28 Addition-Theorems for Complexes.

Article 28
Addition-Theorems for Complexes.

The addition-theorems for $cosh (u + v)$ , etc., where $u$ , $v$ are complex numbers, may be derived as follows. First take $u$ , $v$ as real numbers, then, by Art. 11,

\begin{array}{l} cosh (u + v) & = cosh u cosh v + sinh u sinh v; \\ hence \\ 1 + \frac{1}{2!} {(u + v)}^{2} + \dots & = (1 + \frac{1}{2!} u^{2} + \dots) (1 + \frac{1}{2!} v^{2} + \dots) \\ + (u + \frac{1}{3!} u^{3} + \dots) (v + \frac{1}{3!} v^{3} + \dots) \end{array}

This equation is true when $u$ , $v$ are any real numbers. It must, then, be an algebraic identity. For, compare the terms of the $r$ th degree in the letters $u, v$ on each side. Those on the left are $\frac{1}{r!} {(u + v)}^{r}$ ; and those on the right, when collected, form an $r$ th-degree function which is numerically equal to the former for more than $r$ values of $u$ when $v$ is constant, and for more than $r$ values of $v$ when $u$ is constant. Hence the terms of the $r$ th degree on each side are algebraically identical functions of $u$ and $v$ .⁹ Similarly for the terms of any other degree. Thus the equation above written is an algebraic identity, and is true for all values of $u$ , $v$ , whether real or complex. Then writing for each side its symbol, it follows that

\begin{array}{l} cosh (u + v) & = cosh u cosh v + sinh u sinh v; & (53) \\ and by changing v into - v, \\ cosh (u - v) & = cosh u cosh v - sinh u sinh v . & (54) \end{array}

In a similar manner is found

sinh (u \pm v) = sinh u cosh v \pm cosh u sinh v .

(55)

In particular, for a complex argument,

\begin{aligned} cosh (x \pm i y) & = cosh x cosh i y \pm sinh x sinh i y, \\ sinh (x \pm i y) & = sinh x cosh i y \pm cosh x sinh i y . \end{aligned}\}

(56)

Prob. 79.: Show, by a similar process of generalization,¹⁰ that if $sin u$ , $cos u$ , $exp u$ ¹¹ be deﬁned by their developments in powers of $u$ , then, whatever $u$ may be,

$\begin{array}{l} sin (u + v) & = sin u cos v + cos u sin v, \\ cos (u + v) & = cos u cos v - sin u sin v, \\ exp (u + v) & = exp u exp v . \end{array}$
Prob. 80.: Prove that the following are identities:

$\begin{array}{l} {cosh}^{2} u - {sinh}^{2} u & = 1, \\ cosh u + sinh u & = exp u, \\ cosh u - sinh u & = exp (- u), \\ cosh u & = \frac{1}{2} [exp u + exp (- u)], \\ sinh u & = \frac{1}{2} [exp u - exp (- u)] . \end{array}$

ptail

top

Article 28Addition-Theorems for Complexes.

Article 28
Addition-Theorems for Complexes.