By I.M. Singer
Today, the common undergraduate arithmetic significant unearths arithmetic seriously compartmentalized. After the calculus, he is taking a path in research and a direction in algebra. based upon his pursuits (or these of his department), he's taking classes in unique subject matters. Ifhe is uncovered to topology, it is often elementary element set topology; if he's uncovered to geom etry, it's always classical differential geometry. The intriguing revelations that there's a few solidarity in arithmetic, that fields overlap, that options of 1 box have purposes in one other, are denied the undergraduate. He needs to wait until eventually he's good into graduate paintings to work out interconnections, most likely simply because past he does not comprehend adequate. those notes are an try and get a divorce this compartmentalization, not less than in topology-geometry. What the scholar has realized in algebra and complex calculus are used to end up a few particularly deep effects referring to geometry, topol ogy, and workforce idea. (De Rham's theorem, the Gauss-Bonnet theorem for surfaces, the functorial relation of primary workforce to protecting area, and surfaces of continuing curvature as homogeneous areas are the main be aware valuable examples.) within the first chapters the naked necessities of straightforward aspect set topology are set forth with a few trace ofthe subject's program to useful research.
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Extra resources for Lecture Notes on Elementary Topology and Geometry
Example text
2 Historische Notizen. 15) war Euler sp¨ atestens 1749 gel¨ aufig, vgl. [62, I-15,S. 82]. 13) zum Ausgangspunkt der Theorie, [81, S. 145]. 13) bereits vorweggenommen hatte, [62, I-16, S. 144]; auch Weierstrass gibt noch 1876 Gauss als Entdecker an, [274, S. 91]. ⊓ ⊔ Es hat sich eingeb¨ urgert, – vgl. B. [279, S. 17) nach Weierstrass zu benennen. Es kommt aber bei ihm in dieser Form nicht vor; in [274, S. 91], findet sich allerdings das Produkt ∞ 1+ n=1 x −x log[(n+1)/n] e n f¨ ur die Faktorielle“ 1/Γ (x).
1 ψ(z) = −γ − − z ν=1 1 1 − z+ν ν ; dabei konvergiert die Reihe normal in C. Beweis. Wegen Γ = 1/Δ gilt ψ = −Δ′ /Δ. 5 durch logarithmische Differentiation von Δ(z) = zeγz (1 + ⊓ ⊔ z/ν)e−z/ν . 2. Es gilt Γ ′ (1) = ψ(1) = −γ; ψ(k) = 1 + 1 1 + ··· + − γ f¨ ur k = 2, 3 . . 2 Die Gammafunktion Beweis. Es ist Γ ′ (1) = ψ(1) = −γ − 1 − ν≥1 1 ν+1 − 1 ν 41 = −γ − 1 + 1 = −γ. 18). 3 (Partialbruchdarstellung von ψ ′ (z)). Es gilt ψ ′ (z) = ∞ 1 , (z + ν)2 ν=0 dabei konvergiert die Reihe normal in C. Beweis.
263, S. 42 und 56]). 3]. 4 Eulersche Partitionsprodukte∗ Neben dem Sinusprodukt hat Euler das Produkt Q(z, q) := ν=1 (1 + q ν z) = (1 + qz)(1 + q 2 z)(1 + q 3 z) · . . intensiv studiert. Es ist f¨ ur jedes q ∈ E wegen |q|ν < ∞ normal konvergent in C und also eine ganze Funktion in z, die im Fall q = 0 genau in den Punkten −q −1 , −q −2 , . . Nullstellen, und zwar von erster Ordnung, hat. Aus Q(z, q) entstehen f¨ ur z = 1 bzw. z = −1 die im Einheitskreis holomorphen Produkte (1 + q)(1 + q 2 )(1 + q 3 ) · .