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Article 18
Expansion of Anti-functions.

Since

d(sinh−1x) dx = 1 1 + x2 = (1 + x2)−1 2 = 1 −1 2 ⋅ x2 + 1 2 ⋅3 4 ⋅ x4 −1 2 ⋅3 4 ⋅5 6 ⋅ x6 + …,

hence, by integration,

sinh−1x = x −1 2 ⋅x3 3 + 1 2 ⋅3 4 ⋅x5 5 −1 2 ⋅3 4 ⋅5 6 ⋅x7 7 + …,  (31)

the integration-constant being zero, since sinh−1x vanishes with x. This series is convergent, and can be used in computation, only when x < 1. Another series, convergent when x > 1, is obtained by writing the above derivative in the form

d(sinh−1x) dx = (x2 + 1)−1 2 = 1 x 1 + 1 x2 −1 2 = 1 x 1 −1 2 ⋅ 1 x2 + 1 2 ⋅3 4 ⋅ 1 x4 −1 2 ⋅3 4 ⋅5 6 ⋅ 1 x6 + ⋯, ∴sinh−1 = C + logx + 1 2 ⋅ 1 2x2 −1 2 ⋅3 4 ⋅ 1 4x4 + 1 2 ⋅3 4 ⋅5 6 ⋅ 1 6x6 −⋯,  (32)

where C is the integration-constant, which will be shown in Art. 19 to be equal to loge2.

A development of similar form is obtained for cosh−1x; for

d(cosh−1x) dx = (x2 − 1)−1 2 = 1 x 1 − 1 x2 −1 2 = 1 x 1 + 1 2 ⋅ 1 x2 + 1 2 ⋅3 4 ⋅ 1 x4 + 1 2 ⋅3 4 ⋅5 6 ⋅ 1 x6 + ⋯,

hence

cosh−1x = C + logx −1 2 ⋅ 1 2x2 −1 2 ⋅3 4 ⋅ 1 4x4 −1 2 ⋅3 4 ⋅5 6 ⋅ 1 6x6 −⋯,  (33)

in which C is again equal to loge2 [Art. 19, Prob. 46]. In order that the function cosh−1x may be real, x must not be less than unity; but when x exceeds unity, this series is convergent, hence it is always available for computation.

Again

d(tanh−1x) dx = 1 1 − x2 = 1 + x2 + x4 + x6 + ⋯,  and hence tanh−1x = x + 1 3x3 + 1 5x5 + 1 7x7 + ⋯,  (34)

From (32), (33), (34) are derived:

sech−1x = cosh−1 1 x = C − logx − x2 2 ⋅ 2 −1 ⋅ 3 ⋅ x4 2 ⋅ 4 ⋅ 4 −1 ⋅ 3 ⋅ 5 ⋅ x6 2 ⋅ 4 ⋅ 6 ⋅ 6 −⋯;  (35) csch−1x = sinh−1 1 x = 1 x −1 2 ⋅ 1 3x3 + 1 2 ⋅3 4 ⋅ 1 5x5 −1 2 ⋅3 4 ⋅5 6 ⋅ 1 7x7 + ⋯, = C − logx + x2 2 ⋅ 2 −1 ⋅ 3 ⋅ x4 2 ⋅ 4 ⋅ 4 + 1 ⋅ 3 ⋅ 5 ⋅ x6 2 ⋅ 4 ⋅ 6 ⋅ 6 −⋯;  (36) coth−1x = tanh−1 1 x = 1 x + 1 3x3 + 1 5x5 + 1 7x7 + ⋯.  (37)
Prob. 44.
Show that the series for tanh −1x,coth −1x,sech −1x, are always available for computation.
Prob. 45.
Show that one or other of the two developments of the inverse hyperbolic cosecant is available.

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