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Variations of the Hyperbolic Functions.

Since the values of the hyperbolic functions depend only on the sectorial measure, it is convenient, in tracing their variations, to consider only sectors of one half of a rectangular hyperbola, whose conjugate radii are equal, and to take the principal axis $OA$ as the common initial line of all the sectors. The sectorial measure $u$ assumes every value from $-\infty $, through $0$, to $+\infty $, as the terminal point $P$ comes in from inﬁnity on the lower branch, and passes to inﬁnity on the upper branch; that is, as the terminal line $OP$ swings from the lower asymptotic position $y=-x$, to the upper one, $y=x$. It is here assumed, but is proved in Art. 17, that the sector $AOP$ becomes inﬁnite as $P$ passes to inﬁnity.

Since the functions $coshu,sinhu,tanhu$, for any position of $OP$, are equal to the ratios of $x,y,t$, to the principal radius $a$, it is evident from the ﬁgure that

$$cosh0=1,\phantom{\rule{1em}{0ex}}sinh0=0,\phantom{\rule{1em}{0ex}}tanh0=0,$$ | (15) |

and that as $u$ increases towards positive inﬁnity, $coshu,sinhu$ are positive and become inﬁnite, but $tanhu$ approaches unity as a limit; thus

$$cosh\infty =\infty ,\phantom{\rule{1em}{0ex}}sinh\infty =\infty ,\phantom{\rule{1em}{0ex}}tanh\infty =1.$$ | (16) |

Again, as $u$ changes from zero towards the negative side, $coshu$ is positive and increases from unity to inﬁnity, but $sinhu$ is negative and increases numerically from zero to a negative inﬁnite, and $tanhu$ is also negative and increases numerically from zero to negative unity; hence

$$cosh(-\infty )=\infty ,\phantom{\rule{1em}{0ex}}sinh(-\infty )=-\infty ,\phantom{\rule{1em}{0ex}}tanh(-\infty )=-1.$$ | (17) |

For intermediate values of $u$ the numerical values of these functions can be found from the formulas of Arts. 16, 17, and are tabulated at the end of this chapter. A general idea of their manner of variation can be obtained from the curves in Art. 25, in which the sectorial measure $u$ is represented by the abscissa, and the values of the functions $coshu$, $sinhu$, etc., are represented by the ordinate.

The relations between the functions of $-u$ and of $u$ are evident from the deﬁnitions, as indicated above, and in Art. 8. Thus

- Prob. 12.
- Trace the changes in $sechu,cothu,cschu$, as $u$ passes from $-\infty $ to $+\infty $. Show that $sinhu,coshu$ are inﬁnites of the same order when $u$ is inﬁnite. (It will appear in Art. 17 that $sinhu,coshu$ are inﬁnites of an order inﬁnitely higher than the order of $u$.)
- Prob. 13.
- Applying eq. (12) to ﬁgure, page §, prove $tanh{u}_{1}=tanAOP$.

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