Article 30
Functions of x + iy in the Form X + iY .

By the addition-formulas,

cosh(x + iy) = coshxcoshiy + sinhxsinhiy, sinh(x + iy) = sinhxcoshiy + coshxsinhiy,


coshiy = cosy,sinhiy = isiny,


cosh(x + iy) = coshxcosy + isinhxsiny, sinh(x + iy) = sinhxcosy + icoshxsiny.  (60)

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

a = coshxcosy,b = sinhxsiny, c = sinhxcosy,d = coshxsiny. (61)

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

X2 (coshm)2 + 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

X2 (cosn)2 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 2iπ. For

sinh(u + 2iπ) = sinhucos2π + icoshusin2π = sinhu, cosh(u + 2iπ) = coshucos2π + isinhusin2π = coshu.

The functions sinhu and coshu each change sign when the argument u is increased by the half period iπ. For

sinh(u + iπ) = sinhucosπ + icoshusinπ = sinhu, cosh(u + iπ) = coshucosπ + isinhusinπ = coshu.

The function tanhu has the period iπ. For, it follows from the last two identities, by dividing member by member, that

tanh(u + iπ) = tanhu.

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

sinh(u + 1 2iπ) = icoshu,cosh(u + 1 2iπ) = 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, in which y is less than 1 2π.

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 sinh1m and cosh1m have each an indefinite number of values arranged in a series at intervals of 2iπ. That particular value of sinh1m which has the coefficient of i not greater than 1 2π nor less than 1 2π is called the principal value of sinh1m; and that particular value of cosh1m which has the coefficient of i not greater than π nor less than zero is called the principal value of cosh1m. When it is necessary to distinguish between the general value and the principal value the symbol of the former will be capitalized; thus

 Sinh1m = sinh1m + 2irπ, Cosh1m = cosh1m + 2irπ,  Tanh1m = tanh1m + irπ,

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

x3 ± 3bx = 2c.

Consider first the case in which b has the positive sign. Let x = rsinhu, substitute, and divide by r3, then

sinh3u + 3b r2 sinhu = 2c r3.

Comparison with the formula sinh3u + 3 4 sinhu = 1 4 sinh3u gives

3b r2 = 3 4,2c r3 = sinh3u 4 ,


r = 2b1 2 ,sinh3u = c b3 2 ,u = 1 3sinh1 c b3 2 ;


x = 2b1 2 sinh 1 3sinh1 c b3 2 ,

in which the sign of b1 2 is to be taken the same as the sign of c. Now let the principal value of sinh1 c b3 2 , found from the tables, be n; then two of the imaginary values are n ± 2iπ, hence the three values of x are 2b1 2 sinh n 3 and 2b1 2 sinh n 3 ±2iπ 3 . The last two reduce to b1 2 sinh n 3 ± i3cosh n 3 .

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

2b1 2 sin n 3 ,b1 2 sin n 3 ±3cos n 3 ,  where n = sin1 c b3 2 .

These roots are all real when c b3 2 . If c > b3 2 , the substitution x = rcoshu leads to the solution

x = 2b1 2 cosh 1 3cosh1 c b3 2 ,

which gives the three roots

2b1 2 cosh n 3 , b1 2 cosh n 3 ± i3sinh n 3 ,  wherein n = cosh1 c b3 2 ,

in which the sign of b1 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) = sinh 2x + isin 2y cosh 2x + cos 2y .
Prob. 87.
If cosh(x + iy) = a + ib, be written in the “modulus and amplitude” form as r(cos θ + isin θ),= rexp iθ, then

r2 = a2 + b2 = cosh 2x = sin 2y = cos 2y sinh 2x, tan θ = b a = tanh xtan y.
Prob. 88.
Find the modulus and amplitude of sinh(x + iy).
Prob. 89.
Show that the period of exp 2πu a is ia.
Prob. 90.
When m is real and > 1, cos 1m = icosh 1m, sin 1m = π 2 icosh 1m.

When m is real and < 1, cosh 1m = icos 1m.