17 Exponential Expressions.

Article 17
Exponential Expressions.

Adding and subtracting (27), (28) give the identities

\begin{matrix} \begin{aligned} cosh u + sinh u & = 1 + u + \frac{1}{2!} u^{2} + \frac{1}{3!} u^{3} + \frac{1}{4!} u^{4} + \dots = e^{u}, \\ cosh u - sinh u & = 1 - u + \frac{1}{2!} u^{2} - \frac{1}{3!} u^{3} + \frac{1}{4!} u^{4} - \dots = e^{- u}, \end{aligned} \end{matrix}

hence

\begin{matrix} \begin{aligned} cosh u & = \frac{1}{2} (e^{u} + e^{- u}), & sinh u & = \frac{1}{2} (e^{u} - e^{- u}), \\ tanh u & = \frac{e^{u} - e^{- u}}{e^{u} + e^{- u}}, & sech u & = \frac{2}{e^{u} + e^{- u}}, etc. \end{aligned}\} & (30) \end{matrix}

The analogous exponential expressions for $sin u, cos u$ are

cos u = \frac{1}{2} (e^{u i} + e^{- u i}), sin u = \frac{1}{2 i} (e^{u} - e^{- u i}), (i = \sqrt{- 1})

where the symbol $e^{u i}$ stands for the result of substituting $u i$ for $x$ in the exponential development

e^{x} = 1 + x + \frac{1}{2!} x^{2} + \frac{1}{3!} x^{3} + \dots

This will be more fully explained in treating of complex numbers, Arts. 28, 29.

Prob. 37.

Show that the properties of the hyperbolic functions could be placed on a purely algebraic basis by starting with equations (30) as their deﬁnitions; for example, verify the identities:

\begin{matrix} sinh (- u) = - sinh u, cosh (- u) = cosh u, \\ {cosh}^{2} u - {sinh}^{2} u = 1, sinh (u + v) = sinh u cosh v + cosh u sinh v, \\ \frac{d^{2} (cosh m u)}{d u^{2}} = m^{2} cosh m u, \frac{d^{2} (sinh m u)}{d u^{2}} = m^{2} sinh m u . \end{matrix}

Prob. 38.

Prove

{(cosh u + sinh u)}^{n} = cosh n u + sinh n u

Prob. 39.

Assuming from Art. 14 that

cosh u

sinh u

satisfy the diﬀerential equation

\frac{d^{2} y}{d u^{2}} = y

, whose general solution may be written

y = A e^{n} + B e^{- n}

, where

A

B

are arbitrary constants; show how to determine

A

B

in order to derive the expressions for

cosh u

sinh u

, respectively. [Use eq. (15).]

Prob. 40.

Show how to construct a table of exponential functions from a table of hyperbolic sines and cosines, and vice versa.

Prob. 41.

Prove

u = {log}_{e} (cosh u + sinh u)

Prob. 42.

Show that the area of any hyperbolic sector is inﬁnite when its terminal line is one of the asymptotes.

Prob. 43.

From the relation

2 cosh u = e^{n} + e^{- n}

prove

2^{n - 1} {(cosh u)}^{n} = cosh n u + n cosh (n - 2) u + \frac{1}{2} n (n - 1) cosh (n - 4) u + \dots,

and examine the last term when $n$ is odd or even. Find also the corresponding expression for $2^{n - 1} {(sinh u)}^{n}$ .

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Article 17Exponential Expressions.

Article 17
Exponential Expressions.