Universit
atKarlsruhe Wintersemester2003/04
Institut f
urTheoriederKondensiertenMaterie 17.10.03
Prof.Dr.Ralphv.Baltz, Dr.PhilipHowell http://www-tkm.physik.uni-karlsruhe.de/lehre/
Sprehstunde:Fr13:00{14:00Physikhohhaus10.14 howelltkm.physik.uni-karlsruhe.de
Musterlosung zu
UbungsblattNr. 0zur Theorie A
1 a) Furx!0:
oshx ' 1
2
1+x+ x
2
2
+
1+( x)+ ( x)
2
2
=1+ x
2
2
sinhx ' 1
2
1+x+ x
2
2
1+( x)+ ( x)
2
2
=x
tanhx = sinhx
oshx '
x
1+ x
2
2 '
x
1
=x
Furx!1:
IndiesemBereihgilte x
e x
)oshx' 1
2 e
x
;sinhx' 1
2 e
x
;tanhx'1.
Merke:dae x
>0giltoshx>
1
2 e
x
>sinhx)tanhx<1.
Furx! 1:
In diesem Bereih gilt e x
e x
) oshx ' 1
2 e
x
= 1
2 e
jxj
;sinhx ' 1
2 e
x
=
1
2 e
jxj
;tanhx' 1.
FurdieSkizzehilftes,dieParitatzuuntersuhen:
osh ( x)= 1
2 (e
x
+e x
)=oshx)geradeFunktion.
sinh( x)= 1
2 (e
x
e x
)= sinhx)ungeradeFunktion.
tanh ( x)= sinh( x)
osh( x)
= sinhx
oshx
= tanhx)ungeradeFunktion.
Auhosh (x=0)= 1
2 (e
0
+e 0
)=1;sinh(x=0)= 1
2 (e
0
e 0
)=0.
tanh x
−1 1
sinh x cosh x
1
b) Wirbenutzen d
dx e
ax
=ae ax
.
d
dx
(oshx)= d
d x 1
2 (e
x
+e x
)= 1
2 (e
x
+( e x
))=sinhx
d
(sinhx)= d 1
(e x
e x
)= 1
(e x
( e x
))=oshx
osh 2
x+sinh 2
x= 1
4 (e
x
+e x
) 2
+(e x
e x
) 2
= 1
4
(e 2x
+2+e 2x
)+(e 2x
2+e 2x
)
= 1
4 (2e
2x
+2e 2x
)=osh2x
2sinhxoshx=2 1
2 (e
x
e x
) 1
2 (e
x
+e x
)= 1
2 e
2x
e 2x
=sinh2x
osh 2
x sinh 2
x= 1
4
(e x
+e x
) 2
(e x
e x
) 2
= 1
4
(e 2x
+2+e 2x
) (e 2x
2+e 2x
)
= 1
4 (4)=1
2 a) MitHilfederKettenregelbzw.derQuotientenregel:
d
d x e
sinx
=osxe sinx
d
dr e
r
1+ r 2
= (1+ r
2
)e r
e r
2 r
(1+ r 2
) 2
= e
r
(1+ r 2
2r)
(1+ r 2
) 2
b) Seix=1+ u 2
) d x
d u
=2 u)du=dx 1
2 u
unddaher
Z
du u
1+ u 2
= Z
dx 1
2 u u
1+ u 2
= Z
dx 1
2 x
= 1
2
lnx+= 1
2
ln(1+ u 2
)+
wobeidieIntegrationskonstanteist.
MitdenFormelnaus1(b),
osh2=osh 2
+sinh 2
=(1+sinh 2
)+sinh 2
=1+2sinh 2
)sinh 2
= 1
2
(osh2 1)
) Z
d sinh 2
= 1
2 Z
d(osh2 1)= 1
2
1
2
sinh2
+= 1
4 sinh2
2 +
