Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 3/en/latex
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\zwischenueberschrift{Warm-up-exercises}
\inputexercise
{}
{
Show that the binomial coefficients satisfy the following recursive relation
\mavergleichskettedisp
{\vergleichskette
{ \binom { n+1 } { k}
}
{ =} { \binom { n } { k} + \binom { n } { k-1}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
}
{} {}
\inputexercise
{}
{
Show that the binomial coefficients are natural numbers.
}
{} {}
\inputexercise
{}
{
Prove the formula
\mathdisp {2^n = \sum_{k=0}^n \binom { n } { k}} { . }
}
{} {}
\inputexercise
{}
{
Prove by induction for $n \geq 10$ the inequality
\mathdisp {3^n \geq n^4} { . }
}
{} {}
In the following computing tasks regarding complex numbers, the result must always be in the form $a+bi$, with real numbers $a,b$, and these should be as simple as possible.
\inputexercise
{}
{
Calculate the following expressions in the \definitionsverweis {complex numbers}{}{.} \aufzaehlungsechs{$(5+4i)(3-2i)$. }{$(2+3i)(2-4i) +3(1-i)$. }{$(2i+3)^2$. }{$i^{1011}$. }{$(-2+5i)^{-1}$. }{$\frac{4-3i}{2+i}$. }
}
{} {}
\inputexercise
{}
{
Show that the \definitionsverweis {complex numbers}{}{} constitute a \definitionsverweis {field}{}{.}
}
{} {}
\inputexercise
{}
{
Prove the following statements concerning the
\definitionsverweis {real}{}{} and
\definitionsverweis {imaginary}{}{}
parts of a
\definitionsverweis {complex number}{}{.}
\aufzaehlungfuenf{$z=
\operatorname{Re} \, { \left( z \right) } + \operatorname{Im} \, { \left( z \right) } { \mathrm i}$.
}{$\operatorname{Re} \, { \left( z+w \right) } =
\operatorname{Re} \, { \left( z \right) } +
\operatorname{Re} \, { \left( w \right) }$.
}{$\operatorname{Im} \, { \left( z+w \right) } = \operatorname{Im} \, { \left( z \right) } + \operatorname{Im} \, { \left( w \right) }$.
}{For $r \in \R$ we have
\mathdisp {\operatorname{Re} \, { \left( rz \right) } =r
\operatorname{Re} \, { \left( z \right) } \text{ and } \operatorname{Im} \, { \left( rz \right) } =r \operatorname{Im} \, { \left( z \right) }} { . }
}{ $z =
\operatorname{Re} \, { \left( z \right) }$ if and only if $z \in \R$, and this is exactly the case when $\operatorname{Im} \, { \left( z \right) }=0$.
}
}
{} {}
\inputexercise
{}
{
Prove the following calculating rules for the
\definitionsverweis {complex numbers}{}{.}
\aufzaehlungfuenf{$\betrag { z }= \sqrt{ z \ \overline{ z } }$.
}{$\operatorname{Re} \, { \left( z \right) } = \frac{z+ \overline{ z } }{2}$.
}{$\operatorname{Im} \, { \left( z \right) } = \frac{z - \overline{ z } }{2 { \mathrm i} }$.
}{$\overline{ z }=
\operatorname{Re} \, { \left( z \right) } - { \mathrm i} \operatorname{Im} \, { \left( z \right) }$.
}{For $z \neq 0$ we have
\mathl{z^{-1}= \frac{ \overline{ z } }{ \betrag { z }^2 }}{.}
}
}
{} {}
\inputexercise
{}
{
Prove the following properties of the \definitionsverweis {absolute value}{}{} of a \definitionsverweis {complex number}{}{.} \aufzaehlungsechs{For a real number $z$ its real absolute value and its complex absolute value coincide. }{We have $\betrag { z }=0$ if and only if $z=0$. }{$\betrag { z } = \betrag { \overline{ z } }$. }{$\betrag { zw } = \betrag { z } \betrag { w }$. }{$\betrag { \operatorname{Re} \, { \left( z \right) } }, \betrag { \operatorname{Im} \, { \left( z \right) } } \leq \betrag { z }$. }{For $z \neq 0$ we have $\betrag { 1/z } = 1/ \betrag { z }$. }
}
{} {}
\inputexercise
{}
{
Check the formula we gave in Example 3.15 for the \definitionsverweis {square root}{}{} of a \definitionsverweis {complex number}{}{} $z=a+bi$, in the case $b <0$.
}
{} {}
\inputexercise
{}
{
Determine the two complex solutions of the equation
\mathdisp {z^2+5iz-3=0} { . }
}
{} {}
\zwischenueberschrift{Hand-in-exercises}
\inputexercise
{3}
{
Prove the following formula
\mathdisp {n 2^{n-1} = \sum_{k=0}^n k \binom { n } { k}} { . }
}
{} {}
\inputexercise
{3}
{
Calculate the
\definitionsverweis {complex numbers}{}{}
\mathdisp {(1+i)^n} { }
for $n=1,2,3,4,5$.
}
{} {}
\inputexercise
{3}
{
Prove the following properties of the
\definitionsverweis {complex conjugation}{}{.}
\aufzaehlungsechs{$\overline{ z+w }= \overline{ z } + \overline{ w }$.
}{$\overline{ -z } = - \overline{ z }$.
}{$\overline{ z \cdot w }= \overline{ z } \cdot \overline{ w }$.
}{For $z \neq 0$ we have $\overline{ 1/z } =1/\overline{ z }$.
}{$\overline{ \overline{ z } } =z$.
}{$\overline{ z } =z$ if and only if
\mathl{z \in \R}{}.
}
}
{} {}
\inputexercise
{2}
{
Let $a,b,c \in {\mathbb C}$ with $a \neq 0$. Show that the equation
\mathdisp {az^2+bz+c=0} { }
has at least one complex solution $z$.
}
{} {}
\inputexercise
{3}
{
Calculate the square root, the fourth root and the eighth root of $i$.
}
{} {}
\inputexercise
{4}
{
Find the three complex numbers $z$ such that
\mathdisp {z^3=1} { . }
}
{} {}
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