Wie man sieht, liegen die verwendeten Punkte auf der Linie von Sin[t/2]. Der Fehler (z.B. grosser Imaginäranteil stammt vermutlich davon, dass so nur wenige Koeffizienten berechnet werden können.)

Lösungen

Remove@"Global`*"D

1

Wir verwenden hier die Skalierung nach der Periode 2 Pi. Das vereinfacht die Rechung etwas.

n=10; w = 2 Pi/n;

{x[0],y[0]}={0 w,0};

{x[1],y[1]}={1 w,0.309};

{x[2],y[2]}={2 w,0.588};

{x[3],y[3]}={3 w,0.809};

{x[4],y[4]}={4 w,0.951};

{x[5],y[5]}={5 w,0.99};

{x[6],y[6]}={6 w,0.951};

{x[7],y[7]}={7 w,0.809};

{x[8],y[8]}={8 w,0.588};

{x[9],y[9]}={9 w,0.3};

p[k_]:= {x[k],y[k]};

Table[p[k],{k,0,n-1}]

980, 0<,9€€€€p

5, 0.309=,9€€€€€€€€€2p

5 , 0.588=,9€€€€€€€€€3p

5 , 0.809=,9€€€€€€€€€4p

5 , 0.951=, 8p, 0.99<,96p

€€€€€€€€€

5 , 0.951=,97p

€€€€€€€€€

5 , 0.809=,98p

€€€€€€€€€

5 , 0.588=,99p

€€€€€€€€€

5 , 0.3==

epi=Prepend[Map[Point,Table[p[k],{k,0,n-1}]],PointSize[0.03]]

9PointSize@0.03D, Point@80, 0<D, PointA9p

€€€€5, 0.309=E, PointA92p

€€€€€€€€€

5 , 0.588=E, PointA9€€€€€€€€€3p

5 , 0.809=E, PointA9€€€€€€€€€4p

5 , 0.951=E, Point@8p, 0.99<D, PointA9€€€€€€€€€6p

5 , 0.951=E, PointA9€€€€€€€€€7p

5 , 0.809=E, PointA9€€€€€€€€€8p

5 , 0.588=E, PointA9€€€€€€€€€9p

5 , 0.3=E=

r = E^(-I 2 Pi/n);

c[s_]:= 1/n Sum[y[k] r^(s k),{k,0,n-1}];

Table[c[s],{s,0,10}]//N

80.6295,-0.217264-0.000529007ä,-0.0494452-0.000855951ä,

-0.0232856-0.000855951ä,-0.0178048-0.000529007ä,

-0.0139,-0.0178048+0.000529007ä,-0.0232856+0.000855951ä,

-0.0494452+0.000855951ä,-0.217264+0.000529007ä, 0.6295<

?ExpToTrig

ExpToTrig@exprD converts exponentials in expr to trigonometric functions.Mehr…

fS[t_]:=Sum[c[k] E^(I k t),{k,0,n-1}];

fS[t]

0.6295-H0.217264+0.000529007äLã^ät-H0.0494452+0.000855951äLã^2ät- H0.0232856+0.000855951äLã^3ät-H0.0178048+0.000529007äLã^4ät-

0.0139ã^5ät-H0.0178048-0.000529007äLã^6ät-H0.0232856-0.000855951äLã^7ät- H0.0494452-0.000855951äLã^8ät-H0.217264-0.000529007äLã^9ät

fS[t]//ExpToTrig

0.6295-H0.217264+0.000529007äLCos@tD-H0.0494452+0.000855951äLCos@2 tD- H0.0232856+0.000855951äLCos@3 tD-H0.0178048+0.000529007äLCos@4 tD- 0.0139 Cos@5 tD-H0.0178048-0.000529007äLCos@6 tD-

H0.0232856-0.000855951äLCos@7 tD-H0.0494452-0.000855951äLCos@8 tD- H0.217264-0.000529007äLCos@9 tD+H0.000529007-0.217264äLSin@tD+ H0.000855951-0.0494452äLSin@2 tD+H0.000855951-0.0232856äLSin@3 tD+ H0.000529007-0.0178048äLSin@4 tD-0.0139äSin@5 tD-

H0.000529007+0.0178048äLSin@6 tD-H0.000855951+0.0232856äLSin@7 tD- H0.000855951+0.0494452äLSin@8 tD-H0.000529007+0.217264äLSin@9 tD Plot[Re[fS[t]],{t,0,2Pi}];

1 2 3 4 5 6

0.2 0.4 0.6 0.8 1

Plot[Im[fS[t]],{t,0,2Pi}];

1 2 3 4 5 6

-0.4 -0.2 0.2 0.4

Plot[{Re[fS[t]],Sin[t/2]},{t,0,2Pi},Epilog->epi];

1 2 3 4 5 6

0.2 0.4 0.6 0.8 1

Wie man sieht, liegen die verwendeten Punkte auf der Linie von Sin[t/2]. Der Fehler (z.B. grosser Imaginäranteil stammt vermutlich davon, dass so nur wenige Koeffizienten berechnet werden können.)

2 (Das Problem der Periodengrenzen)

Remove@"Global`*"D

Ÿ f H t L = ã

+ sin H t L

, T = 2 p, t

= €€€€€€

, k = 0, 1, ..., 15

Ÿ Definition der Funktion und Berechnung der Punkte

f@t_D:=E ^ Cos@tD+Sin@tD^ 2;

n=16; w=2 Pi•n;

8x@k_D, y@k_D<=8k 2 Pi•16, f@k 2 Pi•nD<; p@k_D:=8x@kD, y@kD<;

Table@p@kD ••N,8k, 0, n-1<D ••TableForm

0. 2.71828

0.392699 2.66549 0.785398 2.52811 1.1781 2.31977

1.5708 2.

1.9635 1.53558 2.35619 0.993069 2.74889 0.543423 3.14159 0.367879 3.53429 0.543423 3.92699 0.993069 4.31969 1.53558 4.71239 2.

5.10509 2.31977 5.49779 2.52811 5.89049 2.66549

Ÿ Plot der Funktion und der Punkte

epi=Prepend[Map[Point,Table[p[k],{k,0,n-1}]],PointSize[0.03]];

Show[Graphics[epi]];

Plot@f@tD,8t, 0, 2 Pi<, Epilog®epiD;

1 2 3 4 5 6

0.5 1 1.5 2 2.5

Ÿ Berechnung der Koeffizienten mittels der DFT

r = E^(-I 2 Pi/n);

c[s_]:= 1/n Sum[y[k] r^(s k),{k,0,n-1}];

Table[c[s],{s,0,n-1}]//N

81.76607, 0.565159-5.55112´10^-17ä,-0.114252+5.55112´10^-17ä,

0.0221684+2.77556´10^-17ä, 0.00273712+0.ä, 0.000271463+2.77556´10^-17ä, 0.0000224889+6.93889´10^-17ä, 1.60474´10^-6-4.16334´10^-17ä,

1.99212´10^-7, 1.60474´10^-6+4.16334´10^-17ä, 0.0000224889-6.93889´10^-17ä, 0.000271463-2.77556´10^-17ä, 0.00273712+0.ä, 0.0221684-2.77556´10^-17ä,

-0.114252-5.55112´10^-17ä, 0.565159+5.55112´10^-17ä<

Table@c@sD,8s,-n+1, n-1<D ••N••Chop

80.565159,-0.114252, 0.0221684, 0.00273712, 0.000271463, 0.0000224889, 1.60474´10^-6, 1.99212´10^-7, 1.60474´10^-6, 0.0000224889, 0.000271463, 0.00273712, 0.0221684,

-0.114252, 0.565159, 1.76607, 0.565159,-0.114252, 0.0221684, 0.00273712,

0.000271463, 0.0000224889, 1.60474´10^-6, 1.99212´10^-7, 1.60474´10^-6, 0.0000224889, 0.000271463, 0.00273712, 0.0221684,-0.114252, 0.565159<

Ÿ Berechnung der trigonometrischen Reihe durch die gegebenen Punkte (ohne Vereinfachung, aus Zeitgründen)

fS[t_]:=Sum[c[k] E^(I k t),{k,0,n-1}];

fS[t]

€€€€€€€1 16 ã^8äti

kjjj6+ 1

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8E²-SinA€€€€€€€€€3p 8 E²- SinA€€€€€€€€€5p

8 E²-SinA€€€€€€€€€7p

8 E²-SinA€€€€€€€€€9p

8 E²-SinA€€€€€€€€€€€11p

8 E²-SinA€€€€€€€€€€€13p

8 E²-SinA€€€€€€€€€€€15p 8 E²y

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8E²+SinA€€€€€€€€€3p

8 E²+SinA€€€€€€€€€5p 8 E²+ SinA€€€€€€€€€7p

8 E²+SinA€€€€€€€€€9p

8 E²+SinA€€€€€€€€€€€11p

8 E²+SinA€€€€€€€€€€€13p

8 E²+SinA€€€€€€€€€€€15p 8 E²y

{zzz+

€€€€€€€1

16 ã^12äti kjjj2+ 1

€€€€ã + ã -2ã^{-€€€€€€€€€•!!!!!12} -2ã^{€€€€€€€€€•!!!!!12} + äJã^{Cos@€€€€p8D}+SinA€€€€p

8E²N- äi

kjjjã^{CosA€€€€€€€3p8 E}+SinA€€€€€€€€€3p 8 E²y

{zzz+

äi

kjjã^{CosA€€€€€€€5p8 E}+SinA5p

€€€€€€€€€

8 E²y {zz- äi

kjjã^{CosA€€€€€€€7p8 E}+SinA7p

€€€€€€€€€

8 E²y {zz+ äi

kjjjã^{CosA€€€€€€€98pE}+SinA9p

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8 E²y {zzz-

äi

kjjjã^{CosA€€€€€€€€€118pE}+SinA€€€€€€€€€€€11p 8 E²y

{zzz+ äi

kjjjã^{CosA€€€€€€€€€13p8 E}+SinA€€€€€€€€€€€13p 8 E²y

{zzz- äi

kjjjã^{CosA€€€€€€€€€158pE}+SinA€€€€€€€€€€€15p 8 E²y

{zzzy{zzz+

€€€€€€€1 16 ã^4äti

kjjj2+ 1

€€€€ã + ã -2ã^{-€€€€€€€€€•!!!!!12} -2ã^{€€€€€€€€€•!!!!!12} - äJã^{Cos@€€€€p8D}+SinA€€€€p

8E²N+ äi

kjjjã^{CosA€€€€€€€38pE}+SinA€€€€€€€€€3p 8 E²y

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äi

kjjã^{CosA€€€€€€€5p8 E}+SinA€€€€€€€€€5p 8 E²y

{zz+ äi

kjjã^{CosA€€€€€€€7p8 E}+SinA€€€€€€€€€7p 8 E²y

{zz- äi

kjjjã^{CosA€€€€€€€98pE}+SinA€€€€€€€€€9p 8 E²y

{zzz+

äi

kjjjã^{CosA€€€€€€€€€118pE}+SinA€€€€€€€€€€€11p 8 E²y

{zzz- äi

kjjjã^{CosA€€€€€€€€€13p8 E}+SinA€€€€€€€€€€€13p 8 E²y

{zzz+ äi

kjjjã^{CosA€€€€€€€€€158pE}+SinA€€€€€€€€€€€15p 8 E²y

{zzzy {zzz+

€€€€€€€1

16 ã^15äti kjjj-1

€€€€ã + ã + ã^{-€€€€€€€€€€3ä p4} J1

€€€€2 + ã^{-€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€€€€3ä p4} J1

€€€€2 + ã^{-€€€€€€€€€•!!!!!12} N+ ã^{-€€€€€€€ä p4} J1

€€€€2 + ã^{€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€ä p4} J1

€€€€2 + ã^{€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€ä p8} Jã^{Cos@€€€€p8D}+SinA€€€€p

8E²N+ ã^{€€€€€€€€€€3ä p8} i

kjjjã^{CosA€€€€€€€3p8 E}+SinA€€€€€€€€€3p 8 E²y

{zzz+ ã^{€€€€€€€€€€5ä p8} i

kjjã^{CosA€€€€€€€5p8 E}+SinA€€€€€€€€€5p 8 E²y

{zz+ ã^{€€€€€€€€€€7ä p8} i

kjjã^{CosA€€€€€€€7p8 E}+SinA€€€€€€€€€7p 8 E²y

{zz+ ã^{-€€€€€€€€€€7ä p8} i

kjjjã^{CosA€€€€€€€98pE}+SinA€€€€€€€€€9p 8 E²y

{zzz+ ã^{-€€€€€€€€€€5ä p8} i

kjjjã^{CosA€€€€€€€€€118pE}+SinA€€€€€€€€€€€11p 8 E²y

{zzz+ ã^{-€€€€€€€€€€3ä p8} i

kjjjã^{CosA€€€€€€€€€138pE}+SinA€€€€€€€€€€€13p 8 E²y

{zzz+ ã^{-€€€€€€€ä p8} i

kjjjã^{CosA€€€€€€€€€158pE}+SinA€€€€€€€€€€€15p 8 E²y

{zzzy {zzz+

€€€€€€€1 16 ã^äti

kjjj-1

€€€€ã + ã + ã^{-€€€€€€€€€€3ä p4} J1

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€€€€2 + ã^{-€€€€€€€€€•!!!!!12} N+ ã^{-€€€€€€€ä p4} J1

€€€€2 + ã^{€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€ä p4} J1

€€€€2 + ã^{€€€€€€€€€•!!!!!12} N+ ã^{-€€€€€€€ä p8} Jã^{Cos@€€€€p8D}+SinA€€€€p

8E²N+ ã^{-€€€€€€€€€€3ä p8} i

kjjjã^{CosA€€€€€€€38pE}+SinA€€€€€€€€€3p 8 E²y

{zzz+ ã^{-€€€€€€€€€€5ä p8} i

kjjã^{CosA€€€€€€€58pE}+SinA€€€€€€€€€5p 8 E²y

{zz+ ã^{-€€€€€€€€€€7ä p8} i

kjjã^{CosA€€€€€€€78pE}+SinA€€€€€€€€€7p 8 E²y

{zz+ ã^{€€€€€€€€€€7ä p8} i

kjjjã^{CosA€€€€€€€9p8 E}+SinA€€€€€€€€€9p 8 E²y

{zzz+ ã^{€€€€€€€€€€5ä p8} i

kjjjã^{CosA€€€€€€€€€118pE}+SinA€€€€€€€€€€€11p 8 E²y

{zzz+ ã^{€€€€€€€€€€3ä p8} i

kjjjã^{CosA€€€€€€€€€138pE}+SinA€€€€€€€€€€€13p 8 E²y

{zzz+ ã^{€€€€€€€ä p8} i

kjjjã^{CosA€€€€€€€€€158pE}+SinA€€€€€€€€€€€15p 8 E²y

{zzzy{zzz+

€€€€€€€1

16 ã^14äti kjjj-4+ 1

€€€€ã + ã + ã^{€€€€€€€ä p4} Jã^{Cos@€€€€p8D}+SinA€€€€p

8E²N+ ã^{€€€€€€€€€€3ä p4} i

kjjjã^{CosA€€€€€€€3p8 E}+SinA€€€€€€€€€3p 8 E²y

{zzz+

ã^{-€€€€€€€€€€3ä p4} i

kjjã^{CosA€€€€€€€58pE}+SinA€€€€€€€€€5p 8 E²y

{zz+ ã^{-€€€€€€€ä p4} i

kjjã^{CosA€€€€€€€7p8 E}+SinA€€€€€€€€€7p 8 E²y

{zz+ ã^{€€€€€€€ä p4} i

kjjjã^{CosA€€€€€€€98pE}+SinA€€€€€€€€€9p 8 E²y

{zzz+ ã^{€€€€€€€€€€3ä p4} i

kjjjã^{CosA€€€€€€€€€118pE}+SinA€€€€€€€€€€€11p 8 E²y

{zzz+ ã^{-€€€€€€€€€€3ä p4} i

kjjjã^{CosA€€€€€€€€€138pE}+SinA€€€€€€€€€€€13p 8 E²y

{zzz+ ã^{-€€€€€€€ä p4} i

kjjjã^{CosA€€€€€€€€€158pE}+SinA€€€€€€€€€€€15p 8 E²y

{zzzy {zzz+

€€€€€€€1 16 ã^2äti

kjjj-4+ 1

€€€€ã + ã + ã^{-€€€€€€€ä p4} Jã^{Cos@€€€€p8D}+SinAp

€€€€8E²N+ ã^{-€€€€€€€€€€3ä p4} i

kjjjã^{CosA€€€€€€€3p8 E}+SinA3p

€€€€€€€€€

8 E²y {zzz+ ã^{€€€€€€€€€€3ä p4} i

kjjã^{CosA€€€€€€€5p8 E}+SinA€€€€€€€€€5p 8 E²y

{zz+ ã^{€€€€€€€ä p4} i

kjjã^{CosA€€€€€€€78pE}+SinA€€€€€€€€€7p 8 E²y

{zz+ ã^{-€€€€€€€ä p4} i

kjjjã^{CosA€€€€€€€9p8 E}+SinA€€€€€€€€€9p 8 E²y

{zzz+ ã^{-€€€€€€€€€€3ä p4} i

kjjjã^{CosA€€€€€€€€€118pE}+SinA€€€€€€€€€€€11p 8 E²y

{zzz+ ã^{€€€€€€€€€€3ä p4} i

kjjjã^{CosA€€€€€€€€€138pE}+SinA€€€€€€€€€€€13p 8 E²y

{zzz+ ã^{€€€€€€€ä p4} i

kjjjã^{CosA€€€€€€€€€158pE}+SinA€€€€€€€€€€€15p 8 E²y

{zzzy {zzz+

€€€€€€€1

16 ã^13äti kjjj-1

€€€€ã + ã + ã^{-€€€€€€€ä p4} J1

€€€€2 + ã^{-€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€ä p4} J1

€€€€2 + ã^{-€€€€€€€€€•!!!!!12} N+ ã^{-€€€€€€€€€€3ä p4} J1

€€€€2 + ã^{€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€€€€3ä p4} J1

€€€€2 + ã^{€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€€€€3ä p8} Jã^{Cos@€€€€p8D}+SinA€€€€p

8E²N+ ã^{-€€€€€€€€€€7ä p8} i

kjjjã^{CosA€€€€€€€3p8 E}+SinA€€€€€€€€€3p 8 E²y

{zzz+ ã^{-€€€€€€€ä p8} i

kjjã^{CosA€€€€€€€58pE}+SinA€€€€€€€€€5p 8 E²y

{zz+ ã^{€€€€€€€€€€5ä p8} i

kjjã^{CosA€€€€€€€7p8 E}+SinA€€€€€€€€€7p 8 E²y

{zz+ ã^{-€€€€€€€€€€5ä p8} i

kjjjã^{CosA€€€€€€€9p8 E}+SinA€€€€€€€€€9p 8 E²y

{zzz+ ã^{€€€€€€€ä p8} i

kjjjã^{CosA€€€€€€€€€118pE}+SinA€€€€€€€€€€€11p 8 E²y

{zzz+ ã^{€€€€€€€€€€7ä p8} i

kjjjã^{CosA€€€€€€€€€138pE}+SinA€€€€€€€€€€€13p 8 E²y

{zzz+ ã^{-€€€€€€€€€€3ä p8} i

kjjjã^{CosA€€€€€€€€€158pE}+SinA€€€€€€€€€€€15p 8 E²y

{zzzy {zzz+

€€€€€€€1 16 ã^3äti

kjjj-1

€€€€ã + ã + ã^{-€€€€€€€ä p4} J1

€€€€2 + ã^{-€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€ä p4} J1

€€€€2 + ã^{-€€€€€€€€€•!!!!!12} N+ ã^{-€€€€€€€€€€3ä p4} J1

€€€€2 + ã^{€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€€€€3ä p4} J1

€€€€2 + ã^{€€€€€€€€€•!!!!!12} N+ ã^{-€€€€€€€€€€3ä p8} Jã^{Cos@€€€€p8D}+SinA€€€€p

8E²N+ ã^{€€€€€€€€€€7ä p8} i

kjjjã^{CosA€€€€€€€38pE}+SinA€€€€€€€€€3p 8 E²y

{zzz+ ã^{€€€€€€€ä p8} i

kjjã^{CosA€€€€€€€58pE}+SinA€€€€€€€€€5p 8 E²y

{zz+ ã^{-€€€€€€€€€€5ä p8} i

kjjã^{CosA€€€€€€€78pE}+SinA€€€€€€€€€7p 8 E²y

{zz+ ã^{€€€€€€€€€€5ä p8} i

kjjjã^{CosA€€€€€€€9p8 E}+SinA€€€€€€€€€9p 8 E²y

{zzz+ ã^{-€€€€€€€ä p8} i

kjjjã^{CosA€€€€€€€€€118pE}+SinA€€€€€€€€€€€11p 8 E²y

{zzz+ ã^{-€€€€€€€€€€7ä p8} i

kjjjã^{CosA€€€€€€€€€138pE}+SinA13p

€€€€€€€€€€€

8 E²y

{zzz+ ã^{€€€€€€€€€€3ä p8} i

kjjjã^{CosA€€€€€€€€€158pE}+SinA15p

€€€€€€€€€€€

8 E²y {zzzy

{zzz+

€€€€€€€1

16 ã^11äti kjjj-1

€€€€ã + ã + ã^{-€€€€€€€ä p4} J1

€€€€2 + ã^{-€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€ä p4} J1

€€€€2 + ã^{-€€€€€€€€€•!!!!!12} N+ ã^{-€€€€€€€€€€3ä p4} J1

€€€€2 + ã^{€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€€€€3ä p4} J1

€€€€2 + ã^{€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€€€€5ä p8} Jã^{Cos@€€€€p8D}+SinAp

€€€€8E²N+ ã^{-€€€€€€€ä p8} i

kjjjã^{CosA€€€€€€€3p8 E}+SinA3p

€€€€€€€€€

8 E²y {zzz+ ã^{-€€€€€€€€€€7ä p8} i

kjjã^{CosA€€€€€€€58pE}+SinA€€€€€€€€€5p 8 E²y

{zz+ ã^{€€€€€€€€€€3ä p8} i

kjjã^{CosA€€€€€€€7p8 E}+SinA€€€€€€€€€7p 8 E²y

{zz+ ã^{-€€€€€€€€€€3ä p8} i

kjjjã^{CosA€€€€€€€98pE}+SinA€€€€€€€€€9p 8 E²y

{zzz+ ã^{€€€€€€€€€€7ä p8} i

kjjjã^{CosA€€€€€€€€€118pE}+SinA€€€€€€€€€€€11p 8 E²y

{zzz+ ã^{€€€€€€€ä p8} i

kjjjã^{CosA€€€€€€€€€138pE}+SinA€€€€€€€€€€€13p 8 E²y

{zzz+ ã^{-€€€€€€€€€€5ä p8} i

kjjjã^{CosA€€€€€€€€€158pE}+SinA€€€€€€€€€€€15p 8 E²y

{zzzy {zzz+

€€€€€€€1 16 ã^5äti

kjjj-1

€€€€ã + ã + ã^{-€€€€€€€ä p4} J1

€€€€2 + ã^{-€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€ä p4} J1

€€€€2 + ã^{-€€€€€€€€€•!!!!!12} N+ ã^{-€€€€€€€€€€3ä p4} J1

€€€€2 + ã^{€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€€€€3ä p4} J1

€€€€2 + ã^{€€€€€€€€€•!!!!!12} N+ ã^{-€€€€€€€€€€5ä p8} Jã^{Cos@€€€€p8D}+SinA€€€€p

8E²N+ ã^{€€€€€€€ä p8} i

kjjjã^{CosA€€€€€€€3p8 E}+SinA€€€€€€€€€3p 8 E²y

{zzz+ ã^{€€€€€€€€€€7ä p8} i

kjjã^{CosA€€€€€€€5p8 E}+SinA€€€€€€€€€5p 8 E²y

{zz+ ã^{-€€€€€€€€€€3ä p8} i

kjjã^{CosA€€€€€€€7p8 E}+SinA€€€€€€€€€7p 8 E²y

{zz+ ã^{€€€€€€€€€€3ä p8} i

kjjjã^{CosA€€€€€€€9p8 E}+SinA€€€€€€€€€9p 8 E²y

{zzz+ ã^{-€€€€€€€€€€7ä p8} i

kjjjã^{CosA€€€€€€€€€118pE}+SinA€€€€€€€€€€€11p 8 E²y

{zzz+ ã^{-€€€€€€€ä p8} i

kjjjã^{CosA€€€€€€€€€138pE}+SinA€€€€€€€€€€€13p 8 E²y

{zzz+ ã^{€€€€€€€€€€5ä p8} i

kjjjã^{CosA€€€€€€€€€158pE}+SinA€€€€€€€€€€€15p 8 E²y

{zzzy {zzz+

€€€€€€€1

16 ã^10äti kjjj-4+ €€€€1

ã + ã + ã^{€€€€€€€€€€3ä p4} Jã^{Cos@€€€€p8D}+SinA€€€€p

8E²N+ ã^{€€€€€€€ä p4} i

kjjjã^{CosA€€€€€€€3p8 E}+SinA€€€€€€€€€3p 8 E²y

{zzz+

ã^{-€€€€€€€ä p4} i

kjjã^{CosA€€€€€€€5p8 E}+SinA€€€€€€€€€5p 8 E²y

{zz+ ã^{-€€€€€€€€€€3ä p4} i

kjjã^{CosA€€€€€€€7p8 E}+SinA€€€€€€€€€7p 8 E²y

{zz+ ã^{€€€€€€€€€€3ä p4} i

kjjjã^{CosA€€€€€€€9p8 E}+SinA€€€€€€€€€9p 8 E²y

{zzz+ ã^{€€€€€€€ä p4} i

kjjjã^{CosA€€€€€€€€€118pE}+SinA€€€€€€€€€€€11p 8 E²y

{zzz+ ã^{-€€€€€€€ä p4} i

kjjjã^{CosA€€€€€€€€€138pE}+SinA€€€€€€€€€€€13p 8 E²y

{zzz+ ã^{-€€€€€€€€€€3ä p4} i

kjjjã^{CosA€€€€€€€€€158pE}+SinA€€€€€€€€€€€15p 8 E²y

{zzzy {zzz+

€€€€€€€1 16 ã^6äti

kjjj-4+ 1

€€€€ã + ã + ã^{-€€€€€€€€€€3ä p4} Jã^{Cos@€€€€p8D}+SinAp

€€€€8E²N+ ã^{-€€€€€€€ä p4} i

kjjjã^{CosA€€€€€€€3p8 E}+SinA3p

€€€€€€€€€

8 E²y {zzz+ ã^{€€€€€€€ä p4} i

kjjã^{CosA€€€€€€€58pE}+SinA€€€€€€€€€5p 8 E²y

{zz+ ã^{€€€€€€€€€€3ä p4} i

kjjã^{CosA€€€€€€€78pE}+SinA€€€€€€€€€7p 8 E²y

{zz+ ã^{-€€€€€€€€€€3ä p4} i

kjjjã^{CosA€€€€€€€98pE}+SinA€€€€€€€€€9p 8 E²y

{zzz+ ã^{-€€€€€€€ä p4} i

kjjjã^{CosA€€€€€€€€€118pE}+SinA€€€€€€€€€€€11p 8 E²y

{zzz+ ã^{€€€€€€€ä p4} i

kjjjã^{CosA€€€€€€€€€138pE}+SinA€€€€€€€€€€€13p 8 E²y

{zzz+ ã^{€€€€€€€€€€3ä p4} i

kjjjã^{CosA€€€€€€€€€158pE}+SinA€€€€€€€€€€€15p 8 E²y

{zzzy {zzz+

€€€€€€€1 16 ã^9äti

kjjj-1

€€€€ã + ã + ã^{-€€€€€€€€€€3ä p4} J1

€€€€2 + ã^{-€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€€€€3ä p4} J1

€€€€2 + ã^{-€€€€€€€€€•!!!!!12} N+ ã^{-€€€€€€€ä p4} J1

€€€€2 + ã^{€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€ä p4} J1

€€€€2 + ã^{€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€€€€7ä p8} Jã^{Cos@€€€€p8D}+SinA€€€€p

8E²N+ ã^{€€€€€€€€€€5ä p8} i

kjjjã^{CosA€€€€€€€3p8 E}+SinA€€€€€€€€€3p 8 E²y

{zzz+ ã^{€€€€€€€€€€3ä p8} i

kjjã^{CosA€€€€€€€58pE}+SinA€€€€€€€€€5p 8 E²y

{zz+ ã^{€€€€€€€ä p8} i

kjjã^{CosA€€€€€€€78pE}+SinA€€€€€€€€€7p 8 E²y

{zz+ ã^{-€€€€€€€ä p8} i

kjjjã^{CosA€€€€€€€9p8 E}+SinA€€€€€€€€€9p 8 E²y

{zzz+ ã^{-€€€€€€€€€€3ä p8} i

kjjjã^{CosA€€€€€€€€€118pE}+SinA€€€€€€€€€€€11p 8 E²y

{zzz+ ã^{-€€€€€€€€€€5ä p8} i

kjjjã^{CosA€€€€€€€€€138pE}+SinA€€€€€€€€€€€13p 8 E²y

{zzz+ ã^{-€€€€€€€€€€7ä p8} i

kjjjã^{CosA€€€€€€€€€158pE}+SinA€€€€€€€€€€€15p 8 E²y

{zzzy {zzz+

€€€€€€€1 16 ã^7äti

kjjj-1

€€€€ã + ã + ã^{-€€€€€€€€€€3ä p4} J1

€€€€2 + ã^{-€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€€€€3ä p4} J1

€€€€2 + ã^{-€€€€€€€€€•!!!!!12} N+ ã^{-€€€€€€€ä p4} J1

€€€€2 + ã^{€€€€€€€€€•!!!!!12} N+ ã^{€€€€€€€ä p4} J1

€€€€2 + ã^{€€€€€€€€€•!!!!!12} N+ ã^{-€€€€€€€€€€7ä p8} Jã^{Cos@€€€€p8D}+SinA€€€€p

8E²N+ ã^{-€€€€€€€€€€5ä p8} i

kjjjã^{CosA€€€€€€€38pE}+SinA€€€€€€€€€3p 8 E²y

{zzz+ ã^{-€€€€€€€€€€3ä p8} i

kjjã^{CosA€€€€€€€58pE}+SinA€€€€€€€€€5p 8 E²y

{zz+ ã^{-€€€€€€€ä p8} i

kjjã^{CosA€€€€€€€7p8 E}+SinA€€€€€€€€€7p 8 E²y

{zz+ ã^{€€€€€€€ä p8} i

kjjjã^{CosA€€€€€€€9p8 E}+SinA€€€€€€€€€9p 8 E²y

{zzz+ ã^{€€€€€€€€€€3ä p8} i

kjjjã^{CosA€€€€€€€€€118pE}+SinA€€€€€€€€€€€11p 8 E²y

{zzz+ ã^{€€€€€€€€€€5ä p8} i

kjjjã^{CosA€€€€€€€€€138pE}+SinA€€€€€€€€€€€13p 8 E²y

{zzz+ ã^{€€€€€€€€€€7ä p8} i

kjjjã^{CosA€€€€€€€€€158pE}+SinA€€€€€€€€€€€15p 8 E²y

{zzzy {zzz fS[t]//ExpToTrig

€€€€€€€1

16 HCos@8 tD+ äSin@8 tDL i

kjjj6+ ã +Cosh@1D+4 CoshA 1

€€€€€€€€€€•!!!!2 E-CoshACosA€€€€p

8EE-CoshACosA€€€€€€€€€3p

8 EE-CoshACosA€€€€€€€€€5p 8 EE- CoshACosA€€€€€€€€€7p

8 EE-CoshACosA€€€€€€€€€9p

8 EE-CoshACosA€€€€€€€€€€€11p

8 EE-CoshACosA€€€€€€€€€€€13p 8 EE- CoshACosA€€€€€€€€€€€15p

8 EE-SinA€€€€p

8E²-SinA€€€€€€€€€3p

8 E²-SinA€€€€€€€€€5p

8 E²-SinA€€€€€€€€€7p 8 E²- SinA€€€€€€€€€9p

8 E²-SinA€€€€€€€€€€€11p

8 E²-SinA€€€€€€€€€€€13p

8 E²-SinA€€€€€€€€€€€15p

8 E²-Sinh@1D- SinhACosA€€€€p

8EE-SinhACosA€€€€€€€€€3p

8 EE-SinhACosA€€€€€€€€€5p

8 EE-SinhACosA€€€€€€€€€7p 8 EE- SinhACosA€€€€€€€€€9p

8 EE-SinhACosA€€€€€€€€€€€11p

8 EE-SinhACosA€€€€€€€€€€€13p

8 EE-SinhACosA€€€€€€€€€€€15p 8 EEy

{zzz+

€€€€€€€1

16 HCos@12 tD+ äSin@12 tDL i

kjjj2+ ã +Cosh@1D-4 CoshA 1

€€€€€€€€€€•!!!!2 E+ äCoshACosA€€€€p 8EE- äCoshACosA€€€€€€€€€3p

8 EE+ äCoshACosA€€€€€€€€€5p

8 EE- äCoshACosA€€€€€€€€€7p

8 EE+ äCoshACosA€€€€€€€€€9p 8 EE- äCoshACosA€€€€€€€€€€€11p

8 EE+ äCoshACosA€€€€€€€€€€€13p

8 EE- äCoshACosA€€€€€€€€€€€15p

8 EE+ äSinA€€€€p

8E²- äSinA€€€€€€€€€3p 8 E²+ äSinA€€€€€€€€€5p

8 E²- äSinA€€€€€€€€€7p

8 E²+ äSinA€€€€€€€€€9p

8 E²- äSinA€€€€€€€€€€€11p

8 E²+ äSinA€€€€€€€€€€€13p

8 E²- äSinA€€€€€€€€€€€15p 8 E²- Sinh@1D+ äSinhACosA€€€€p

8EE- äSinhACosA€€€€€€€€€3p

8 EE+ äSinhACosA€€€€€€€€€5p

8 EE- äSinhACosA€€€€€€€€€7p 8 EE+ y {