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Functions of $x+iy$ in the Form $X+iY$.

By the addition-formulas,

$$\begin{array}{cc}\begin{array}{ccc}\hfill cosh(x+iy)& =coshxcoshiy+sinhxsinhiy,\hfill & \hfill \\ \hfill sinh(x+iy)& =sinhxcoshiy+coshxsinhiy,\hfill \end{array}& \end{array}$$but

$$\begin{array}{cc}coshiy=cosy,\phantom{\rule{1em}{0ex}}sinhiy=isiny,& \\ & \end{array}$$hence

$$\begin{array}{cc}\left.\begin{array}{ccc}\hfill cosh(x+iy)& =coshxcosy+isinhxsiny,\hfill & \hfill \\ \hfill sinh(x+iy)& =sinhxcosy+icoshxsiny.\hfill \end{array}\right\}& \text{(60)}\end{array}$$

Thus if $cosh(x+iy)=a+ib$, $sinh(x+iy)=c+id$, then

From these expressions the complex tables at the end of this chapter have been computed.

Writing $coshz=Z$, where $z=x+iy$, $Z=X+iY$; let the complex numbers $z,Z$ be represented on Argand diagrams, in the usual way, by the points whose coordinates are $(x,y)$, $(X,Y)$; and let the point $z$ move parallel to the $y$-axis, on a given line $x=m$, then the point $Z$ will describe an ellipse whose equation, obtained by eliminating $y$ between the equations $X=coshmcosy$, $Y=sinhmsiny$, is

$$\frac{{X}^{2}}{{(coshm)}^{2}}+\frac{{Y}^{2}}{{(sinhm)}^{2}}=1,$$ |

and which, as the parameter $m$ varies, represents a series of confocal ellipses, the distance between whose foci is unity. Similarly, if the point $z$ move parallel to the $x$-axis, on a given line $y=n$, the point $Z$ will describe an hyperbola whose equation, obtained by eliminating the variable $x$ from the equations. $X=coshxcosn$, $Y=sinhxsinn$, is

$$\frac{{X}^{2}}{{(cosn)}^{2}}-\frac{{Y}^{2}}{{(sinn)}^{2}}=1,$$ |

and which, as the parameter $n$ varies, represents a series of hyperbolas confocal with the former series of ellipses.

These two systems of curves, when accurately drawn at close intervals on the
$Z$
plane, constitute a chart of the hyperbolic cosine; and the numerical value of
$cosh(m+in)$
can be read oﬀ at the intersection of the ellipse whose parameter is
$m$ with the hyperbola whose
parameter is $n$.^{13} A similar
chart can be drawn for $sinh(x+iy)$,
as indicated in Prob. 85.

Periodicity of Hyperbolic Functions.—The functions $sinhu$ and $coshu$ have the pure imaginary period $2i\pi $. For

$$\begin{array}{llll}\hfill sinh(u+2i\pi )& =sinhucos2\pi +icoshusin2\pi =sinhu,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill cosh(u+2i\pi )& =coshucos2\pi +isinhusin2\pi =coshu.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$The functions $sinhu$ and $coshu$ each change sign when the argument $u$ is increased by the half period $i\pi $. For

$$\begin{array}{llll}\hfill sinh(u+i\pi )& =sinhucos\pi +icoshusin\pi =-sinhu,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill cosh(u+i\pi )& =coshucos\pi +isinhusin\pi =-coshu.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$The function $tanhu$ has the period $i\pi $. For, it follows from the last two identities, by dividing member by member, that

$$tanh(u+i\pi )=tanhu.$$ |

By a similar use of the addition formulas it is shown that

$$sinh(u+\frac{1}{2}i\pi )=icoshu,\phantom{\rule{1em}{0ex}}cosh(u+\frac{1}{2}i\pi )=isinhu.$$ |

By means of these periodic, half-periodic, and quarter-periodic relations, the hyperbolic functions of $x+iy$ are easily expressible in terms of functions of $x+iy\prime $, in which $y\prime $ is less than $\frac{1}{2}\pi $.

The hyperbolic functions are classed in the modern function-theory of a complex variable as functions that are singly periodic with a pure imaginary period, just as the circular functions are singly periodic with a real period, and the elliptic functions are doubly periodic with both a real and a pure imaginary period.

Multiple Values of Inverse Hyperbolic Functions.—It follows from the periodicity of the direct functions that the inverse functions ${sinh}^{-1}m$ and ${cosh}^{-1}m$ have each an indeﬁnite number of values arranged in a series at intervals of $2i\pi $. That particular value of ${sinh}^{-1}m$ which has the coeﬃcient of $i$ not greater than $\frac{1}{2}\pi $ nor less than $-\frac{1}{2}\pi $ is called the principal value of ${sinh}^{-1}m$; and that particular value of ${cosh}^{-1}m$ which has the coeﬃcient of $i$ not greater than $\pi $ nor less than zero is called the principal value of ${cosh}^{-1}m$. When it is necessary to distinguish between the general value and the principal value the symbol of the former will be capitalized; thus

$$\begin{array}{cc}{\text{Sinh}}^{-1}m={sinh}^{-1}m+2ir\pi ,\phantom{\rule{1em}{0ex}}{\text{Cosh}}^{-1}m={cosh}^{-1}m+2ir\pi ,& \\ {\text{Tanh}}^{-1}m={tanh}^{-1}m+ir\pi ,& \end{array}$$in which $r$ is any integer, positive or negative.

Complex Roots of Cubic Equations.—It is well known that when the roots of a cubic equation are all real they are expressible in terms of circular functions. Analogous hyperbolic expressions are easily found when two of the roots are complex. Let the cubic, with second term removed, be written

$${x}^{3}\pm 3bx=2c.$$ |

Consider ﬁrst the case in which $b$ has the positive sign. Let $x=rsinhu$, substitute, and divide by ${r}^{3}$, then

$${sinh}^{3}u+\frac{3b}{{r}^{2}}sinhu=\frac{2c}{{r}^{3}}.$$ |

Comparison with the formula ${sinh}^{3}u+\frac{3}{4}sinhu=\frac{1}{4}sinh3u$ gives

$$\begin{array}{cc}\frac{3b}{{r}^{2}}=\frac{3}{4},\phantom{\rule{1em}{0ex}}\frac{2c}{{r}^{3}}=\frac{sinh3u}{4},& \\ & \end{array}$$whence

$$\begin{array}{cc}r=2{b}^{\frac{1}{2}},\phantom{\rule{1em}{0ex}}sinh3u=\frac{c}{{b}^{\frac{3}{2}}},\phantom{\rule{1em}{0ex}}u=\frac{1}{3}{sinh}^{-1}\frac{c}{{b}^{\frac{3}{2}}};& \\ & \end{array}$$therefore

$$\begin{array}{cc}x=2{b}^{\frac{1}{2}}sinh\left(\frac{1}{3}{sinh}^{-1}\frac{c}{{b}^{\frac{3}{2}}}\right),& \end{array}$$in which the sign of ${b}^{\frac{1}{2}}$ is to be taken the same as the sign of $c$. Now let the principal value of ${sinh}^{-1}\frac{c}{{b}^{\frac{3}{2}}}$, found from the tables, be $n$; then two of the imaginary values are $n\pm 2i\pi $, hence the three values of $x$ are $2{b}^{\frac{1}{2}}sinh\frac{n}{3}$ and $2{b}^{\frac{1}{2}}sinh\left(\frac{n}{3}\pm \frac{2i\pi}{3}\right)$. The last two reduce to $-{b}^{\frac{1}{2}}sinh\left(\frac{n}{3}\pm i\sqrt{3}cosh\frac{n}{3}\right)$.

Next, let the coeﬃcient of $x$ be negative and equal to $-3b$. It may then be shown similarly that the substitution $x=rsin\theta $ leads to the three solutions

$$-2{b}^{\frac{1}{2}}sin\frac{n}{3},\phantom{\rule{1em}{0ex}}{b}^{\frac{1}{2}}\left(sin\frac{n}{3}\pm \sqrt{3}cos\frac{n}{3}\right),\phantom{\rule{1em}{0ex}}\text{where}n={sin}^{-1}\frac{c}{{b}^{\frac{3}{2}}}.$$ |

These roots are all real when $c\ngtr {b}^{\frac{3}{2}}$. If $c>{b}^{\frac{3}{2}}$, the substitution $x=rcoshu$ leads to the solution

$$x=2{b}^{\frac{1}{2}}cosh\left(\frac{1}{3}{cosh}^{-1}\frac{c}{{b}^{\frac{3}{2}}}\right),$$ |

which gives the three roots

$$2{b}^{\frac{1}{2}}cosh\frac{n}{3},\phantom{\rule{1em}{0ex}}-{b}^{\frac{1}{2}}\left(cosh\frac{n}{3}\pm i\sqrt{3}sinh\frac{n}{3}\right),\phantom{\rule{1em}{0ex}}\text{wherein}n={cosh}^{-1}\frac{c}{{b}^{\frac{3}{2}}},$$ |

in which the sign of ${b}^{\frac{1}{2}}$ is to be taken the same as the sign of $c$.

- Prob. 85.
- Show that the chart of $cosh(x+iy)$ can be adapted to $sinh(x+iy)$, by turning through a right angle; also to $sin(x+iy)$.
- Prob. 86.
- Prove the identity
$$tanh(x+iy)=\frac{sinh2x+isin2y}{cosh2x+cos2y}.$$ - Prob. 87.
- If $cosh(x+iy)=a+ib$,
be written in the “modulus and amplitude” form as
$r(cos\theta +isin\theta ),=rexpi\theta $,
then
- Prob. 88.
- Find the modulus and amplitude of $sinh(x+iy)$.
- Prob. 89.
- Show that the period of $exp\frac{2\pi u}{a}$ is $ia$.
- Prob. 90.
- When $m$ is
real and $>1$,
${cos}^{-1}m=i{cosh}^{-1}m$,
${sin}^{-1}m=\frac{\pi}{2}-i{cosh}^{-1}m$.
When $m$ is real and $<1$, ${cosh}^{-1}m=i{cos}^{-1}m$.

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