The set c is usually referred to as the extended

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The set C ∪{∞} is usually referred to as the extended complex plane . Because of the correspondence, some like to use S 2 to denote the same set. If S 2 is used, then the extended complex plane is sometimes also referred to as the Riemann sphere . Now to analyze how the neighborhoods of a point are transformed under stereographic projection, we study the correspondence of the boundary circles. A circle on S 2 is the intersection of S 2 with a plane Au + Bv + Cw = D . This corresponds to ( C - D )( x 2 + y 2 ) + 2 Ax + 2 By = C + D, which is a circle on C if C = D line on C if C = D . Conversely, a circle x 2 + y 2 + ax + by = c on C corresponds to a circle au + bv + ( c + 1) w = c - 1 on S 2 not containing N ; a line ax + by = c on C correspond to a circle au + bv + cw = c on S 2 containing N. In particular, the circle from S 2 intersecting the plane w = w 0 < 1 corresponds to x 2 + y 2 = 1 + w 0 1 - w 0 , i.e. the boundary of a neighborhood of N corresponds to a circle centered at the origin. Then deleted neighborhoods of N correspond to sets of the form { z : | z | > r } . By abuse of notation, we will let B ( , r ) = { z : | z | > r } . The remark following the definition of lim w z f ( w ) suggests how limits involving can be defined. Definitions. (1) z n → ∞ iff | z n | → ∞ iff for any ε > 0, there exists N ε N such that n N ε ⇒ | z n | > ε . (2) lim z →∞ f ( z ) = c C iff for any ε > 0, there exists δ > 0 such that | z | > δ ⇒ | f ( z ) - c | < ε iff lim w 0 f ( 1 w ) = c . In that case, we define f ( ) = c and say f ( z ) is continuous at . (3) lim z a C f ( z )= iff for any ε > 0, there exists δ > 0 such that 0 < | z - a | ⇒ | f ( z ) | > ε iff lim z a 1 f ( z ) = 0. (4) lim z →∞ f ( z ) = iff for any ε > 0, there exists δ > 0 such that | z | > δ ⇒ | f ( z ) | > ε . 10
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Definitions. Let U be a subset of C and f : U C be a function. (1) f is differentiable at z iff there exists B ( z, r ) U such that f ( z ) = lim h 0 f ( z + h ) - f ( z ) h C , where h B (0 , r ) . f is differentiable at iff U contains some B ( , r ) , f is continuous at and g ( z ) = f (1 /z ) if 0 < | z | < 1 /r f ( ) if z = 0 is differentiable at 0. In that case, we define f ( ) = g (0) . (2) f is holomorphic or analytic (or regular ) on a set S iff f is differentiable at every point of some open set containing S . In case S = { z } , f is holomorphic at z iff f is differentiable at every point of some open set (like B ( z, r )) containing z . (3) f is univalent (or schlicht ) on an open set iff f is differentiable and injective there. (4) f is entire (or integral ) iff f is differentiable on the complex plane. Of course, as usual, the sum, difference, product, quotient and chain rules are valid in (complex) differentiation. Some common functions are differentiable, e.g. polynomials are entire functions and ( z n ) = nz n - 1 for integer n . However, some are not, e.g. the conjugate function z is not differentiable anywhere because lim h 0 ,h C z + h - z h = lim h 0 ,h C h h doesn’t exist, as can be seen by the following computations : lim h 0 ,h R h h = lim h 0 ,h R h h = 1 and lim h = it 0 ,t R h h = lim t 0 - it it = - 1 .
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