The exponential, circular and hyperbolic functions, whether direct or inverse, like the real constants π and e, would be not much useful if we could not calculate their values.
However, it must be clear that, since they are real numbers, in general it isn't possible to thoroughly express these values, like we do with natural numbers. In general, to calculate a real number α means to find an algorithm which generates a sequence of rational numbers ai converging to α. The more i increases, the more ai approximates α. In the practical applications we use sufficiently good approximations.
Maybe the more simple example of such sequences is the sequence of the powers with natural exponent i of a rational number x such that |x|<1. The more i increases, the more the terms xi of this sequence approach 0. We may formally express this by saying that for all rational number x, such that |x|<1
(the limit of the sequence xi equals 0)
if and only if for every positive real number ε there is a natural
number iε such that
for every i>iε.
This could be a good starting point to find other useful sequences converging the real numbers generated by the transcendental functions.
It can easily checked that, for every real number x,
and that, if x≠1,
The sum in (7.2) is a geometric series, because the sequence of its addends is a geometric sequence, that is the ratio of any two successive addends is constant.
The limit for n→∞ of the series equals the limit of the fraction:
If |x|<1, using a simplified notation,
From (7.3) we have also
The fraction
is the derivative with respect to x of ln(1+x).
Hence, for |x|<1 and remembering that ln1=0,
we can vice versa say that ln(1+x) is the antiderivative
of the sum
and therefore
So it is possible to approximate the natural logarithm of the numbers in the interval ]0;2[ and the approximation will be better the more one increases the number of the addends in the sum.
The sum (7.5) is said Mercator's series. This series has a limited convergence domain and converges very slowly, but from it we can deduce other series converging over all the logarithm domain.
In fact, from the Mercator's series we have
Subtracting term by term the (7.6) from the (7.5) and remembering the basic properties of the logarithm we obtain
For example, to calculate ln10
The (7.7) can be rewritten as
The first term in (7.8) equals the hyperbolic arctangent, so
From the (7.4) we can write
Integrating both the terms and remembering that arctan(0)=0, we have
From the (7.10), remembering that
we obtain
an algorithm to approximate π
This way to approximate π is known as Leibniz's series.
If we consider the Mercator's series (7.5), we could find that the coefficients
ci of the powers xi in the sum are such that
,
where f(i)(0) denotes the value of the n-th derivative of the
function applied to 0 and i! is the factorial of the index i.
To can apply the given expression of ci also when i=0,
we assume that f(0)(x) is the function itself and 0!=1.
We could find the same thing for the arctangent functions (7.9) and (7.10) and, in general, for every other infinitely differentiable function f(x), if the function and its derivatives are calculable for x=0. In fact, from
we can get
So for every other infinitely differentiable function f(x), if the function and its derivatives are calculable for x=0, we have
the (7.13) is said MacLaurin's series expansion
The more immediate series expansion we can get using the (7.13) is that of the natural exponential ex for which all the derivatives coincide with the function itself which, when x=0, has value 1.
The series (7.14) allows to approximate the number e=e1
From the (7.14) we have also
From the (7.14), (7.16) and (3.9), we get the series for the hyperbolic cosine and sine, which could however be directly deduced from the (7.13)
From the (7.13) we can easily deduce the series expansion of the circular cosine and sine
If we use the (7.17) to expand coshix, where i is the imaginary unit, we get
If we use the (7.18) to expand sinhix, we get
From the (7.21) and the (3.9) we get
From the (7.22) and the (3.9) we get
Gathering all together
These are the well known and very important Euler's formulas.
From the (7.23) we have also
which allows to express a complex number z, with modulus ρ and argument θ, in both the following ways:
Given the function
real for x<1, its successive derivatives are
Since f(0)=1, the MacLaurin's series expansion of f(x) results
Moreover, for -1<x<1, we have
The antiderivative G(x) of g(x), such that G(0)=0, is the arcsine of x, so
and , finally
The antiderivative Γ(x) of -g(x) such that Γ(0)=π/2 is the arccosine of x, so
We can obtain the expansion (7.24) in a more direct way if we expand the function (7.23) using the generalized binomial coefficients. Infact from
we have
Other very useful series expansions can be found in Fourier series expansion.