Article 29
Functions of Pure Imaginaries.

In the defining identities

put for $u$ the pure imaginary $iy$, then

These formulas serve to interchange hyperbolic and circular functions. The hyperbolic cosine of a pure imaginary is real, and the hyperbolic sine and tangent are pure imaginaries.

The following table exhibits the variation of $sinhu$, $coshu$. $tanhu$, $expu$, as $u$ takes a succession of pure imaginary values.

Prob. 81.

Prove the following identities:

Prob. 82.

Equating the respective real and imaginary parts on each side of the equation , express $cosny$ in powers of $cosy$, $siny$; and hence derive the corresponding expression for $coshny$.

Prob. 83.

Show that, in the identities (57) and (58), $y$ may be replaced by a general complex, and hence that

Prob. 84.

From the product-series for $sinx$ derive that for $sinhx$:

$$\begin{array}{llll}\hfill sinx& =x\left(1-\frac{x^2}{{\pi }^2}\right)\left(1-\frac{x^2}{2^2{\pi }^2}\right)\left(1-\frac{x^2}{3^2{\pi }^2}\right)\dots ,& \hfill & \\ \hfill sinhx& =x\left(1+\frac{x^2}{{\pi }^2}\right)\left(1+\frac{x^2}{2^2{\pi }^2}\right)\left(1+\frac{x^2}{3^2{\pi }^2}\right)\dots .& \hfill & \end{array}$$