A conjecture in arithmetic theory of differential equations by Katz N.M.

By Katz N.M.

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Extra resources for A conjecture in arithmetic theory of differential equations (Bull. Soc. Math. Fr. 1982)

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Wechselspannung oder Wechselstrom, soll die symbolische Methode erläutert und eingeführt werden. y ( t ) =  ˜ sin ( Z ˜ t  M ) = A ˜ sin ( Z ˜ t  M ) = A0 ˜ sin ( Z ˜ t  M ) (mechanische Schwingung) (2-1) u ( t) = Û ˜ sin Z ˜ t  M u = Umax ˜ sin Z ˜ t  M u = U0 ˜ sin Z ˜ t  M u (Wechselspannung) (2-2) i ( t) = Î ˜ sin Z ˜ t  M i = Imax ˜ sin Z ˜ t  M i = I0 ˜ sin Z ˜ t  M i (Wechselstrom) (2-3) Die physikalischen Größen bedeuten: Â, A, A0 Schwingungsamplitude mit der entsprechenden Einheit Û, Umax, U0 Scheitelwert der Spannung in V Î, Imax, I0 Scheitelwert des Stromes in A Z= 2 .

5 1 1 ˜cos ( Z˜t)  xz1  xz2  x2 t x Abb. 2 Seite 40 3 4 Komplexe Zahlen und Funktionen Der Zeiger befindet sich zum Zeitpunkt t 1 = 0 in der Ausgangsposition. Sein Richtungswinkel gegenüber der Bezugsachse (t-Achse) ist der Phasenwinkel (Nullphasenwinkel) M. In der Zeit t2 dreht sich der Zeiger um Z t2 weiter. Sein Richtungswinkel gegenüber der Bezugsachse ist nunmehr Z t2 + M. Der Ordinatenwert der Zeigerspitze entspricht dabei dem augenblicklichen Funktionswert y(t). Bei der Rotation des Zeigers um den Nullpunkt durchläuft daher die Ordinate nacheinander alle Funktionswerte der Sinusschwingung.

01  S Zeiger in der Gaußschen Ebene 10 8 6 z 4 yz 2 z1 y z1 y( M ) 10 8 6 4 2 0 2 4 6 8 2 4 6 8 10 x z  xz1  x( M ) Abb. 7 Gegeben ist folgende komplexe Zahl: z = (3 ; 30°). Gesucht ist der Zeiger z 2 , der aus der Streckung von z um das k = 3 fache (z 1 = k z) und der Drehung um den Winkel von 60 ° hervorgeht. 01  S Zeiger in der Gaußschen Ebene 10 z2 8 z1 6 yz 4 y z1 y z2 z 2 10 8 6 4 2 0 2 4 6 8 10 Abb. 785 3. M Argument 45 Grad Die imaginäre Einheit j lässt sich in Exponentialform darstellen: j˜90˜Grad j = 1˜ e j˜90˜Grad z1  e j˜ oder j = 1 ˜ e j ˜45˜Grad ˜ r˜ e S 2 .

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