Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 12/latex
\setcounter{section}{12}
\zwischenueberschrift{Power series}
\inputdefinition
{ }
{
Let \mathl{{ \left( c_n \right) }_{n \in \N }}{} be a sequence of
real numbers
and $x$ another real number. Then the
series
\mathdisp {\sum _{ n= 0}^\infty c_n x^{ n }} { }
}
For a power series, it is important that $x$ varies and that the power series represents in some \stichwort {convergence interval} {} a function in $x$. Every polynomial is a power series, but one for which all coefficients starting with a certain member are $0$. In this case, the convergence is everywhere.
We have encountered an important power series earlier, the geometric series \mathl{\sum_{n=0}^\infty x^n}{} (here all coefficients equal $1$), which converges for
\mavergleichskette
{\vergleichskette
{ \betrag { x }}
{ < }{ 1}
{ }{}
{ }{}
{ }{}
}
{}{}{}
and represents the function \mathl{1/(1-x)}{,} see
Theorem 9.13
.
Another important power series is the \stichwort {exponential series} {,} which for every real number converges and represents the \stichwort {real exponential function} {.} Its inverse function is the \stichwort {natural logarithm} {.}
The behavior of convergence of a power series is given by the following theorem.
\inputfaktbeweis
{Real power series/Convergence/Continuous function/Fact}
{Theorem}
{}
{
\faktsituation {}
\faktvoraussetzung {Let
\mavergleichskettedisp
{\vergleichskette
{ f(x)
}
{ \defeq} {\sum_{n = 0}^\infty c_n x^n
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
be a
power series
and suppose that there exists some
\mavergleichskette
{\vergleichskette
{ x_0
}
{ \neq }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
such that \mathl{\sum_{n=0}^\infty c_n x_0^n}{} converges.}
\faktfolgerung {Then there exists a positive $R$
(where
\mavergleichskette
{\vergleichskette
{ R
}
{ = }{ \infty
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
is allowed)
such that for all
\mavergleichskette
{\vergleichskette
{ x
}
{ \in }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
fulfilling
\mavergleichskette
{\vergleichskette
{ \betrag { x }
}
{ < }{ R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
the series converges absolutely. On such an
\zusatzklammer {open} {} {}
interval of convergence, the power series \mathl{f(x)}{} represents a
continuous function.}
\faktzusatz {}
}
{Real power series/Convergence/Continuous function/Fact/Proof
If two functions are given by power series, then their sum is simply given by the
\zusatzklammer {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.
\inputdefinition
{ }
{
For two
series
\mathkor {} {\sum_{ i = 0}^\infty a_{ i }} {and} {\sum_{ j = 0}^\infty b_{ j }} {}
of
real numbers,
the series
\mathdisp {\sum_{ k = 0}^\infty c_{ k } \text{ with } c_k \defeq \sum_{i = 0}^k a_i b_{k-i}} { }
}
Also, for the following statement we do not provide a proof.
\inputfakt
{Real series/Cauchy-product/Absolute convergence/Fact}
{Lemma}
{}
{
\faktsituation {Let
\mathdisp {\sum_{ k = 0}^\infty a_{ k } \text{ and } \sum_{ k = 0}^\infty b_{ k }} { }
be
absolutely convergent
series
of
real numbers.}
\faktfolgerung {Then also the
Cauchy product
\mathl{\sum_{ k = 0}^\infty c_{ k }}{} is absolutely convergent and for its sum the equation
\mavergleichskettedisp
{\vergleichskette
{ \sum_{ k = 0}^\infty c_{ k }
}
{ =} { { \left( \sum_{ k = 0}^\infty a_{ k } \right) } \cdot { \left( \sum_{ k = 0}^\infty b_{ k } \right) }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
holds.}
\faktzusatz {}
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
.
\zwischenueberschrift{Exponential series and exponential function}
We discuss another important power series, the exponential series and the exponential function defined by it.
\inputdefinition
{ }
{
For every
\mavergleichskette
{\vergleichskette
{x
}
{ \in }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the
series
\mathdisp {\sum_{ n =0}^\infty \frac{ x^{ n } }{n!}} { }
}
So this is just the series
\mathdisp {1+x+ { \frac{ x^2 }{ 2 } } +{ \frac{ x^3 }{ 6 } } + { \frac{ x^4 }{ 24 } } +{ \frac{ x^5 }{ 120 } } + \ldots} { . }
\inputfaktbeweis
{Exponential series/Real/Absolute convergence/Fact}
{Theorem}
{}
{
\faktsituation {For every
\mavergleichskette
{\vergleichskette
{ x
}
{ \in }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the
exponential series
\mathdisp {\sum_{ n =0}^\infty \frac{ x^{ n } }{n!}} { }
}
\faktfolgerung {is
absolutely convergent.}
\faktzusatz {}
}
{
For
\mavergleichskette
{\vergleichskette
{ x
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the statement is clear. Else, we consider the fraction
\mavergleichskettedisp
{\vergleichskette
{ \betrag { \frac{ \frac{x^{n+1} }{(n+1)!} }{\frac{x^n}{n!} } }
}
{ =} { \betrag { \frac{x}{n+1} }
}
{ =} { \frac{ \betrag { x } }{n+1}
}
{ } {
}
{ } {
}
}
{}{}{.}
This is, for
\mavergleichskette
{\vergleichskette
{ n
}
{ \geq }{ 2 \betrag { x }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
smaller than \mathl{1/2}{.} By the
ratio test,
we get
convergence.
Due to this property, we can define the real exponential function.
\inputdefinition
{ }
{
The
function
\mathdisp {\R \longrightarrow \R
, x \longmapsto \exp x \defeq \sum_{ n =0}^\infty \frac{ x^{ n } }{n!}} { , }
is called the (real)
}
The following statement is called the \stichwort {functional equation for the exponential function} {.}
\inputfaktbeweis
{Exponential series/Real/Functional equation/Fact}
{Theorem}
{}
{
\faktsituation {}
\faktvoraussetzung {For
real numbers
\mavergleichskette
{\vergleichskette
{ x,y
}
{ \in }{\R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}}
\faktfolgerung {the equation
\mavergleichskettedisp
{\vergleichskette
{ \exp { \left( x+y \right) }
}
{ =} { \exp x \cdot \exp y
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
holds.}
\faktzusatz {}
}
{
The
Cauchy product
of the two exponential series is
\mathdisp {\sum_{ n = 0}^\infty c_{ n }} { , }
where
\mavergleichskettedisp
{\vergleichskette
{ c_n
}
{ =} { \sum_{i = 0}^n \frac{x^{i} }{i!} \cdot \frac{ y^{n-i } }{ (n-i)!}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
This series is due to
Lemma 12.4
absolutely convergent
and the
limit
is the product of the two limits. Furthermore, the $n$-th summand of the exponential series of \mathl{x+y}{} equals
\mavergleichskettedisp
{\vergleichskette
{ \frac{(x+y)^n}{n!}
}
{ =} { \frac{1}{n!} \sum_{i = 0}^n \binom { n } { i } x^{i} y^{n-i}
}
{ =} { c_n
}
{ } {
}
{ } {}
}
{}{}{,}
so that both sides coincide.
\inputfaktbeweis
{Exponential series/Real/Elementary properties/Fact}
{Corollary}
{}
{
\faktsituation {}
\faktvoraussetzung {The
exponential function
\mathdisp {\R \longrightarrow \R
, x \longmapsto \exp x} { , }
}
\faktuebergang {fulfills the following properties.}
\faktfolgerung {\aufzaehlungsechs {
\mavergleichskette
{\vergleichskette
{ \exp 0
}
{ = }{ 1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
} {For every
\mavergleichskette
{\vergleichskette
{ x
}
{ \in }{\R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
we have
\mavergleichskette
{\vergleichskette
{ \exp { \left( -x \right) }
}
{ = }{ ( \exp x )^{-1}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
In particular
\mavergleichskette
{\vergleichskette
{ \exp x
}
{ \neq }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
} {For integers
\mavergleichskette
{\vergleichskette
{ n
}
{ \in }{ \Z
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the relation
\mavergleichskette
{\vergleichskette
{ \exp n
}
{ = }{ ( \exp 1)^n
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds.
} {For every $x$, we have
\mavergleichskette
{\vergleichskette
{ \exp x
}
{ \in }{ \R_+
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
} {For
\mavergleichskette
{\vergleichskette
{ x
}
{ > }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
we have
\mavergleichskette
{\vergleichskette
{ \exp x
}
{ > }{ 1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and for
\mavergleichskette
{\vergleichskette
{ x
}
{ < }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
we have
\mavergleichskette
{\vergleichskette
{ \exp x
}
{ < }{ 1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
} {The real exponential function is
strictly increasing.
}}
\faktzusatz {}
}
{
(1) follows directly from the definition.
(2) follows from
\mavergleichskettedisp
{\vergleichskette
{ \exp x \cdot \exp { \left( -x \right) }
}
{ =} { \exp { \left( x-x \right) }
}
{ =} { \exp 0
}
{ =} { 1
}
{ } {
}
}
{}{}{}
using
Theorem 12.8
.
(3) follows for
\mavergleichskette
{\vergleichskette
{ n
}
{ \in }{ \N
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
from
Theorem 12.8
by induction, and from that it follows with the help of (2) also for negative $n$.
(4). Nonnegativity follows from
\mavergleichskettedisp
{\vergleichskette
{ \exp x
}
{ =} { \exp { \left( { \frac{ x }{ 2 } } + { \frac{ x }{ 2 } } \right) }
}
{ =} { \exp { \frac{ x }{ 2 } } \cdot \exp { \frac{ x }{ 2 } }
}
{ =} { { \left(\exp { \frac{ x }{ 2 } }\right) }^2
}
{ \geq} { 0
}
}
{}{}{.}
(5). For real $x$ we have
\mavergleichskette
{\vergleichskette
{ \exp x \cdot \exp { \left( -x \right) }
}
{ = }{ 1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
so that because of (4), one factor must be $\geq 1$ and the other factor must be $\leq 1$. For
\mavergleichskette
{\vergleichskette
{ x
}
{ > }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
we have
\mavergleichskettedisp
{\vergleichskette
{ \exp x
}
{ =} { \sum_{n = 0}^\infty \frac{1}{n!} x^n
}
{ =} {1+x+ { \frac{ 1 }{ 2 } } x^2 + \ldots
}
{ >} {1
}
{ } {
}
}
{}{}{}
as only positive numbers are added.
(6). For real
\mavergleichskette
{\vergleichskette
{ y
}
{ > }{ x
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
we have
\mavergleichskette
{\vergleichskette
{ y-x
}
{ > }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and therefore, because of (5)
\mavergleichskette
{\vergleichskette
{ \exp { \left( y-x \right) }
}
{ > }{ 1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
hence
\mavergleichskettedisp
{\vergleichskette
{ \exp y
}
{ =} { \exp { \left( y-x + x \right) }
}
{ =} { \exp { \left( y-x \right) } \cdot \exp x
}
{ >} { \exp x
}
{ } {
}
}
{}{}{.}
With the help of the exponential series, we also define \stichwort {Euler's number} {.}
\inputdefinition
{ }
{
The real number
\mavergleichskettedisp
{\vergleichskette
{e
}
{ \defeq} { \sum_{k = 0}^\infty { \frac{ 1 }{ k! } }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
}
So we have
\mavergleichskette
{\vergleichskette
{ e
}
{ = }{ \exp 1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Its numerical value is
\mavergleichskettedisp
{\vergleichskette
{e
}
{ =} { 1+1+ { \frac{ 1 }{ 2 } } + { \frac{ 1 }{ 6 } } + { \frac{ 1 }{ 24 } } + \ldots
}
{ \cong} { 2,71 ...
}
{ } {
}
{ } {
}
}
{}{}{.}
\inputremark {}
{
For Euler's number there is also the description
\mavergleichskettedisp
{\vergleichskette
{e
}
{ =} { \lim_{n \rightarrow \infty} { \left( 1+ { \frac{ 1 }{ n } } \right) }^n
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
so that $e$ can also be introduced as the limit of this sequence. However, the convergence in the exponential series is much faster.
}
We will write also \mathl{e^x}{} instead of \mathl{\exp x}{.} This is, for
\mavergleichskette
{\vergleichskette
{x
}
{ \in }{ \Z
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
compatible with the usual meaning of powers in the sense of the fourth lecture due to
Corollary 12.9
.
The compatibility with arbitrary roots
\zusatzklammer {if the exponents are rational} {} {}
follows from
Remark 12.17
and
Exercise 12.17
.
\inputfaktbeweis
{Real exponential function/Continuity and image/Fact}
{Theorem}
{}
{
\faktsituation {The
real exponential function
\mathdisp {\R \longrightarrow \R
, x \longmapsto \exp x} { , }
}
\faktfolgerung {is
continuous
and defines a bijection between $\R$ and $\R_+$.}
\faktzusatz {}
}
{
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 $\R_+$, 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 $\R_+$. Injectivity follows from Corollary 12.9 , in connection with Exercise 5.38 .
\zwischenueberschrift{Logarithms}
\inputdefinition
{ }
{
The \definitionswort {natural logarithm}{}
\mathdisp {\ln \colon \R_+ \longrightarrow \R
, x \longmapsto \ln x} { , }
is defined as the
inverse function
of the
}
\inputfaktbeweis
{Natural logarithm/Functional equation/Bijection/Continuity/Monotonicity/Fact}
{Theorem}
{}
{
\faktsituation {The
natural logarithm
\mathdisp {\ln \colon \R_+ \longrightarrow \R
, x \longmapsto \ln x} { , }
}
\faktfolgerung {is a
continuous
strictly increasing function, which defines a bijection between
\mathkor {} {\R_+} {and} {\R} {.}
Moreover, the functional equation
\mavergleichskettedisp
{\vergleichskette
{ \ln (x \cdot y)
}
{ =} { \ln x + \ln y
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
holds for all
\mavergleichskette
{\vergleichskette
{x,y
}
{ \in }{ \R_+
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}}
\faktzusatz {}
}
{
This follows from Theorem 12.12 , Theorem 11.7 , Corollary 12.9 and Corollary 12.9 .
\inputdefinition
{ }
{
For a positive real number
\mavergleichskette
{\vergleichskette
{ b
}
{ > }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the
\definitionswort {exponential function for the base}{}
$b$ is defined as
\mavergleichskettedisp
{\vergleichskette
{ b^x
}
{ \defeq} { \exp (x \ln b )
}
{ } {
}
{ } {
}
{ } {
}
}
}
\inputfaktbeweis
{Real exponential function/Base/Properties/Fact}
{Theorem}
{}
{
\faktsituation {Let $b$ denote a
positive
real number.}
\faktuebergang {Then the
exponential function
\mathdisp {f \colon \R \longrightarrow \R
, x \longmapsto b^x} { , }
fulfills the following properties.}
\faktfolgerung {\aufzaehlungacht {We have
\mavergleichskette
{\vergleichskette
{ b^{x+x'}
}
{ = }{ b^x \cdot b^{x'}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
for all
\mavergleichskette
{\vergleichskette
{x,x'
}
{ \in }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
} {We have
\mavergleichskette
{\vergleichskette
{ b^{-x}
}
{ = }{ { \frac{ 1 }{ b^x } }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
} {For
\mavergleichskette
{\vergleichskette
{b
}
{ > }{1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and
\mavergleichskette
{\vergleichskette
{x
}
{ > }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
we have
\mavergleichskette
{\vergleichskette
{b^x
}
{ > }{1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
} {For
\mavergleichskette
{\vergleichskette
{b
}
{ < }{1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and
\mavergleichskette
{\vergleichskette
{x
}
{ > }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
we have
\mavergleichskette
{\vergleichskette
{b^x
}
{ < }{1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
} {For
\mavergleichskette
{\vergleichskette
{b
}
{ > }{1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the function $f$ is
strictly increasing.
} {For
\mavergleichskette
{\vergleichskette
{b
}
{ < }{1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the function $f$ is
strictly decreasing.
} {We have
\mavergleichskette
{\vergleichskette
{ (b^{x})^{x'}
}
{ = }{ b^{ x \cdot x'}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
for all
\mavergleichskette
{\vergleichskette
{x,x'
}
{ \in }{\R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
} {For
\mavergleichskette
{\vergleichskette
{a
}
{ \in }{ \R_+
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
we have
\mavergleichskette
{\vergleichskette
{ (ab)^x
}
{ = }{ a^x \cdot b^x
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
}}
\faktzusatz {}
{See Exercise 12.9 . }
\inputremark {}
{
There is another way to introduce the exponential function \mathl{x \mapsto a^x}{} to base
\mavergleichskette
{\vergleichskette
{a
}
{ > }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
For a natural number
\mavergleichskette
{\vergleichskette
{ n
}
{ \neq }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
one takes the $n$th product of $a$ with itself as definition for $a^n$. For a negative integer $x$, one sets
\mavergleichskette
{\vergleichskette
{a^x
}
{ \defeq }{ (a^{-x} )^{-1}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
For a positive rational number
\mavergleichskette
{\vergleichskette
{x
}
{ = }{r/s
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
one sets
\mavergleichskettedisp
{\vergleichskette
{ a^x
}
{ \defeq} { \sqrt[s] { a^r }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
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 $x$, one takes a sequence $q_n$ of rational numbers converging to $x$, and defines
\mavergleichskettedisp
{\vergleichskette
{ a^x
}
{ \defeq} { \lim_{n \rightarrow \infty} a^{q_n}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
For this, one has to show that these limits exist and that they are independent of the chosen rational sequence. For the passage from $\Q$ to $\R$, the concept of
uniform continuity
is crucial.
}
\inputdefinition
{ }
{
For a positive real number
\mathbed {b>0} {}
{b \neq 1} {}
{} {} {} {,}
the
\definitionswort {logarithm to base}{}
$b$ of
\mavergleichskette
{\vergleichskette
{x
}
{ \in }{ \R_+
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
is defined by
\mavergleichskettedisp
{\vergleichskette
{ \log_{ b } x
}
{ \defeq} { { \frac{ \ln x }{ \ln b } }
}
{ } {
}
{ } {
}
{ } {
}
}
}
\inputfaktbeweis
{Logarithm/Base/Rules/Fact}
{Theorem}
{}
{
\faktsituation {The
logarithms to base
$b$}
\faktuebergang {fulfill the following rules.}
\faktfolgerung {\aufzaehlungvier {We have
\mathkor {} {\log_b(b^x) =x} {and} {b^{\log_b(y)} =y} {,}
this means that the logarithm to Base $b$ is the inverse function for the
exponential function to base
$b$.
} {We have
\mavergleichskette
{\vergleichskette
{ \log_{ b } (y \cdot z)
}
{ = }{ \log_{ b } y + \log_{ b } z
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
} {We have
\mavergleichskette
{\vergleichskette
{\log_{ b } y^u
}
{ = }{u \cdot \log_{ b } y
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
for
\mavergleichskette
{\vergleichskette
{u
}
{ \in }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
} {We have
\mavergleichskettedisp
{\vergleichskette
{ \log_{ a } y
}
{ =} { \log_{ a } { \left( b^{ \log_{ b } y } \right) }
}
{ =} { \log_{ b } y \cdot \log_{ a } b
}
{ } {
}
{ } {
}
}
{}{}{.}
}}
\faktzusatz {}
{See Exercise 12.25 . }