93. Periodicity of Modocyclic Functions. In Art. 81 it was shown that all exponentials are periodic having the period 2iκπ. The forms ew/k, eiw and ew have therefore the periods 2iкπ, 2, and 2ir respectively; hence also the modocyclic, circular and hyperbolic functions, whose definitions render them explicitly in terms of ew/K, eiw and ev (Art. 90) have the same respective periods, so that, if f stand for any one of the symbols sink, cos tan, etc., g for sin, cos, etc., and h for sinh, cosh, etc., the law is, n being any integer: For modocyclic functions ƒ(w + 2ni«π)=ƒ(w),

have also the shorter period iπ.

(2). Range of Values. Compute the following table of the range of values of the modocyclic functions:

cotk w ti∞/K to ±10/K+i0/K to ix/Kix/K to Fi0/K F10/K to i∞ok

(3). Tabulate, as above, the range of values of the circular and hyperbolic functions.

(12).

Write similar formulæ for all modocyclic functions of w \ ίκπ, ίκπ+w, w+ίκπί2, ἐκπ/2 +w, w +3εκπ|2, зiкπ/2±w, and 2iкπ ±w.

95. The Inverse Functions. The modocyclic, circular and hyperbolic functions, being defined in terms of exponentials, are direct; and corresponding to them, through inversion (Art. 83) we have the inverse modocyclic, circular, and hyperbolic functions.

Applying the operation of inversion to the three

equations

z=ƒ(w+2nikπ)=ƒ(w),

z=g (w + 2n′′) =g(w),

z= h (w + 2nin) = h (w),

2=

of Art. 93 we have the corresponding inverse forms:

in which n is any integer whatever. Thus, like logarithms, the various forms of inverse cyclic functions are manyvalued to an infinite extent.

We may however define the interval over which the values of w shall range in such a way as to make it onevalued within the interval considered. Care must be taken to do this in writing the formulæ of inverse functions.

By

96. Agenda. Formulæ of Inverse Functions. inverting the corresponding direct formulæ prove the following, and assign in each case the interval for which the formula is true.

= sin7' (x 1/1 + k ̄2y2±y√1+k ̄2x2).

=

(4). singsing1y

= cos ̧2 (V/ (1 + k ̃ ̄2x2) (1+k ̄2 j2) ±k ̄2xy).

=

(8). 2 sin x=sing' (2x1/1 + x ̄2x2).

(14). Deduce the corresponding formulæ for inverse circular and hyperbolic functions by assigning to the values + i and + 1 successively.

97. Logarithmic Forms of Inverse Cyclic Functions. The inverse cyclic functions are obviously logarithmic. We may obtain them as such in the following manner. Let u, v, x, y be four quantities, in general complex, such that

and

u = cosxsin1y=log1 (x + x ̄3y)

= ln (x + K ̄'y),

x = ± √ I + Kay2, k ̃y=± √ x2 — I.

... sing1y=«ln (y/κ ± √y2/«2 + I),

cos' x = ln (x ± √x2 − 1 ).

(Art. 91, 3.)

The logarithmic equivalent of tan' may be deduced from the definition tan,w=к (ew/K-e-w/K) / (ew/K + e−w/K)=%, by solving this equation as a quadratic in ew/K. Thus,

Since cot, w= 1/x, the corresponding formula for cot'z is

The forms for secx and cscy are also obtained from those for cosx and singly by changing x and y into 1/x and 1/y respectively. They are

sec1x = k ln (1/x ± √ ́ ́1/v2 − 1).

cscy == × ln (1/ky ± VI / k2 y2 + 1).