By Irving Adler

This richly exact assessment surveys the evolution of geometrical principles and the improvement of the options of contemporary geometry from precedent days to the current. subject matters contain projective, Euclidean, and non-Euclidean geometry in addition to the position of geometry in Newtonian physics, calculus, and relativity. Over a hundred workouts with solutions. 1966 edition.

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**Example text**

1. Meromorphic functions on compact Riemann surfaces. 1. Let X be a compact Riemann surface. If f ∈ M (X), then Ordp (f ) = 0. p∈X Proof. Let p ∈ X such that f (p) = 0 or ∞. Let D be a small neighborhood of p such that (i) f has no zeros or poles in D and (ii) D is the image, under the inverse of some complex chart ϕ, of a disk in the plane with center ϕ(p). 9 shows that ∂D df = 0 and f Ordp (f ) = p∈X Ordp (f ) = ∂(X−D) p∈X−D df . f But ∂(X−D) df =− f ∂D df . f This completes the proof. 2. Meromorphic functions on P1 .

Our hypotheses imply that F (0, y) = yg(y) for some holomorphic function g that does not vanish at the origin. Thus n(0) = 1 and by continuity there exists ε < ε such that n(x) = 1 for all |x| ≤ ε . Thus, by the Argument Principle, for each such x there is a unique solution y = f (x) of the equation F (x, y) = 0. Moreover, by the Residue Theorem we have the formula Fy (x, y) 1 f (x) = dy, y 2πi |y|=ε F (x, y) which shows that f is holomorphic. 3. P LANE CURVES . 11. An affine plane curve is the zero locus of a holomorphic function f ∈ O(C2 ).

5. The Poisson Formula. If f ∈ O(D) then by the Cauchy Formula f (z) = |ζ|=1 f (ζ) dζ √ , ζ − z 2π −1 z ∈ D. On the other hand, if w ∈ C − D, then 0= |ζ|=1 dζ f (ζ) √ . ζ − w 2π −1 If we now take w = 1/¯ z , then by using repeatedly that on the circle |ζ| = 1, ¯ ζ = 1/ζ we obtain f (z) = = = = |ζ|=1 f (ζ) dζ √ − ζ − z 2π −1 |ζ|=1 dζ ζf (ζ) √ − ζ − z 2π −1ζ |ζ|=1 |ζ|=1 |ζ|=1 f (ζ) dζ √ ζ − z¯−1 2π −1 |ζ|=1 f (ζ) z¯ ζ + ¯ ζ − z ζ − z¯ f (ζ) dζ 1 − |z|2 √ . |ζ − z|2 2π −1ζ f (ζ) dζ √ −1 1 − ζ¯z¯ 2π −1ζ dζ √ 2π −1ζ Now, on any circle centered at the origin, dζ/ζ is pure imaginary, since 0 = dζ¯ d log |ζ|2 = dζ ζ + ζ¯ .