From the results obtained in the previous section we can easily deduce
Using the (3.9) to express the hyperbolic cotangent, we have
If we explicitly write some addends of the sum we have
Finally, using a more formal and compact notation:
The series expansion of the circular cotangent can be obtained from the (9.1), (7.21) and (7.22)
In a more compact way:
To deduce the series expansion of the hyperbolic tangent we can use the equality
In fact
So, using twice the series expansion of the hyperbolic cotangent, we get
If we calculate explicitly the coefficients, we have
In a more compact way:
To have the series expansion of the circular tangent we use the equality
We get
If we calculate explicitly the coefficients, we have
In a more compact way:
Since
the series expansion of the hyperbolic cosecant can be obtained using the series expansions of the hyperbolic tangent and cotangent. From (9.1) and (9.3) we get
More explicitly
From (7.22) e (9.7) we get
and, in more compact way
The hyperbolic secant can be expressed as a function of the exponential
This expression of the hyperbolic secant coincides with that of the function s(x) defined in (8.2) of the previous page, then
where En is the n-th Euler's number. Since, for odd indices these numbers are zero, we can more economically write
From the Euler's formulas, the circular secant can be expressed as
The even powers of the imaginary unit oscillate between 1 and -1, then
