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By Louis B. Rall (eds.)

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F : N → N : n → n2 − 6n + 10; g : R → R : x → |x + 1| − 3; h : R → R : x → x3 . m. ¨ 4. Uberpr¨ ufen Sie, dass die folgenden Funktionen D → B auf den angegebenen Bereichen bijektiv sind und berechnen Sie die Umkehrfunktion: y = −2x + 3, y = x2 + 1, y = x2 − 2x − 1, D = R, B = R; D = (−∞, 0] , B = [1, ∞) ; D = [1, ∞) , B = [−2, ∞) . 5. Gehen Sie in mathe online zu Galerie – Funktionen 1 und l¨ osen Sie die unter den Applets Funktionen erkennen 1 und Graphen erkennen 1 gestellten Aufgaben. Erl¨ autern Sie Ihre Ergebnisse.

Ein Beispiel: z 2 = 2i = 2 eiπ/2 hat die beiden L¨osungen z= und √ 2 eiπ/4 = 1 + i √ z = − 2 eiπ/4 = −1 − i. Euler’sche Formeln. Diese erm¨ oglichen eine Darstellung der reellen Winkelfunktionen durch die komplexe Exponentialfunktion. Sie besagen, dass 1 iϕ e + e−iϕ 2 1 iϕ sin ϕ = e − e−iϕ 2i cos ϕ = ist, was sich durch Addition bzw. Subtraktion der Beziehungen eiϕ = cos ϕ + i sin ϕ e−iϕ = cos ϕ − i sin ϕ sofort herleiten l¨asst. 3 Abbildungseigenschaften komplexer Funktionen In diesem Abschnitt werfen wir einen Blick auf Abbildungseigenschaften komplexer Funktionen, das heißt, wie ihre Wirkung geometrisch beschrieben werden kann.

6. 28 3 Trigonometrie Das Bogenmaß des Winkels α (in Grad) ist die L¨ange des zugeh¨origen Einheitskreisbogens mit dem Vorzeichen von α. Das Bogenst¨ uck am Einheitskreis hat keine physikalische Einheit. Man spricht aber manchmal von Radianten (rad), um den Unterschied zu Grad hervorzuheben. 141592653589793... ≈ 22 . 7 Somit gilt f¨ ur die Umrechnung 360◦ ↔ 2π [rad], beziehungsweise α◦ ↔ π α [rad] 180 Beispielsweise entsprechen 90◦ ↔ wir Winkel stets im Bogenmaß. und π 2 [rad] ↔ ◦ 180 π . und −270◦ ↔ − 3π 2 .

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