2.4.3 Euler’s Form of Complex Numbers
As will be derived later, the exponential of ξ may be exactly expressed as
as per Taylor’s series expanded around ξ = 0; the sine of ξ may likewise appear as
while cos ξ abides to
After setting ξ ≡ ιθ, Eq. (2.551) becomes
– where splitting of the summation between even and odd values of k was possible based on ι2 = −1 and ι4 = 1 as algebraic rules (supporting switch between −1 and 1 along consecutive terms), and ι and ι3 = −ι (that switches between ι and −ι along consecutive terms). In view of
(2.555)
based on Eq. (2.552), and
(2.556)
stemming from Eq. (2.553), one may rewrite Eq. (2.554) as
– thus supporting Euler’s (exponential) form of a complex number a + ιb, as long as a2 + b2 = 1. By the same token,
(2.558)
based on Eq. (2.557) after replacing θ by −θ – where the even nature of cosine as per Eq. (2.296) and the odd nature of sine as per Eq. (2.295) allow further transformation to
Ordered addition of Eqs. (2.557) and (2.559) unfolds
whereas ordered subtraction thereof gives rise to
division of both sides by 2 yields
and
from Eqs. (2.560) and (2.561), respectively – or else
from Eq. (2.562) after combination with Eq. (2.473), as well as
from Eq. (2.563) following combination with Eq. (2.472). The two basic relationships between hyperbolic and circular functions are consequently conveyed by Eqs. (2.564) and (2.565).
2.4.4 Inverse Functions
Inverse hyperbolic functions are often useful, and will accordingly be discussed next; in the case of sinh−1 x, one should start by setting
so application of hyperbolic sine to both sides gives rise to
– since composition of a function with its inverse cancels out their mutual effects. Insertion of Eq. (2.567) allows transformation of Eq. (2.496) – rewritten in (dummy) variable y, to
(2.568)
so adding x2 to both sides and taking square roots thereof afterward generate