Many important functions, like the exponential function or the trigonometric functions, are represented by a power series. The following theorem shows that these functions are differentiable, and that the derivative of a power series is itself a power series, given by differentiating the individual terms of the series.
denote a
power series which converges
on the
open interval, and represents there a function
. Then the formally differentiated power series
is convergent on . The function is
differentiable
in every point of the interval, and
holds.
Proof
The proof requires a detailed study of power series.
In the formulation of the theorem, we have distinguished between for the power series and for the function, defined by the series, in order to stress the roles they play. This distinction is now not necessary anymore.
holds, due to
Theorem 16.4
.
Hence, there is a proportional relationship between the function and its derivative , and
is the factor. This is still true if is multiplied with a constant. If we consider as a function depending on time , then describes the growing behavior at that point of time. The equation
means that the instantaneous growing rate is always proportional with the magnitude of the function. Such an increasing behavior
(or decreasing behavior, if
)
occurs in nature for a population, if there is no competition for resources, and if the dying rate is neglectable
(the number of mice is then proportional with the number of mice born).
A condition of the form
is an example of a differential equation. This is an equation for a function, which expresses a condition for the derivative. A solution for such a differential equation is a differentiable function which fulfills the condition on its derivative. The differential equation just mentioned are fulfilled by the functions
We will study differential equations in the second semester.
The derivative of the cotangent function follows in the same way.
The number
The number is the area and half of the circumference of a circle with radius . But, in order to build a precise definition for this number on this, we would have first to establish measure theory or the theory of the length of curves. Also, the trigonometric functions have an intuitive interpretation at the unit circle, but also this requires the concept of the arc length. An alternative approach is to define the functions sine and cosine by their power series, and then to define the number with the help of them, and establishing finally the relation with the circle.
Hence, due to
the intermediate value theorem,
there exists at least one zero in the given interval.
To prove uniqueness, we consider the
derivative
of cosine, which is
due to
Theorem 16.8
.
Hence, it is enough to show that sine is positive in the interval , because then cosine is
strictly decreasing
by
Theorem 15.7
in the interval and there is only one zero. Now, for
,
we have
we have
,
hence
,
because of the reasoning in the proof of
Lemma 16.10
.
From that we deduce, with the help of
the addition theorems,
the relations between sine and cosine as mentioned in (3), e.g.
Hence it is enough to prove the statements for cosine. All statements follow from the definition of and from (3).