Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 12
- Power series
Definition
Let be a sequence of real numbers and another real number. Then the series
For a power series, it is important that varies and that the power series represents in some convergence interval a function in . Every polynomial is a power series, but one for which all coefficients starting with a certain member are . In this case, the convergence is everywhere.
We have encountered an important power series earlier, the geometric series (here all coefficients equal ), which converges for and represents the function , see Theorem 9.13 . Another important power series is the exponential series, which for every real number converges and represents the real exponential function. Its inverse function is the natural logarithm.
The behavior of convergence of a power series is given by the following theorem.
Theorem
Let
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be a power series and suppose that there exists some such that converges. Then there exists a positive (where is allowed) such that for all fulfilling the series converges absolutely. On such an (open) interval of convergence, the power series represents a continuous function.
Proof
If two functions are given by power series, then their sum is simply given by the
(componentwise defined)
sum of the power series. It is not clear at all by which power series the product of two power series is described. The answer is given by the Cauchy-product of series.
Definition
For two series and of real numbers, the series
Also, for the following statement we do not provide a proof.
Lemma
Let
be absolutely convergent series of real numbers. Then also the Cauchy product is absolutely convergent and for its sum the equation
holds.
From this we can infer that the product of power series is given by the power series whose coefficients are those which arise by the multiplication of polynomials, see
Exercise 12.3
.
- Exponential series and exponential function
We discuss another important power series, the exponential series and the exponential function defined by it.
So this is just the series
Theorem
Proof
For , the statement is clear. Else, we consider the fraction
This is, for , smaller than . By the ratio test we get convergence.
Due to this property, we can define the real exponential function.
The following statement is called the functional equation for the exponential function.
Theorem
Proof
The Cauchy product of the two exponential series is
where
This series is due to Lemma 12.4 absolutely convergent and the limit is the product of the two limits. Furthermore, the -th summand of the exponential series of equals
so that both sides coincide.
Corollary
- .
- For every , we have . In particular .
- For integers , the relation holds.
- For every , we have .
- For we have , and for we have .
- The real exponential function is strictly increasing.
Proof
(1) follows directly from the definition.
(2) follows from
using
Theorem 12.8
.
(3) follows for
from
Theorem 12.8
by induction, and from that it follows with the help of (2) also for negative .
(4). Nonnegativity follows from
(5). For real we have , so that because of (4), one factor must be and the other factor must be . For , we have
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as only positive numbers are added.
(6). For real
,
we have
,
and therefore, because of (5)
,
hence
With the help of the exponential series, we also define Euler's number.
Definition
The real number
So we have . Its numerical value is
Remark
For Euler's number there is also the description
so that can also be introduced as the limit of this sequence. However, the convergence in the exponential series is much faster.
We will write also instead of . This is, for , compatible with the usual meaning of powers in the sense of the fourth lecture due to Corollary 12.9 . The compatibility with arbitrary roots (if the exponents are rational) follows from Remark 12.17 and Exercise 12.17 .
Theorem
Proof
The continuity follows from Theorem 12.2 , since the exponential function is defined with the help of a power series. Due to Corollary 12.9 , the image lies in , and the image is, because of the intermediate value theorem an interval. The unboundedness of the image follows from Corollary 12.9 . This implies, because of Corollary 12.9 , that also arbitrary small positive real numbers are obtained. Thus the image is . Injectivity follows from Corollary 12.9 , in connection with Exercise 5.38 .
- Logarithms
Theorem
continuous strictly increasing function, which defines a bijection between and . Moreover, the functional equation
holds for all .
Proof
This follows from Theorem 12.12 , Theorem 11.7 , Corollary 12.9 and Corollary 12.9 .
Definition
For a positive real number , the exponential function for the base is defined as
Theorem
Let denote a positive real number. Then the exponential function
- We have for all .
- We have .
- For and , we have .
- For and , we have .
- For , the function is strictly increasing.
- For , the function is strictly decreasing.
- We have for all .
- For , we have .
Proof
Remark
There is another way to introduce the exponential function to base . For a natural number , one takes the th product of with itself as definition for . For a negative integer , one sets . For a positive rational number , one sets
where one has to show that this is independent of the chosen representation as a fraction. For a negative rational number, one takes again the inverse. For an arbitrary real number , one takes a sequence of rational numbers converging to , and defines
For this, one has to show that these limits exist and that they are independent of the chosen rational sequence. For the passage from to , the concept of uniform continuity is crucial.
Definition
For a positive real number , , the logarithm to base of is defined by
Theorem
- We have and , this means that the logarithm to Base is the inverse function for the exponential function to base .
- We have .
- We have for .
- We have
Proof
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