f @ t_ D : = t ^ 2 − t ê ; H − Pi ê 2 ≤ t && t ≤ Pi ê 2 L ; f @ t_ D : = f @ Pi ê 2 D ê ; H Pi ê 2 < t && t ≤ 3 Pi ê 2 L ; f @ t_ D : = f @ t + 2 Pi D ê ; H − 5 Pi ê 2 ≤ t && t < − Pi ê 2 L ; f @ t_ D := f @ t − 2 Pi D ê ; H 3 Pi ê 2 < t && t ≤ 7 Pi ê 2 L

1

Remove @ "Global` ∗ " D

f @ t_ D : = t ^ 2 − t ê ; H − Pi ê 2 ≤ t && t ≤ Pi ê 2 L ; f @ t_ D : = f @ Pi ê 2 D ê ; H Pi ê 2 < t && t ≤ 3 Pi ê 2 L ; f @ t_ D : = f @ t + 2 Pi D ê ; H − 5 Pi ê 2 ≤ t && t < − Pi ê 2 L ; f @ t_ D := f @ t − 2 Pi D ê ; H 3 Pi ê 2 < t && t ≤ 7 Pi ê 2 L ; f1 @ t_ D := t ^ 2 − t;

f2 @ t_ D : = f1 @ Pi ê 2 D ;

ü a Skizze

Plot @ f @ t D , 8 t, − Pi ê 2, 3 Pi ê 2 < , PlotRange → 8 − 0.5, 4.5 <D

-1 1 2 3 4

1 2 3 4

Plot @ f @ t D , 8 t, − 5 Pi ê 2, 7 Pi ê 2 < , PlotRange → 8 − 0.5, 4.5 <D

-5 5 10

1 2 3 4

ü b Koeffizienten

T = 2 Pi;

cc = − Pi ê 2;

ω = 2 Pi ê T;

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

a @ k_ D : = 2 ê T NIntegrate @ f @ t D Cos @ k ω t D , 8 t, cc, cc + T <D ; b @ k_ D : = 2 ê T NIntegrate @ f @ t D Sin @ k ω t D , 8 t, cc, cc + T <D ; H ∗ c @ k_ D : = 1 ê T Integrate @ f @ t D E^ H − I k ω t L , 8 t,cc,cc + T <D ; ∗ L

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

H ∗ ffk @ t_ D : = Sum @ c @ n D E^ H I n ω t L , 8 n, − Infinity,Infinity <D ; ∗ L H ∗ ffk @ t_,h_ D : = Sum @ c @ n D E^ H I n ω t L , 8 n, − h,h <D ; ∗ L

g @ t_ D := H ff @ u, 4 D ê . u → t L êê Simplify; g @ t D NIntegrate::ncvb :

NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in t near

t

=

1.57379

. NIntegrate obtained 0.22222220216854388` and 5.505613236842656`*^-7 for the integral and error estimates.

NIntegrate::ncvb :

NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in t near

t

=

1.57379

. NIntegrate obtained 0.392699101752458` and 5.503283727217613`*^-7 for the integral and error estimates.

0.859536 − 0.27324 Cos @ t D − 0.5 Cos @ 2 t D − 0.286176 Cos @ 3 t D + 0.125 Cos @ 4 t D − 0.63662 Sin @ t D − 1. Cos @ t D Sin @ t D + 0.0707355 Sin @ 3 t D + 0.25 Sin @ 4 t D Infinity ist hier doch zu weit.

0.859536 − 0.27324 Cos @ t D − 0.5 Cos @ 2 t D − 0.286176 Cos @ 3 t D + 0.125 Cos @ 4 t D − 0.63662 Sin @ t D − 1. Cos @ t D Sin @ t D + 0.0707355 Sin @ 3 t D + 0.25 Sin @ 4 t D a1 @ 0 D : =

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

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

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

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

g1 @ t_ D : = H ff1 @ u, 4 D ê . u → t L ; g1 @ t D êê Expand

− π 4

+ π

6

+ Cos @ t D − 4 Cos @ t D π

− 1 2

Cos @ 2 t D − 1 3

Cos @ 3 t D + 4 Cos @ 3 t D

27 π + 1

8

Cos @ 4 t D − 2 Sin @ t D π

− 1 2

Sin @ 2 t D + 2 Sin @ 3 t D 9 π

+ 1 4

Sin @ 4 t D g1N @ t_ D : = H ff1 @ u, 4 D ê . u → t L êê N;

g1N @ t D

0.859536 − 0.27324 Cos @ t D − 0.5 Cos @ 2. t D − 0.286176 Cos @ 3. t D + 0.125 Cos @ 4. t D − 0.63662 Sin @ t D − 0.5 Sin @ 2. t D + 0.0707355 Sin @ 3. t D + 0.25 Sin @ 4. t D

g1N @ t D êê Expand

0.859536 − 0.27324 Cos @ t D − 0.5 Cos @ 2. t D − 0.286176 Cos @ 3. t D + 0.125 Cos @ 4. t D − 0.63662 Sin @ t D − 0.5 Sin @ 2. t D + 0.0707355 Sin @ 3. t D + 0.25 Sin @ 4. t D

a1 @ 0 D

π³ 12

+ π J −

2

+

4

N

π

H ∗ a0 ê 2, a0 ∗ L 8 0.859535903450778`, 2 × 0.859535903450778` <

8 0.859536, 1.71907 <

H ∗ ak ∗ L 8 − 0.27323954473516276`, − 0.5` , − 0.286176313157957` , + 0.125` <

8 − 0.27324, − 0.5, − 0.286176, 0.125 <

H ∗ bk ∗ L 8 0.6366197723675814`, − 0.5` , +0.07073553026306459` , +0.25` <

8 0.63662, − 0.5, 0.0707355, 0.25 <

ü c: Gute Näherung schon mit wenigen Koeffizienten

g1 @ Pi ê 2 D 5

8

− 20 9 π

+

π³

12

+ π J −

2

+

4

N 2 π f @ Pi ê 2 D

− π 2

+ π

4

% êê N 0.896605 f @ 3 Pi ê 2 D

− π 2

+ π

4

% êê N 0.896605

Abs @ g1 @ Pi ê 2 D − f @ Pi ê 2 DD

− 5 8

+ 20 9 π

− π 2

+ π

4

−

π³ 12

+ π J −

2

+

4

N 2 π Abs @ g1N @ Pi ê 2 D − f @ Pi ê 2 DD 0.119424

Abs @ g1 @ 3 Pi ê 2 D − f @ Pi ê 2 DD 5

8 + 20

9 π +

π 2

− π

4 +

π³ 12

+ π J −

2

+

4

N 2 π Abs @ g1N @ 3 Pi ê 2 D − f @ 3 Pi ê 2 DD 1.29529

Abs @ g1 @ 3 Pi ê 2 D − f @ 3 Pi ê 2 DD 5

8 + 20

9 π +

π 2

− π

4 +

π³ 12

+ π J −

2

+

4

N

2 π

ü d:

Plot @ Evaluate @8 f @ t D , g @ t D<D , 8 t, − Pi ê 2, 3 Pi ê 2 < , PlotPoints → 60 D NIntegrate::ncvb :

NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in t near 8t<

81.57379<. NIntegrate obtained 0.22222220216854388` and 5.505613236842656`*^-7 for the integral and error estimates.

NIntegrate::ncvb :

NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in t near 8t<

81.57379<. NIntegrate obtained 0.392699101752458` and 5.503283727217613`*^-7 for the integral and error estimates.

-1 1 2 3 4

0.5 1.0 1.5 2.0 2.5

Plot @ Evaluate @8 f @ t D , g1 @ t D<D , 8 t, − Pi ê 2, 3 Pi ê 2 <D

-1 1 2 3 4

0.5 1.0 1.5 2.0 2.5

ü e

8 f @ Pi ê 2 D , g1 @ Pi ê 2 D<

9 − π 2

+ π

4 ,

5 8

− 20 9 π

+

π³

12

+ π J −

2

+

4

N

2 π =

N @ % D

8 0.896605, 0.777181 <

In π 2

+ π

4

= f @ Pi ê 2 D = g1 @ Pi ê 2 D

lässt sich π auf eine Seite der Gleichung bringen und so isolieren,

also berechnen.

2

ü a

Remove @ "Global` ∗ " D ; T = Pi;

cc = − Pi ê 2;

f @ t_ D : = Abs @ t D + t;

ω = 2 Pi ê T;

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

a @ k_ D : = 2 ê T Integrate @ f @ t D Cos @ k ω t D , 8 t, cc, cc + T <D ; b @ k_ D : = 2 ê T Integrate @ f @ t D Sin @ k ω t D , 8 t, cc, cc + T <D ;

ff @ s_, h_ D : = a @ 0 D ê 2 + Sum @ a @ n D Cos @ n ω s D + b @ n D Sin @ n ω s D , 8 n, 1, h <D ; ff @ s, 10 D

π 4

− 2 Cos @ 2 s D π

− 2 Cos @ 6 s D 9 π

− 2 Cos @ 10 s D 25 π

− 2 Cos @ 14 s D 49 π

− 2 Cos @ 18 s D

81 π

+ Sin @ 2 s D − 1 2

Sin @ 4 s D + 1 3

Sin @ 6 s D − 1 4

Sin @ 8 s D + 1 5

Sin @ 10 s D − 1

6

Sin @ 12 s D + 1 7

Sin @ 14 s D − 1 8

Sin @ 16 s D + 1 9

Sin @ 18 s D − 1 10

Sin @ 20 s D N @ % D

0.785398 − 0.63662 Cos @ 2. s D − 0.0707355 Cos @ 6. s D − 0.0254648 Cos @ 10. s D − 0.0129922 Cos @ 14. s D − 0.0078595 Cos @ 18. s D + Sin @ 2. s D − 0.5 Sin @ 4. s D + 0.333333 Sin @ 6. s D − 0.25 Sin @ 8. s D + 0.2 Sin @ 10. s D − 0.166667 Sin @ 12. s D + 0.142857 Sin @ 14. s D − 0.125 Sin @ 16. s D + 0.111111 Sin @ 18. s D − 0.1 Sin @ 20. s D ff @ 4, 10 D

π 4

− 2 Cos @ 8 D π

− 2 Cos @ 24 D 9 π

− 2 Cos @ 40 D 25 π

− 2 Cos @ 56 D 49 π

− 2 Cos @ 72 D 81 π

+ Sin @ 8 D − Sin @ 16 D 2

+ Sin @ 24 D

3

− Sin @ 32 D 4

+ Sin @ 40 D 5

− Sin @ 48 D 6

+ Sin @ 56 D 7

− Sin @ 64 D 8

+ Sin @ 72 D 9

− Sin @ 80 D 10 ff @ s, 10 D ê . s → t

π 4

− 2 Cos @ 2 t D π

− 2 Cos @ 6 t D 9 π

− 2 Cos @ 10 t D 25 π

− 2 Cos @ 14 t D 49 π

− 2 Cos @ 18 t D

81 π

+ Sin @ 2 t D − 1 2

Sin @ 4 t D + 1 3

Sin @ 6 t D − 1 4

Sin @ 8 t D + 1 5

Sin @ 10 t D − 1

6

Sin @ 12 t D + 1 7

Sin @ 14 t D − 1 8

Sin @ 16 t D + 1 9

Sin @ 18 t D − 1 10

Sin @ 20 t D Plot @ Evaluate @8 f @ s D , ff @ s, 10 D<D , 8 s, − Pi ê 2, Pi ê 2 <D

-1.5 -1.0 -0.5 0.5 1.0 1.5

0.5 1.0 1.5 2.0 2.5 3.0

ü b

Remove @ "Global` ∗ " D ;

Ersetze t durch - t und Abs[t]+t durch -(Abs[-t]-t). Schiebe dann die Funktion um 3.

f @ t_ D : = Abs @ t D + t;

f1 @ t_ D := 3 − f @ −t D ; f1 @ t D 3 + t − Abs @ t D

T = Pi;

cc = − Pi ê 2;

ω = 2 Pi ê T;

a @ 0 D : = 2 ê T Integrate @ f1 @ t D , 8 t, cc, cc + T <D ;

a @ k_ D : = 2 ê T Integrate @ f1 @ t D Cos @ k ω t D , 8 t, cc, cc + T <D ; b @ k_ D : = 2 ê T Integrate @ f1 @ t D Sin @ k ω t D , 8 t, cc, cc + T <D ;

ff1 @ s_, h_ D : = a @ 0 D ê 2 + Sum @ a @ n D Cos @ n ω s D + b @ n D Sin @ n ω s D , 8 n, 1, h <D ; ff1 @ s, 10 D êê Simplify

3 − π 4

+ 2 Cos @ 2 s D π

+ 2 Cos @ 6 s D 9 π

+ 2 Cos @ 10 s D 25 π

+ 2 Cos @ 14 s D 49 π

+ 2 Cos @ 18 s D

81 π

+ Sin @ 2 s D − 1 2

Sin @ 4 s D + 1 3

Sin @ 6 s D − 1 4

Sin @ 8 s D + 1 5

Sin @ 10 s D − 1

6

Sin @ 12 s D + 1 7

Sin @ 14 s D − 1 8

Sin @ 16 s D + 1 9

Sin @ 18 s D − 1 10

Sin @ 20 s D N @ % D

2.2146 + 0.63662 Cos @ 2. s D + 0.0707355 Cos @ 6. s D + 0.0254648 Cos @ 10. s D + 0.0129922 Cos @ 14. s D + 0.0078595 Cos @ 18. s D + Sin @ 2. s D − 0.5 Sin @ 4. s D + 0.333333 Sin @ 6. s D − 0.25 Sin @ 8. s D + 0.2 Sin @ 10. s D − 0.166667 Sin @ 12. s D + 0.142857 Sin @ 14. s D − 0.125 Sin @ 16. s D + 0.111111 Sin @ 18. s D − 0.1 Sin @ 20. s D Plot @ Evaluate @8 f1 @ s D , ff1 @ s, 10 D<D , 8 s, − Pi ê 2, Pi ê 2 <D

-1.5 -1.0 -0.5 0.5 1.0 1.5

0.5 1.0 1.5 2.0 2.5 3.0

3 “Beidseitig”

Remove @ "Global` ∗ " D ;

ü a

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

n=4; 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,3};

{x[-1],y[-1]}={-1 w,3};

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

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

{x[-4],y[-4]}={-4 w,2};

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

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

99 − 3 π 2

, 2 = , 8 − π , 3 < , 9 − π 2

, 3 = , 8 0, 2 < , 9 π 2

, 2 = , 8 π , 3 < , 9 3 π 2

, 3 ==

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

9 PointSize @ 0.03 D , Point @8 − 2 π , 2 <D , Point B9 − 3 π 2

, 2 =F , Point @8 − π , 3 <D ,

Point B9 − π 2

, 3 =F , Point @8 0, 2 <D , Point B9 π 2

, 2 =F , Point @8 π , 3 <D , Point B9 3 π 2

, 3 =F=

epi1 = Prepend @ Map @ Point, Table @8 k, y @ k D< , 8 k, 0, n − 1 <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, 3 <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 2.5, − 0.25 + 0.25 , 0., − 0.25 − 0.25 , 2.5,

− 0.25 + 0.25 , 0., − 0.25 − 0.25 , 2.5, − 0.25 + 0.25 , 0. <

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

fS[t]

5 2

− 1 4

+ 4

− 1 4

− 4

% êê ExpandAll 5

2

− 1 4

+ 4

− 1 4

− 4

fS1 @ s_ D : = fS @ s 2 Pi ê n D ; fS1 @ s D

5 2

− 1 4

+ 4

1 2

πs

− 1 4

− 4

πs 2

% êê ExpandAll 5

2

− 1 4

+ 4

1 2

πs

− 1 4

− 4

πs 2

% êê N êê Simplify

2.5 − H 0.25 + 0.25 L

− H 0.25 − 0.25 L

fS[t]//ExpToTrig

5 2

− Cos @ t D 2

− Sin @ t D 2

% êê ExpandAll 5

2

− Cos @ t D 2

− Sin @ t D

2

% êê N

2.5 − 0.5 Cos @ t D − 0.5 Sin @ t D fS1 @ s D êê ExpToTrig

5 2

− 1 2

Cos B π s 2 F − 1

2

Sin B π s 2 F

% êê ExpandAll 5

2

− 1 2

Cos B π s 2 F − 1

2

Sin B π s 2 F

% êê N

2.5 − 0.5 Cos @ 1.5708 s D − 0.5 Sin @ 1.5708 s D Plot[Re[fS[t]],{t,0,2Pi}]

1 2 3 4 5 6

2.0 2.2 2.4 2.6 2.8 3.0 3.2

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

1 2 3 4 5 6

-1.0 -0.5 0.5 1.0

Man beachte im letzten Plot die Grösse der Amplitude.

Plot[{Re[fS[t]]},{t,-2Pi,2Pi},Epilog->epi]

-6 -4 -2 2 4 6

2.0 2.2 2.4 2.6 2.8 3.0 3.2

Plot @8 Re @ fS1 @ s DD< , 8 s, 0, 4 < , Epilog → epi1 D

1 2 3 4

2.0 2.2 2.4 2.6 2.8 3.0 3.2

ü b

Um eine FFT machen zu können, braucht man eine 2-er Potenz als Anzahl der Intervalle.

ü c

Man muss z.B. 4 Messungen in einer Periode haben..

4

Remove @ "Global` ∗ " D ;

ü a

f1 @ t_ D : = UnitStep @ x + 3 D − UnitStep @ x − 3 D ; Plot @ f1 @ x D , 8 x, − 4, 4 <D

-4 -2 2 4

0.2 0.4 0.6 0.8 1.0

ü b

f2 @ t_ D : = H 9 − x ^ 2 L H UnitStep @ x + 3 D − UnitStep @ x − 3 DL ; Plot @ f2 @ x D , 8 x, − 4, 4 <D

-4 -2 2 4

2 4 6 8

ü c (Achtung Faktor

π

bei anderer Definition der Fouriertransformation!!!)

FourierTransform @ f1 @ x D , x, ω D

2

π

Sin @ 3 ω D ω

ü d (Achtung Faktor

π

bei anderer Definition der Fouriertransformation!!!)

FourierTransform @ f2 @ x D , x, ω D

−3 ω 2 π

ω

−

3 ω 2 π

ω

−

3

π

ω

−

3

π

ω

FourierTransform @ f2 @ x D , x, ω D êê Simplify

−

π

I − + 3 ω +

H + 3 ω LM

ω

% êê ExpToTrig êê Simplify

−

2

H 3 ω Cos @ 3 ω D − Sin @ 3 ω DL ω

ü e

InverseFourierTransform @H Cos @ Ω D − Sin @ Ω DL ê Ω , Ω , x D 1

2

− 2

π 2

Sign @ − 1 + x D − 1 2

+ 2

π 2

Sign @ 1 + x D

InverseFourierTransform @H Cos @ Ω D − Sin @ Ω DL ê Ω, Ω, x D êê Simplify 1

2

− 2

π

2 H Sign @ − 1 + x D − Sign @ 1 + x DL

5

ü a

Remove @ "Global` ∗ " D ;

f @ x_ D : = Cos @ 4 x D + I Sin @ 4 x D

FourierTransform @ f @ x D , x, ω D êê Simplify 2 π DiracDelta @ 4 + ω D

1 í 2 π FourierTransform @ f @ x D , x, ω D êê Simplify DiracDelta @ 4 + ω D

2 π FourierTransform @ f @ x D , x, ω D êê Simplify 2 π DiracDelta @ 4 + ω D

ü b

Remove @ "Global` ∗ " D ;

f @ x_ D : = Sin @ 4 x D + I Cos @ 4 x D

FourierTransform @ f @ x D , x, ω D êê Simplify 2 π DiracDelta @ − 4 + ω D

1 í 2 π FourierTransform @ f @ x D , x, ω D êê Simplify DiracDelta @ − 4 + ω D

2 π FourierTransform @ f @ x D , x, ω D êê Simplify 2 π DiracDelta @ − 4 + ω D

ü c

Remove @ "Global` ∗ " D ;

fHat @ x_ D : = Cos @ 4 ω D + I Sin @ 4 ω D

InverseFourierTransform @ fHat @ x D , ω , x D êê Simplify 2 π DiracDelta @ − 4 + x D

2 π InverseFourierTransform @ fHat @ x D , ω , x D êê Simplify

2 π DiracDelta @ − 4 + x D

1 í 2 π InverseFourierTransform @ fHat @ x D , ω , x D êê Simplify DiracDelta @ − 4 + x D

1 í 2 π FourierTransform B 2 π InverseFourierTransform @ fHat @ x D , ω , x D , x, ω F êê Simplify

êê ExpToTrig Cos @ 2 ω D + Sin @ 2 ω D

6

ü Modul (in Zusatzfenster betreiben!)

Remove @ "Global`∗" D ;

four @ fkt_, var_, perT_, start0Int_, n_, druck_ D : =

Module @8 fktInt, tInt, nInt, znInt < , Print @ " " D ; Print @ "Output:" D ; Print @ " " D ; Print @ "Ausgabe: ω , fktInt @ var D , a @ 0 D , a @ k D , b @ k D , c @ k D , Fourierreihen

ff @ var,n D , ff @ var D , ffExp @ var D , ffKomplexTrig @ var,n D , ffKomplexExp @ var,n D , ffKomplex @ var D , Plot: z.B.

Plot @ Evaluate @ ff @ t,n DD , 8 t,perT,perT + start0Int < ,PlotPoints → 50 D " D ; ω = 2 Pi ê perT; If @ druck == 1, Print @ " ω = ", ω D , " " D ;

fktInt @ tInt_ D : = Function @ fkt @ DD@ tInt D ;

If @ druck == 1, Print @ "Funktion @ ", var, " D = ", fktInt @ var DD , " " D ; a @ 0 D = 2 ê T Integrate @ fktInt @ var D , 8 var, start0Int, start0Int + perT <D ; If @ druck == 1, Print @ "a @ 0 D = ", a @ 0 DD , " " D ;

a @ k_ D : = 2 ê T Integrate @ Cos @ k ω var D fktInt @ var D , 8 var, start0Int, start0Int + perT <D ; If @ druck == 1, Print @ "a @ k D = ", a @ k DD , " " D ; b @ k_ D : = 2 ê T Integrate @ Sin @ k ω var D fktInt @ var D , 8 var, start0Int,

start0Int + perT <D ; If @ druck == 1, Print @ "b @ k D = ", b @ k DD , " " D ; c @ k_ D := 1 ê T Integrate @ fktInt @ var D E ^ H −I k ω var L ,

8 var, start0Int, start0Int + perT <D ; If @ druck == 1, Print @ "c @ k D = ", c @ k DD , " " D ; ff @ tInt_, znInt_ D : = a @ 0 D ê 2 + Sum @ a @ nInt D Cos @ nInt ω tInt D +

b @ nInt D Sin @ nInt ω tInt D , 8 nInt, 1, znInt <D ;

If @ druck == 1, Print @ "Fourierreihe @ ", var, ", ", n, " D = ", ff @ var, n DD , " " D ; If @ druck == 1,

Print @ "Num. Fourierreihe @ ", var, ", ", n, " D = ", ff @ var, n D êê N D , " " D ; ff @ tInt_ D : = a @ 0 D ê 2 + Sum @ a @ nInt D Cos @ nInt ω tInt D + b @ nInt D Sin @ nInt ω tInt D ,

8 nInt, 1, Infinity <D ;

If @ druck == 1, Print @ "Unendliche Fourierreihe @ ", var, " D = ", ff @ var DD , " " D ; ffExp @ tInt_ D : = ExpToTrig @ a @ 0 D ê 2 +

Sum @ a @ nInt D Cos @ nInt ω tInt D + b @ nInt D Sin @ nInt ω tInt D , 8 nInt, 1, Infinity <DD ; If @ druck == 1, Print @ "Unendliche Fourierreihe komplex @ ",

var, " D = ", ffExp @ var DD , " " D ;

ffKomplexTrig @ tInt_, znInt_ D := ExpToTrig @

Sum @ c @ nInt D E ^ H I nInt ω tInt L , 8 nInt, − znInt, znInt <DD ;

If @ druck == 1, Print @ "Komplexe Fourierreihe wieder trigonometrisch @ ", var, ", ", n, " D = ", ffKomplexTrig @ var, n DD , " " D ;

ffKomplexExp @ tInt_, znInt_ D : = TrigToExp @

Sum @ c @ nInt D E ^ H I nInt ω tInt L , 8 nInt, − znInt, znInt <DD ; If @ druck == 1, Print @ "Komplexe Fourierreihe @ ", var, ", ",

n, " D = ", ffKomplexExp @ var, n DD , " " D ;

ffKomplex @ tInt_ D := Sum @ c @ nInt D E ^ H I nInt ω tInt L , 8 nInt, −Infinity, Infinity <D ; If @ druck == 1, Print @ "Komplexe Fourierreihe @ ", var, " D = ", ffKomplex @ var DD , " " D ; If @ druck == 1, Print @ "Plot" D ;

Plot @ Evaluate @8 fktInt @ var D , ff @ var, n D<D , 8 var, start0Int, start0Int + perT <D ;, " " D D ;

four @ f, t, T, t0, 6, 0 D

Output:

Ausgabe: ω , fktInt @ var D , a @ 0 D , a @ k D , b @ k D , c @ k D , Fourierreihen ff @ var,n D , ff @ var D , ffExp @ var D , ffKomplexTrig @ var,n D , ffKomplexExp @ var,n D , ffKomplex @ var D ,

Plot: z.B. Plot @ Evaluate @ ff @ t,n DD , 8 t,perT,perT + start0Int < ,PlotPoints → 50 D

f @ t_ D : = Abs @ t − Pi D + Sin @ t ê 2 D ; T = 2 Pi;

t0 = − Pi;

H ∗ four @ fkt_,var_,perT_,start0Int_,n_,druck_ D ∗ L four @ f, t, T, t0, 6, 1 D

Output:

Ausgabe: ω , fktInt @ var D , a @ 0 D , a @ k D , b @ k D , c @ k D , Fourierreihen ff @ var,n D , ff @ var D , ffExp @ var D , ffKomplexTrig @ var,n D , ffKomplexExp @ var,n D , ffKomplex @ var D ,

Plot: z.B. Plot @ Evaluate @ ff @ t,n DD , 8 t,perT,perT + start0Int < ,PlotPoints → 50 D ω = 1

Funktion @ t D = Abs @ − π + t D + Sin B t 2 F a @ 0 D = 2 π

a @ k D = 2 Sin @ k π D k

b @ k D = 2 I − 4 k

Cos @ k π D − k π Cos @ k π D + 4 k

π Cos @ k π D + Sin @ k π D − 4 k

Sin @ k π DM k

I − 1 + 4 k

M π

c @ k D =

I 1 −

− 4 k

+ 4

k

+ 4 k

+ 4

k

+ 2

k π − 8

k

π M 2 k

I − 1 + 4 k

M π

Fourierreihe @ t, 6 D = π − 2 H − 4 + 3 π L Sin @ t D 3 π

+ H − 16 + 15 π L Sin @ 2 t D 15 π

− 2 H − 36 + 35 π L Sin @ 3 t D 105 π

+ H − 64 + 63 π L Sin @ 4 t D

126 π

− 2 H − 100 + 99 π L Sin @ 5 t D 495 π

+ H − 144 + 143 π L Sin @ 6 t D 429 π

Num. Fourierreihe @ t, 6 D = 3.14159 − 1.15117 Sin @ t D + 0.660469 Sin @ 2. t D − 0.448397 Sin @ 3. t D + 0.338319 Sin @ 4. t D − 0.27139 Sin @ 5. t D + 0.226488 Sin @ 6. t D Unendliche Fourierreihe @ t D =

π + 1 2 π

t

2

K 2 ArcTan B

t

2

F − 2

ArcTan B

t

2

F + 2 ArcTan B

t 2

F − 2

ArcTan B

t 2

F + 2

t

2

π Log A 1 +

E − 2

t

2

π Log A

I 1 +

MEO Unendliche Fourierreihe komplex @ t D =

π + 1

2 π Cos B t

2 F − Sin B t

2 F 2 ArcTan B Cos B t

2 F − Sin B t 2 FF + 2 ArcTan B Cos B t

2 F + Sin B t

2 FF − 2 ArcTan B Cos B t

2 F − Sin B t

2 FF Cos @ t D − 2 ArcTan B Cos B t

2 F + Sin B t

2 FF Cos @ t D + 2 π Cos B t

2 F Log @ 1 + Cos @ t D + Sin @ t DD − 2 π Cos B t

2 F Log A Cos @ t D + Cos @ t D

− Sin @ t D + Sin @ t D

E − 2 π Log @ 1 + Cos @ t D + Sin @ t DD Sin B t

2 F + 2 π Log A Cos @ t D + Cos @ t D

− Sin @ t D + Sin @ t D

E Sin B t 2 F + 2 ArcTan B Cos B t

2 F − Sin B t

2 FF Sin @ t D + 2 ArcTan B Cos B t

2 F + Sin B t

2 FF Sin @ t D

Komplexe Fourierreihe wieder trigonometrisch @ t, 6 D = π − 2 Sin @ t D + 8 Sin @ t D

3 π

+ Sin @ 2 t D − 16 Sin @ 2 t D 15 π

− 2 3

Sin @ 3 t D + 24 Sin @ 3 t D 35 π

+ 1

2

Sin @ 4 t D − 32 Sin @ 4 t D 63 π

− 2 5

Sin @ 5 t D + 40 Sin @ 5 t D 99 π

+ 1 3

Sin @ 6 t D − 48 Sin @ 6 t D 143 π Komplexe Fourierreihe @ t, 6 D =

−

+

+ 1 2

−2t

− 1 2

2t

− 1 3

−3t

+ 1 3

3t

+ 1 4

−4t

− 1 4

4t

− 1 5

−5t

+ 1

5

5t

+ 1 6

−6t

− 1 6

6t

+ 4

3 π

− 4

3 π

− 8

15 π

+ 8

15 π

+ 12

35 π

− 12

35 π

− 16

63 π

+ 16

63 π

+ 20

99 π

− 20

99 π

− 24

143 π

+ 24

143 π

+ π Komplexe Fourierreihe @ t D =

π + 1 3 π

−^t

2

− 3

t

2

π + 3 π ArcTan B

t

2

F − 3

π ArcTan B

t 2

F − 4

3t

2

Hypergeometric2F1 B 1 2

, 2, 5

2

, −

F + 4

3t

2

π Hypergeometric2F1 B 1 2

, 2, 5 2

, −

F + 3

t

2

π Log A 1 +

E − 1

3 π

−t

− 3

π − 3

t

2

π ArcTan B

t 2

F + 3

3t

2

π ArcTan B

t

2

F − 4 Hypergeometric2F1 B 1

2 , 2,

5 2

, −

F + 4 π Hypergeometric2F1 B 1 2

, 2, 5 2

, −

F + 3

π Log A

I 1 +

ME Plot

Plot @ f @ t D , 8 t, − Pi, Pi <D

-3 -2 -1 1 2 3

2 3 4 5

Plot @ Evaluate @ ff @ t, 4 DD , 8 t, − Pi, Pi < , PlotPoints → 50 D

-3 -2 -1 1 2 3

2 3 4 5

7

Remove @ "Global` ∗ " D ;

ü a

f1 @ x_ D : = 1 ê 2 E ^ H − 2 x ^ 2 L ;

fTransf1 @ Ω _ D : = 1 ê H 2 Pi L Integrate @ f1 @ λ D E ^ H − I λ Ω L , 8 λ , − Infinity, Infinity <D ; fTransf1 @ Ω D

Ω2 8

4 2 π

f2 @ x_ D : = f1 @ x D ê Sqrt @ 2 Pi D ;

fTransf2 @ Ω _ D : = 1 ê Sqrt @ 2 Pi D Integrate @ f2 @ λ D E ^ H − I λ Ω L , 8 λ , − Infinity, Infinity <D ; fTransf2 @ Ω D

Ω2 8

4 2 π

1 ê Sqrt @ 2 Pi D Integrate @ Evaluate @ fTransf2 @ Ω D E ^ H I Ω x LD , 8 Ω , − Infinity, Infinity <D

2 2 π

1 ê Sqrt @ 2 Pi D Integrate @ Evaluate @ fTransf2 @ Ω D E ^ H I Ω x LD , 8 Ω , − Infinity, Infinity <D f2 @ x D True

FourierTransform @ f1 @ t D , t, Ω D 1

4

Ω² 8

FourierTransform @ f2 @ t D , t, Ω D

Ω² 8

4 2 π

ü b

Plot @ Evaluate @ fTransf1 @ Ω DD , 8 Ω , − 6, 6 <D

-6 -4 -2 2 4 6

0.02 0.04 0.06 0.08 0.10

Plot @ Evaluate @ fTransf2 @ Ω DD , 8 Ω , − 6, 6 <D

-6 -4 -2 2 4 6

0.02 0.04 0.06 0.08 0.10

ü c

Plot @ Evaluate @8 f1 @ x D , fTransf1 @ x D<D , 8 x, − 6, 6 <D

-6 -4 -2 2 4 6

0.05 0.10 0.15 0.20

Plot @ Evaluate @8 f1 @ x D , f2 @ x D , fTransf1 @ x D , fTransf2 @ x D<D , 8 x, − 6, 6 < , PlotRange → 8 0, 1 <D

-6 -4 -2 0 2 4 6

0.2 0.4 0.6 0.8 1.0

8 f1 @ x D , f2 @ x D , fTransf1 @ x D , fTransf2 @ x D<

: 1 2

,

2 2 π

,

x² 8

4 2 π ,

x² 8

4 2 π

>

fTransf1 @ x D == fTransf2 @ x D True

Bemerkenswert: Alles Gauss-Glocken.