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f @ t_ D := t ^ 2 • ; H-Pi • 2 £ t && t £ Pi • 2 L ; f @ t_ D := f @ Pi • 2 D • ; H Pi • 2 < t && t £ 3 Pi • 2 L ; f @ t_ D := f @ t + 2 Pi D • ; H-5 Pi • 2 £ t && t < -Pi • 2 L ; f @ t_ D := f @ t - 2 Pi D • ; H 3 Pi • 2 < t && t £ 7 Pi • 2 L ; f1 @ t_ D

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Academic year: 2021

Aktie "f @ t_ D := t ^ 2 • ; H-Pi • 2 £ t && t £ Pi • 2 L ; f @ t_ D := f @ Pi • 2 D • ; H Pi • 2 < t && t £ 3 Pi • 2 L ; f @ t_ D := f @ t + 2 Pi D • ; H-5 Pi • 2 £ t && t < -Pi • 2 L ; f @ t_ D := f @ t - 2 Pi D • ; H 3 Pi • 2 < t && t £ 7 Pi • 2 L ; f1 @ t_ D"

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(1)

Lösungen

1

Remove @ "Global`*" D

f @ t_ D := t ^ 2 • ; H-Pi • 2 £ t && t £ Pi • 2 L ; f @ t_ D := f @ Pi • 2 D • ; H Pi • 2 < t && t £ 3 Pi • 2 L ; f @ t_ D := f @ t + 2 Pi D • ; H-5 Pi • 2 £ t && t < -Pi • 2 L ; f @ t_ D := f @ t - 2 Pi D • ; H 3 Pi • 2 < t && t £ 7 Pi • 2 L ; f1 @ t_ D := t ^ 2;

f2 @ t_ D := f1 @ Pi • 2 D ;

Plot @ f @ t D , 8 t, -Pi • 2, 3 Pi • 2 <D ;

-1 1 2 3 4

0.5 1 1.5 2 2.5

Plot @ f @ t D , 8 t, -5 Pi • 2, 7 Pi • 2 <D ;

-7.5 -5 -2.5 2.5 5 7.5 10

0.5

1

1.5

2

2.5

(2)

Ÿ a Koeffizienten

T = 2 Pi;

cc = -Pi • 2;

w = 2 Pi • T;

a @ 0 D := 2 • T NIntegrate @ f @ t D , 8 t, cc, cc + T <D ;

a @ k_ D := 2 • T NIntegrate @ f @ t D Cos @ k w t D , 8 t, cc, cc + T <D ; b @ k_ D := 2 • T NIntegrate @ f @ t D Sin @ k w t D , 8 t, cc, cc + T <D ; H* c @ k_ D :=1 • T Integrate @ f @ t D E^ H-I k w t L , 8 t,cc,cc+T <D ; *L

ff @ t_ D := a @ 0 D • 2 + Sum @ a @ n D Cos @ n w t D + b @ n D Sin @ n w t D , 8 n, 1, Infinity <D ; ff @ t_, h_ D := a @ 0 D • 2 + Sum @ a @ n D Cos @ n w t D + b @ n D Sin @ n w t D , 8 n, 1, h <D •• Chop;

H * ffk @ t_ D :=Sum @ c @ n D E^ H I n w t L , 8 n,-Infinity,Infinity <D ; * L H* ffk @ t_,h_ D :=Sum @ c @ n D E^ H I n w t L , 8 n,-h,h <D ; *L

g @ t_ D := H ff @ u, 4 D • . u ® t L •• Simplify; g @ t D

1.64493 - 1.27324 Cos @ t D - 0.5 Cos @ 2 t D + 0.047157 Cos @ 3 t D + 0.125 Cos @ 4 t D a1 @ 0 D :=

2 • T H Integrate @ f1 @ t D , 8 t, cc, cc + T • 2 <D + Integrate @ f2 @ t D , 8 t, cc + T • 2, cc + T <DL ; a1 @ k_ D := 2 • T H Integrate @ f1 @ t D Cos @ k w t D , 8 t, cc, cc + T • 2 <D +

Integrate @ f2 @ t D Cos @ k w t D , 8 t, cc + T • 2, cc + T <DL ; b1 @ k_ D := 2 • T H Integrate @ f1 @ t D Sin @ k w t D , 8 t, cc, cc + T • 2 <D +

Integrate @ f2 @ t D Sin @ k w t D , 8 t, cc + T • 2, cc + T <DL ;

ff1 @ u_, h_ D := a1 @ 0 D • 2 + Sum @ a1 @ n D Cos @ n w u D + b1 @ n D Sin @ n w u D , 8 n, 1, h <D •• Chop;

g1 @ t_ D := H ff1 @ u, 4 D • . u ® t L ; g1 @ t D •• Simplify

p

2

€€€€€€€

6 - €€€€€€€€€€€€€€€€€€€€€€€ 4 Cos p @ t D - 1

€€€€ 2 Cos @ 2 t D + €€€€€€€€€€€€€€€€€€€€€€€€€€€ 4 Cos @ 3 t D 27 p + 1

€€€€ 8 Cos @ 4 t D g1 @ t_ D := H ff1 @ u, 4 D • . u ® t L •• N;

g1 @ t D

1.64493 - 1.27324 Cos @ t D - 0.5 Cos @ 2. t D + 0.047157 Cos @ 3. t D + 0.125 Cos @ 4. t D a1 @ 0 D

p

2

€€€€€€€

3

H* a0 • 2, a0 *L 8 1.6449340668482262`, 2 1.6449340668482262` <

8 1.64493, 3.28987 <

H * ak * L 8 -1.2732395447351628, -0.5`, +0.047157020175376325`, 0.125` <

8-1.27324, -0.5, 0.047157, 0.125 <

H * bk * L 8 0, 0, 0, 0 <

8 0, 0, 0, 0 <

(3)

Ÿ b

Abs @ g @ Pi • 2 D - f @ Pi • 2 DD 0.197467

Abs @ g1 @ Pi • 2 D - f @ Pi • 2 DD 0.197467

Abs @ g @ 3 Pi • 2 D - f @ 3 Pi • 2 DD 0.197467

Abs @ g1 @ 3 Pi • 2 D - f @ 3 Pi • 2 DD 0.197467

Ÿ c: Gute Näherung schon mit wenigen Koeffizienten

Plot @ Evaluate @8 f @ t D , g @ t D<D , 8 t, -Pi • 2, 3 Pi • 2 <D ;

-1 1 2 3 4

0.5 1 1.5 2 2.5

Plot @ Evaluate @8 f @ t D , g1 @ t D<D , 8 t, -Pi • 2, 3 Pi • 2 <D ;

-1 1 2 3 4

0.5

1

1.5

2

2.5

(4)

Ÿ d

f @ Pi • 2 D p

2

€€€€€€€

4 N @ % D 2.4674 a1 @ 0 D

p

2

€€€€€€€

3

8 Cos @ Pi • 2 D , Cos @ 2 Pi • 2 D , Cos @ 3 Pi • 2 D , Cos @ 4 Pi • 2 D<

8 0, -1, 0, 1 <

In p

2

€€€€€€€

4 == p

2

€€€€€€€

6 + I - €€€€€

p22

+ €€€

12

H -8 + p

2

LM * 0

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€€

p - 1

€€€€ 2 * H -1 L + I €€€€€

p62

+ €€€€€

541

H 8 - 9 p

2

LM * 0

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€€€€€€€€€€€

p + 1

€€€€ 8 * 1 + ....

lässt sich p auf eine Seite der Gleichung bringen und so isolieren, also berechnen

ff1 @ u, Infinity D p

2

€€€€€€€

6 + 1

€€€€€€€€€

2 p H p PolyLog @ 2, -ä ã

-äu

D + p PolyLog @ 2, ä ã

-äu

D +

p PolyLog @ 2, -ä ã

äu

D + p PolyLog @ 2, ä ã

äu

D - 2 ä PolyLog @ 3, -ä ã

-äu

D + 2 ä PolyLog @ 3, ä ã

-äu

D - 2 ä PolyLog @ 3, -ä ã

äu

D + 2 ä PolyLog @ 3, ä ã

äu

DL

2

Remove @ "Global`*" D g1 @ t_ D := -t;

Ÿ a Koeffizienten

T = 2;

cc = -1;

w = 2 Pi • T;

a @ 0 D := 2 • T Integrate @ g1 @ t D , 8 t, cc, cc + T <D ;

a @ k_ D := 2 • T Integrate @ g1 @ t D Cos @ k w t D , 8 t, cc, cc + T <D ; b @ k_ D := 2 • T Integrate @ g1 @ t D Sin @ k w t D , 8 t, cc, cc + T <D ; H* c @ k_ D :=1 • T Integrate @ g1 @ t D E^ H-I k w t L , 8 t,cc,cc+T <D ; *L

gg @ t_ D := a @ 0 D • 2 + Sum @ a @ n D Cos @ n w t D + b @ n D Sin @ n w t D , 8 n, 1, Infinity <D ; gg @ t_, h_ D := a @ 0 D • 2 + Sum @ a @ n D Cos @ n w t D + b @ n D Sin @ n w t D , 8 n, 1, h <D •• Chop;

H* ffk @ t_ D :=Sum @ c @ n D E^ H I n w t L , 8 n,-Infinity,Infinity <D ; *L H* ffk @ t_,h_ D :=Sum @ c @ n D E^ H I n w t L , 8 n,-h,h <D ; *L

h4 @ t_ D := H gg @ u, 4 D • . u ® t L •• Simplify; ExpandAll @ h4 @ t DD - €€€€€€€€€€€€€€€€€€€€€€€€€€€ 2 Sin p @ p t D + €€€€€€€€€€€€€€€€€€€€€€€€€€€ Sin @ p 2 p t D - €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 2 Sin @ 3 p t D

3 p + €€€€€€€€€€€€€€€€€€€€€€€€€€€ Sin @ 4 p t D

2 p

(5)

h20 @ t_ D := H gg @ u, 20 D • . u ® t L •• Simplify;

w = 2 Pi • T p

h4N @ t_ D := h4 @ t D •• N •• Simplify; h4N @ t D

-0.63662 Sin @ 3.14159 t D + 0.31831 Sin @ 6.28319 t D - 0.212207 Sin @ 9.42478 t D + 0.159155 Sin @ 12.5664 t D 8 -0.6366197723675814`, +0.3183098861837907`,

-0.2122065907891938` , +0.15915494309189535` <

8-0.63662, 0.31831, -0.212207, 0.159155 <

Plot @ Evaluate @8 g1 @ t D , h4N @ t D<D , 8 t, -1, 1 <D ;

-1 -0.5 0.5 1

-1 -0.5 0.5 1

Ÿ b

3 h4 @ t D + 2 •• ExpandAll 2 - €€€€€€€€€€€€€€€€€€€€€€€€€€€ 6 Sin @ p t D

p + €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 3 Sin @ 2 p t D

p - €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 2 Sin @ 3 p t D

p + €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 3 Sin @ 4 p t D 2 p g2 @ t_ D := 3 g1 @ t D + 2;

h2 @ t_ D := 3 h4N @ t D + 2 •• Simplify; 8 g2 @ t D , h2 @ t D<

8 2 - 3 t, 2. - 1.90986 Sin @ 3.14159 t D +

0.95493 Sin @ 6.28319 t D - 0.63662 Sin @ 9.42478 t D + 0.477465 Sin @ 12.5664 t D<

8 2.`, -1.909859317102744`, +0.954929658551372`, -0.6366197723675814`, +0.477464829275686` <

8 2., -1.90986, 0.95493, -0.63662, 0.477465 <

(6)

Plot @ Evaluate @8 g2 @ t D , h2 @ t D<D , 8 t, -1, 1 <D ;

-1 -0.5 0.5 1

-1 1 2 3 4 5

Ÿ c Integration der Fourierreihe!

Expand @ h4 @ u DD

- €€€€€€€€€€€€€€€€€€€€€€€€€€€ 2 Sin p @p u D + €€€€€€€€€€€€€€€€€€€€€€€€€€€ Sin @ p 2 p u D - €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 2 Sin @ 3 p u D

3 p + €€€€€€€€€€€€€€€€€€€€€€€€€€€ Sin @ 4 p u D 2 p

Integrate @ Evaluate @ Expand @ h4 @ u DDD , 8 u, 0, t <D •• TrigReduce •• Expand

- 115

€€€€€€€€€€€€€€

72 p

2

+ €€€€€€€€€€€€€€€€€€€€€€€€€€€ 2 Cos p @p

2

t D - €€€€€€€€€€€€€€€€€€€€€€€€€€€ Cos @ 2 p t D

2 p

2

+ €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 2 Cos @ 3 p t D

9 p

2

- €€€€€€€€€€€€€€€€€€€€€€€€€€€ Cos @ 4 p t D 8 p

2

g3 @ t_ D := Integrate @ g1 @ u D , 8 u, 0, t <D ; g3 @ t D

- t

2

€€€€€€€

2

g31 @ t_ D := Integrate @ h4N @ u D , 8 u, 0, t <D ; g31 @ t D

-0.161832 + 0.202642 Cos @ 3.14159 t D - 0.0506606 Cos @ 6.28319 t D + 0.0225158 Cos @ 9.42478 t D - 0.0126651 Cos @ 12.5664 t D

8 -0.16183244609540065`, +0.20264236728467558` ,

-0.050660591821168895` , +0.022515818587186175` , -0.012665147955292224` <

8-0.161832, 0.202642, -0.0506606, 0.0225158, -0.0126651 <

Plot @ Evaluate @8 g3 @ t D , g31 @ t D<D , 8 t, -1, 1 <D ;

-1 -0.5 0.5 1

-0.5

-0.4

-0.3

-0.2

-0.1

(7)

Ÿ d Differentiation führt zu keiner Konstante: Oszillation, Gibbs!

g1 ' @ t D -1

h4N ' @ t D

-2. Cos @ 3.14159 t D + 2. Cos @ 6.28319 t D - 2. Cos @ 9.42478 t D + 2. Cos @ 12.5664 t D Plot @ Evaluate @8 g1 ' @ t D , h4N ' @ t D<D , 8 t, -1, 1 <D ;

-1 -0.5 0.5 1

-2 2 4

Plot @ Evaluate @8 g1 ' @ t D , h20 ' @ t D<D , 8 t, -1, 1 <D ;

-1 -0.5 0.5 1

-6 -4 -2 2 4

3

Remove @ "Global`*" D ;

Ÿ a

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

(8)

n=6; w = 2 Pi/n;

{x[0],y[0]}={0 w,2};

{x[1],y[1]}={1 w,2};

{x[2],y[2]}={2 w,3};

{x[3],y[3]}={3 w,2};

{x[4],y[4]}={4 w,3.5};

{x[5],y[5]}={5 w,7};

{x[6],y[6]}={6 w,2};

{x[-1],y[-1]}={-1 w,7};

{x[-2],y[-2]}={-2 w,3.5};

{x[-3],y[-3]}={-3 w,2};

{x[-4],y[-4]}={-4 w,3};

{x[-5],y[-5]}={-5 w,2};

{x[-6],y[-6]}={-6 w,2};

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

Table[p[k],{k,-(n-1),(n-1)}]

99- €€€€€€€€€ 5 p

3 , 2 = , 9- €€€€€€€€€ 4 p

3 , 3 = , 8-p, 2 < , 9- €€€€€€€€€ 2 p

3 , 3.5 = , 9- €€€€ p 3 , 7 = , 8 0, 2 < , 9 €€€€ p

3 , 2 = , 9 €€€€€€€€€ 2 p

3 , 3 = , 8 p, 2 < , 9 €€€€€€€€€ 4 p

3 , 3.5 = , 9 €€€€€€€€€ 5 p 3 , 7 ==

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

9 PointSize @ 0.03 D , Point @8 0, 2 <D , Point A9 €€€€ p

3 , 2 =E , Point A9 €€€€€€€€€ 2 p 3 , 3 =E , Point @8p, 2 <D , Point A9 €€€€€€€€€ 4 p

3 , 3.5 =E , Point A9 €€€€€€€€€ 5 p

3 , 7 =E , Point @8 2 p, 2 <D=

epi1 = Prepend @ Map @ Point, Table @8 k, y @ k D< , 8 k, 0, n <DD , PointSize @ 0.03 DD 8 PointSize @ 0.03 D , Point @8 0, 2 <D , Point @8 1, 2 <D , Point @8 2, 3 <D ,

Point @8 3, 2 <D , Point @8 4, 3.5 <D , Point @8 5, 7 <D , Point @8 6, 2 <D<

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

c[s_]:= 1/n Sum[y[k] r^(s k),{k,-Floor[(n-1)/2],n-1-Floor[(n-1)/2]}];

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

8 3.25, 0.208333 + 0.793857 ä, -0.625 + 0.649519 ä,

-0.416667, -0.625 - 0.649519 ä, 0.208333 - 0.793857 ä, 3.25,

0.208333 + 0.793857 ä, -0.625 + 0.649519 ä, -0.416667, -0.625 - 0.649519 ä<

Ÿ a c[s]

8 3.25`, 0.20833333333333373` + 0.7938566201357355` ä, -0.6249999999999993` + 0.649519052838329` ä,

-0.41666666666666663`, -0.6249999999999993` - 0.649519052838329` ä, 0.20833333333333373` - 0.7938566201357355` ä,

3.25`, 0.20833333333333373` + 0.7938566201357355` ä, -0.6249999999999993` + 0.649519052838329` ä,

-0.41666666666666663`, -0.6249999999999993` - 0.649519052838329` ä<

8 3.25, 0.208333 + 0.793857 ä, -0.625 + 0.649519 ä,

-0.416667, -0.625 - 0.649519 ä, 0.208333 - 0.793857 ä, 3.25,

0.208333 + 0.793857 ä, -0.625 + 0.649519 ä, -0.416667, -0.625 - 0.649519 ä <

(9)

fS[t_]:=Sum[c[k] E^(I k t),{k,-Floor[(n-1)/2],n-1-Floor[(n-1)/2]}];

fS[t]

3.25 + H 0.208333 - 0.793857 ä L ã

-ät

+ H 0.208333 + 0.793857 ä L ã

ät

- H 0.625 + 0.649519 äL ã

-2ät

- H 0.625 - 0.649519 äL ã

2ät

- 0.416667 ã

3ät

% •• ExpandAll

3.25 + H 0.208333 - 0.793857 ä L ã

-ät

+ H 0.208333 + 0.793857 ä L ã

ät

- H 0.625 + 0.649519 ä L ã

-2ät

- H 0.625 - 0.649519 ä L ã

2ät

- 0.416667 ã

3ät

fS1 @ s_ D := fS @ s 2 Pi • n D ;

fS1 @ s D

3.25 + H 0.208333 - 0.793857 ä L ã

-€€€€13ä ps

+ H 0.208333 + 0.793857 ä L ã

€€€€€€€€€€ä p3s

- H 0.625 + 0.649519 äL ã

-€€€€23ä ps

- H 0.625 - 0.649519 äL ã

€€€€€€€€€€€€€€2ä ps3

- 0.416667 ã

ä ps

% •• ExpandAll

3.25 + H 0.208333 - 0.793857 äL ã

-€€€€13ä ps

+ H 0.208333 + 0.793857 äL ã

€€€€€€€€€€ä p3s

- H 0.625 + 0.649519 ä L ã

-€€€€23ä ps

- H 0.625 - 0.649519 ä L ã

€€€€€€€€€€€€€€2ä ps3

- 0.416667 ã

ä ps

% •• N •• Simplify

3.25 + H 0.208333 - 0.793857 ä L ã

-1.0472äs

+ H 0.208333 + 0.793857 ä L ã

1.0472äs

- H 0.625 + 0.649519 ä L ã

-2.0944äs

- H 0.625 - 0.649519 ä L ã

2.0944äs

- 0.416667 ã

3.14159äs

fS[t]//ExpToTrig

3.25 + H 0.416667 + 0. äL Cos @ t D - H 1.25 + 0. äL Cos @ 2 t D - 0.416667 Cos @ 3 t D - H 1.58771 + 0. ä L Sin @ t D - H 1.29904 + 0. ä L Sin @ 2 t D - 0.416667 ä Sin @ 3 t D

% •• ExpandAll

3.25 + H 0.416667 + 0. äL Cos @ t D - H 1.25 + 0. äL Cos @ 2 t D - 0.416667 Cos @ 3 t D - H 1.58771 + 0. äL Sin @ t D - H 1.29904 + 0. äL Sin @ 2 t D - 0.416667 ä Sin @ 3 t D

% •• N

3.25 + H 0.416667 + 0. äL Cos @ t D - H 1.25 + 0. äL Cos @ 2. t D - 0.416667 Cos @ 3. t D - H 1.58771 + 0. äL Sin @ t D - H 1.29904 + 0. äL Sin @ 2. t D - H 0. + 0.416667 äL Sin @ 3. t D

fS1 @ s D •• ExpToTrig

3.25 + H 0.416667 + 0. ä L Cos A p s

€€€€€€€€€

3 E - H 1.25 + 0. ä L Cos A 2 p s

€€€€€€€€€€€€€

3 E - 0.416667 Cos @ p s D - H 1.58771 + 0. äL Sin A p s

€€€€€€€€€

3 E - H 1.29904 + 0. äL Sin A 2 p s

€€€€€€€€€€€€€

3 E - 0.416667 ä Sin @p s D

% •• ExpandAll

3.25 + H 0.416667 + 0. äL Cos A p s

€€€€€€€€€

3 E - H 1.25 + 0. äL Cos A 2 p s

€€€€€€€€€€€€€

3 E - 0.416667 Cos @p s D - H 1.58771 + 0. ä L Sin A p s

€€€€€€€€€

3 E - H 1.29904 + 0. ä L Sin A 2 p s

€€€€€€€€€€€€€

3 E - 0.416667 ä Sin @ p s D

% •• N

3.25 + H 0.416667 + 0. ä L Cos @ 1.0472 s D - H 1.25 + 0. ä L Cos @ 2.0944 s D - 0.416667 Cos @ 3.14159 s D - H 1.58771 + 0. ä L Sin @ 1.0472 s D -

H 1.29904 + 0. ä L Sin @ 2.0944 s D - H 0. + 0.416667 ä L Sin @ 3.14159 s D

(10)

fS1 @ s D •• ExpToTrig •• Chop 3.25 + 0.416667 Cos A p s

€€€€€€€€€

3 E - 1.25 Cos A 2 p s

€€€€€€€€€€€€€

3 E - 0.416667 Cos @ p s D - 1.58771 Sin A p s

€€€€€€€€€

3 E - 1.29904 Sin A 2 p s

€€€€€€€€€€€€€

3 E - 0.416667 ä Sin @p s D 8 3.25`, 8 0.41666666666666746` , -1.2499999999999987` < ,

8 -0.41666666666666663` , -1.587713240271471` , -1.299038105676658` < , 8 -0.41666666666666663` <<

8 3.25, 8 0.416667, -1.25 < , 8-0.416667, -1.58771, -1.29904 < , 8-0.416667 <<

Ÿ b

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

1 2 3 4 5 6

1 2 3 4 5 6 7

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

1 2 3 4 5 6

-0.4 -0.2 0.2 0.4

Man beachte im letzten Plot die Grösse der Amplitude.

(11)

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

1 2 3 4 5 6

1 2 3 4 5 6 7

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.)

Plot @8 Re @ fS1 @ s DD< , 8 s, 0, n < , Epilog ® epi1 D ;

1 2 3 4 5 6

1 2 3 4 5 6 7

Ÿ c

Re @ fS1 @ 3.5 DD 1.93301

Ÿ d

linInt @ t_ D := 2 + H t - 3 L H 3.5 - 2 L ; Plot @ linInt @ t D , 8 t, 2, 5 <D ;

2.5 3 3.5 4 4.5 5

1

2

3

4

5

(12)

linInt @ 3.5 D - Re @ fS1 @ 3.5 DD 0.816987

4

Remove @ "Global`*" D ;

Ÿ a Skizze

f @ t_ D := Cos @ 2 t • 3 D + Sin @ 0.4 t - 1 D ; Plot @ f @ t D , 8 t, 0, 100 <D ;

20 40 60 80 100

-2 -1 1 2

Ÿ b

8 2 t • 3, 2 t • 3 + 2 Pi <

9 2 t

€€€€€€€€€

3 , 2 p + 2 t

€€€€€€€€€

3 =

8 2 t • 3, 2 t • 3 + 2 Pi < 3 • 2 •• ExpandAll 8 t, 3 p + t <

8 4 • 10 t, 4 • 10 t + 2 Pi <

9 2 t

€€€€€€€€€

5 , 2 p + 2 t

€€€€€€€€€

5 =

8 4 • 10 t, 4 • 10 t + 2 Pi < 5 • 2 •• ExpandAll 8 t, 5 p + t <

T = LCM @ 5 , 3 D p 15 p

? GCD

GCD @ n1, n2, ... D gives the greatest common divisor of the integers ni. Mehr…

(13)

0 Š Simplify @ f @ t + 15 Pi D - f @ t DD •• Chop True

T = 15 Pi;

Ÿ c

cc = 0;

w = 2 Pi • T;

a @ 0 D := 2 • T Integrate @ f @ t D , 8 t, cc, cc + T <D ;

a @ k_ D := 2 • T Integrate @ f @ t D Cos @ k w t D , 8 t, cc, cc + T <D ; b @ k_ D := 2 • T Integrate @ f @ t D Sin @ k w t D , 8 t, cc, cc + T <D ; c @ k_ D := 1 • T Integrate @ f @ t D E ^ H-I k w t L , 8 t, cc, cc + T <D ;

ff @ t_ D := a @ 0 D • 2 + Sum @ a @ n D Cos @ n w t D + b @ n D Sin @ n w t D , 8 n, 1, Infinity <D ; ff @ t_, h_ D := a @ 0 D • 2 + Sum @ a @ n D Cos @ n w t D + b @ n D Sin @ n w t D , 8 n, 1, h <D ; ffk @ t_ D := Sum @ c @ n D E ^ H I n w t L , 8 n, -Infinity, Infinity <D ;

ffk @ t_, h_ D := Sum @ c @ n D E ^ H I n w t L , 8 n, -h, h <D ; ff @ t, 6 D •• Chop

-0.841471 Cos A 2 t

€€€€€€€€€

5 E + 1. Cos A 2 t

€€€€€€€€€

3 E + 0.540302 Sin A 2 t

€€€€€€€€€

5 E Abs @ f @ 10 D - ff @ 10, 6 DD

1.11022 ´ 10

-15

Ÿ d

Plot @ Evaluate @8 f @ t D , ff @ t, 6 D •• Chop <D , 8 t, 0, T <D ;

10 20 30 40

-2 -1 1 2

Die Genauigkeit der beiden Funktionen f und ff6 ist anhand der Skizze nicht mehr unterscheidbar.

(14)

Plot @ Evaluate @8 f @ t D , ff @ t, 6 D •• Chop <D , 8 t, 0, 2 T <D ;

20 40 60 80

-2 -1 1 2

Ÿ e

ff2 @ t_, h_ D := a @ 0 D ^ 2 • 2 + Sum @ a @ n D ^ 2 Cos @ n w t D + b @ n D ^ 2 Sin @ n w t D , 8 n, 1, h <D ; Plot @ Evaluate @8 f @ t D , ff2 @ t, 6 D •• Chop <D , 8 t, 0, 30 Pi <D ;

20 40 60 80

-2 -1 1 2

Plot @ Evaluate @8 f @ t D ^ 2, ff @ t, 6 D •• Chop <D , 8 t, 0, 30 Pi <D ;

20 40 60 80

-2 -1 1 2 3 4

Evaluate @H f @ t D ^ 2 - ff @ t, 6 DL • ff @ t, 6 D •• Chop D • . t ® 15.

-2.798

(15)

Plot @ Evaluate @H f @ t D ^ 2 - ff @ t, 6 DL • ff @ t, 6 D •• Chop D , 8 t, 0, T <D ;

10 20 30 40

-3 -2 -1 1

Der Ausdruck schwanktje nach t. Die maximale absolute relative Differenz ist etwa 3.

5 Varianten!

Ÿ a

Remove @ "Global`*" D ;

f @ x_ D := Cos @ 2 x D + I Sin @ 2 x D

FourierTransform @ f @ x D , x, wD •• Simplify

•!!!!!!!! 2 p DiracDelta @ 2 + w D

1 ‘ •!!!!!!!! 2 p FourierTransform @ f @ x D , x, w D •• Simplify DiracDelta @ 2 + wD

•!!!!!!!! 2 p FourierTransform @ f @ x D , x, w D •• Simplify

2 p DiracDelta @ 2 + wD

Ÿ b

Remove @ "Global`*" D ;

f @ x_ D := Sin @ 2 x D + I Cos @ 2 x D

FourierTransform @ f @ x D , x, wD •• Simplify ä •!!!!!!!! 2 p DiracDelta @ -2 + w D

1 ‘ •!!!!!!!! 2 p FourierTransform @ f @ x D , x, wD •• Simplify

ä DiracDelta @-2 + wD

(16)

•!!!!!!!!

2 p FourierTransform @ f @ x D , x, wD •• Simplify 2 ä p DiracDelta @-2 + wD

Ÿ c

Remove @ "Global`*" D ;

fHat @ x_ D := Cos @ 2 wD + I Sin @ 2 wD

InverseFourierTransform @ fHat @ x D , w, x D •• Simplify

•!!!!!!!! 2 p DiracDelta @-2 + x D

•!!!!!!!!

2 p InverseFourierTransform @ fHat @ x D , w, x D •• Simplify 2 p DiracDelta @ -2 + x D

1 ‘ •!!!!!!!! 2 p InverseFourierTransform @ fHat @ x D , w, x D •• Simplify DiracDelta @-2 + x D

1 ‘ •!!!!!!!! 2 p FourierTransform A•!!!!!!!! 2 p InverseFourierTransform @ fHat @ x D , w, x D , x, wE ••

Simplify ã

2ä w

ã

2ä w

•• ExpToTrig Cos @ 2 wD + ä Sin @ 2 wD

6 Varianten!

Remove @ "Global`*" D ;

Ÿ a

H @ x_ D := UnitStep @ x D ; Plot @ H @ x D , 8 x, -3, 3 <D ;

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

(17)

f @ x_ D := Pi H H @ x + Pi D - H @ x - Pi DL ; Plot @ f @ x D , 8 x, -5, 5 <D ;

-4 -2 2 4

0.5 1 1.5 2 2.5 3

FourierTransform @ f @ x D , x, w D •• Simplify

•!!!!!!!! 2 p Sin @p wD

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

w

1 ‘ •!!!!!!!! 2 p FourierTransform @ f @ x D , x, wD •• Simplify Sin @ p w D

€€€€€€€€€€€€€€€€€€€€€€€

w

•!!!!!!!! 2 p FourierTransform @ f @ x D , x, w D •• Simplify

2 p Sin @p wD

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

w

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