infty) = - \frac{1}{2} π, \\ {gd}^{- 1} 0 & = 0, {gd}^{- 1} (\frac{1}{2} π) = \infty, {gd}^{- 1} (- \frac{1}{2} π) = - \infty . \end{array}$
Prob. 52.: Show that $gd u$ and ${gd}^{- 1} v$ are odd functions of $u, v$ .
Prob. 53.: From the ﬁrst identity in 4, Prob. 17, derive the relation $tanh \frac{1}{2} u = tan \frac{1}{2} v$ .
Prob. 54.: Prove ${tanh}^{- 1} (tan u) = \frac{1}{2} gd 2 u,$ and ${tan}^{- 1} (tanh x) = \frac{1}{2} {gd}^{- 1} 2 x .$

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Article 21Circular Functions of Gudermanian.

Article 21
Circular Functions of Gudermanian.