Article 27
Functions of Complex Numbers.

As vector quantities are of frequent occurrence in Mathematical Physics; and as the numerical measure of a vector in terms of a standard vector is a complex number of the form $x+iy$, in which $x,y$ are real, and $i$ stands for $\sqrt{-1}$; it becomes necessary in treating of any class of functional operations to consider the meaning of these operations when performed on such generalized numbers.⁶ The geometrical definitions of $coshu$, $sinhu$, given in Art. 7, being then no longer applicable, it is necessary to assign to each of the symbols , , a suitable algebraic meaning, which should be consistent with the known algebraic values of $coshx$, $sinhx$, and include these values as a particular case when $y=0$. The meanings assigned should also, if possible, be such as to permit the addition-formulas of Art. 11 to be made general, with all the consequences that flow from them.

Such definitions are furnished by the algebraic developments in Art. 16, which are convergent for all values of $u$, real or complex. Thus the definitions of , are to be

From these series the numerical values of , could be computed to any degree of approximation, when $x$ and $y$ are given. In general the results will come out in the complex form⁷

The other functions are defined as in Art. 7, eq. (9).

Prob. 79.

Prove from these definitions that, whatever $u$ may be,